Questions on test answered by real tutors!

Tutors Answer Your Questions about test (FREE)

Question 558195: 2x=√(1-3x)
Answer by MathTherapy(10806)

You can put this solution on YOUR website!

2x=√(1-3x)
***************
The other person's solutions,  are WRONG!!


For the RADICAND, 1 - 3x, we have: 
So, we get: , with 
      <==== Up to this point is CORRECT!
(x + 1)(4x - 1) = 0
                x + 1 = 0    OR    4x - 1 = 0
                      
Although both solutions , x = - 1 is EXTRANEOUS, and so, ONLY VALID/ACCEPTABLE solution is:

Question 271147: A BOX MEASURES 40CM*50CM*45CM .FIND THE AREA OF THE CARD BOARD REQUIRED TO MAKE 350BOXES
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
All six surfaces, 1 box

For 350 of these boxes, area amount of cardboard

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
A BOX MEASURES 40CM*50CM*45CM .FIND THE AREA OF THE CARD BOARD REQUIRED TO MAKE 350BOXES
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The problem is posed in inaccurate way. It should be retold (reformulated) in other way.
        Simply speaking, they want you find the surface area of one single box and then
        multiplied the single area by 350.

        The solution by @mananth is fatally wrong. For example, he writes
        "In a rectangle or square there are 8 faces."
        It is the case when the artificial intelligence works perpendicularly to common sense.

        I came to bring a correct solution.

The surface area of one rectangular box is 2*(ab + ac + bc) = 2*(40*50 + 40*45 + 50*45) = 12100 cm^2. The surface area of 350 boxes is 350 * 12100 = 4235000 cm^2 = 423.5 m^2. ANSWER

Solved correctly.

Question 1032795: Julie wants to build a sidewalk of uniform width around her garden. Her garden is rectangular, and its dimensions are 20 feet by 30 feet. She has enough pavers to cover 900 square feet and wants to use all the pavers. Complete the following statement. Round to the nearest tenth. Julie should make the width of the sidewalk _ _ _ _ _ feet.
Found 2 solutions by timofer, ikleyn:
Answer by timofer(155)   (Show Source):
You can put this solution on YOUR website!
Simplest equation found after some setup and some steps looks like .
Expecting to use quadratic formula solution and take the one with the plus-square root.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Julie wants to build a sidewalk of uniform width around her garden. Her garden is rectangular,
and its dimensions are 20 feet by 30 feet. She has enough pavers to cover 900 square feet
and wants to use all the pavers. Complete the following statement. Round to the nearest tenth.
Julie should make the width of the sidewalk _ _ _ _ _ feet.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In the post by @mananth, the value of x2 is determined by @mananth incorrectly
as -15.5.

The correct value for x2 is about -30.

For this problem, the precise value of x2 does not matter, since it is negative.

But I do not think, that your teacher will be glad by seeing wrong value in your solution.

Question 494362: It takes 4 men 14 days to do a certain job. How long should it take 7 men working at the same rate to do the same job?
Found 3 solutions by ikleyn, josgarithmetic, MathTherapy:
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
It takes 4 men 14 days to do a certain job.
How long should it take 7 men working at the same rate to do the same job?
~~~~~~~~~~~~~~~~~~~~

                This can be solved MENTALLY and MOMENTARILY.

(1)   The whole job is   4 * 14 = 56 man-days.

(2)   7 men will make it in   56/7 = 8 days.         ANSWER

Solved.

It is how my teachers taught us solving such problems 65 years ago.

Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
r, work rate of each man
x, the unknown time for the 7 men
1, the amount of jobs

See, each expression on the left equals to 1, so

Answer by MathTherapy(10806)   (Show Source):
You can put this solution on YOUR website!

It takes 4 men 14 days to do a certain job. How long should it take 7 men working at the same rate to do the same job? ************************************ The other person's answer, that it takes 7 men, 24 days to complete a job that 4 men take 14 days to do, makes ABSOLUTELY no sense, especially seeing that both groups are working at the same rate. How can a larger group of men take MORE TIME to a job than a smaller group, when both groups are working at the same rate? RIDICULOUS and NONSENSICAL!! It'll take the 7 men to do the same job that it takes 4 men, 14 days to complete.

Question 1155046: It's a question on indices
Solve for x and y in:
1)3^x-2y*5^y+x
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
It's a question on indices
Solve for x and y in:
1)3^x-2y*5^y+x
~~~~~~~~~~~~~~~~~~~~~~

This post is a compote of words with no mathematical sense.

Every word taken separately makes sense, but taken altogether, they are mathematically non-sensical.

Question 1176752: Using k as the constant of variation, write the equation of variation for each following.
1. The current l varies directly as the electromotive force E and inversely as as the resistance R.
2. The acceleration a of an object varies directly as the force f exerted and inversely as it's mass m.
3. The stiffness s of a beam varies directly as it's depth d and inversely as the square of the length l.
4. The time t required for an elevator to lift a weight varies directly with the distance d through which it is to be lifted and inversely with the power p of the motor.
5. The force f of attraction between two bodies varies directly as the product of their weights w and inversely as the square of the distance d between them.
6. The electrical resistance R of wire varies directly as it's length l and inversely as the square of its diameter d.
7. The acceleration A of a moving objects varies directly as the distance d inversely as the square of the time t.
Answer by n2(79)   (Show Source):
You can put this solution on YOUR website!
.

Formulation #5 is incorrect.

To understand why it's incorrect, imagine two objects (two masses) in outer space.

There is no weight because there is weightlessness (the weight is zero),
but there is a force of attraction.

The error in the formulation is that instead of the concept of weight, the concept of mass should be used.

Question 1210577: Find the derivative of y=x using the first principle of differenciation
2) y=2x²—x
Answer by KMST(5345)   (Show Source):
You can put this solution on YOUR website!
I assume that in your class what was called the first principle of differentiation is that the derivative of a function is
the limit of when tends to zero.

For or , , ,
but does not depend on ,
so the limit of when tends to zero is and or

For or ,
,
,
, and the limit of when tends to zero is , so

Question 1183412: A sector with an angle 110 degrees at the center of a circle is cut away from a circular piece of paper of radius 70cm. The remaining part is folded to form a cone. Find 1. the vertical angle of the cone 2. The angle of the sector.

Found 4 solutions by n3, n2, CPhill, ikleyn:
Answer by n3(7)   (Show Source):
You can put this solution on YOUR website!
.

        To the managers of this project !
    Attention !!     Attention !!     ATTENTION !!

Today, I had a chance to review the bunch of solutions produced by @CPhill to Math problems at this forum.

               A list of the links follows

https://www.algebra.com/algebra/homework/word/travel/Travel_Word_Problems.faq.question.1182443.html

https://www.algebra.com/algebra/homework/logarithm/logarithm.faq.question.244998.html

https://www.algebra.com/algebra/homework/word/travel/Travel_Word_Problems.faq.question.1182444.html

https://www.algebra.com/algebra/homework/Systems-of-equations/Systems-of-equations.faq.question.1210543.html

https://www.algebra.com/algebra/homework/word/travel/Travel_Word_Problems.faq.question.1182591.html

https://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.1182886.html

https://www.algebra.com/algebra/homework/playground/test.faq.question.1183412.html

https://www.algebra.com/algebra/homework/playground/test.faq.question.1210545.html

https://www.algebra.com/algebra/homework/Systems-of-equations/Systems-of-equations.faq.question.1210543.html

https://www.algebra.com/algebra/homework/formulas/Geometric_formulas.faq.question.1201106.html

My impressions about his skills as a developer of the AI for solution of school Math problems are at very low level.

This person has no necessary knowledge and understanding Math to work on the AI project
in the area of solution of Math problems.

He doesn't know what kinds of mathematical problems are possible and which ones are not.
He also doesn't know what kinds of solutions to mathematical problems are possible,
and which ones are not and should not be.

Very often he posts nonsense to the forum under the guise of mathematical problems,
and very often he posts nonsense to the forum under the guise of solutions to mathematical problems.

He also has no necessary skills to work in such a project NEITHER as an individual NOR as a member of a team.
For the team work, he has no necessary respect to the work of specialists in this area.

So, if you, the managers, want to continue such a project successfully, you should consider replacing this person
to a more appropriate candidate.

Yours well-wisher, @ikleyn

Answer by n2(79)   (Show Source):
You can put this solution on YOUR website!
.
A sector with an angle 110 degrees at the center of a circle is cut away from a circular piece of paper
of radius 70cm. The remaining part is folded to form a cone. Find
1. the vertical angle of the cone
2. The angle of the sector.
~~~~~~~~~~~~~~~~~~~~~~~~~~~

This post by @CPhill is a copy-paste of the post by @mananth.

They both are very suspicious, because they both contain this phrase
"The area of major arc is the circumference of the base of cone".

As you see this gibberish, you should throw this solution to a garbage box
immediately without any doubts - it does not deserve further consideration.

In addition, the solution in both their posts are incomplete.

In my post (as @ikleyn) at this spot, I gave another, fully correct and complete solution to this problem,
So, you can disregard both posts by @mananth and by @CPhill.

Don't let them cloud your brain with flawed "quasi"-solutions.

Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
Length of arc = +%28%28theta%29%2F360%29%2A+%28pi%29%2Ad%29
length of arc p = 110/360 (2*pi*70)
p= 1540/36 * pi
area of major arc = (140 *pi -(1540/36)*(pi))
= 97.2 *pi
The area of major arc is the circumference of the base of cone
97.2*pi = d *pi
d= 97.2
r = 48.6
Now the radius of the circle becomes the slant height of the cone
height of cone h=+sqrt%28l%5E2-r%5E2%29
height of cone = sqrt%2870%5E2-48.6%5E2%29
height of cone = 50.37
we know radius and height angle of cone can be calculated

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
A sector with an angle 110 degrees at the center of a circle is cut away from a circular piece of paper of radius 70cm.
The remaining part is folded to form a cone. Find 1. the vertical angle of the cone 2. The angle of the sector.

After cutting away the sector of 110°, the remaining pert of the circle is the sector of 360° - 110° = 250°. The length of the arc of the sector of (250°, R = 70 cm) is = = 305.4323611 cm. The radius of the base of the cone 'r' can be defined from = 305.4323611, r = = 48.61111... cm. It is the same as to use equation = , r = = = 48.61111... cm. Now the cone has the slant height of 70 cm and the radius of 48.61111... cm. From the right-angled triangle, for the angle 'a' between the slant height and the cone axis we have sin(a) = = 0.694444... Thus angle 'a' is a = arcsin(0.694444), or about 44°. Vertical angle of the cone is about 2*44° = 88°.

Solved.

Question 1210545: Gorimapa Nigeria plc has just received an order for it's bathroom cabinet which is made up of two kinds that is standard and deluxe. The order is for at least 200 bathroom cabinets of either varieties and including at least 60 of the deluxe kind. The standard model takes 4 hours of the assembling time and has a valuable cost of #4000 whereas the deluxe model takes 5 hours of assembling time and has a valuable cost of #6000 . There are 400 hours available for assembling time. The equipment can be used to assemble either kind of cabinet in any combination. The company's manager engages you as an expert and wishes to minimize the company's cost of this special order. You are required as an expert to formulate this problem in a linear programming form and using the graphical method advise the manager on the best product that will enable his firm to minimize it's cost
Found 3 solutions by n2, CPhill, ikleyn:
Answer by n2(79)   (Show Source):
You can put this solution on YOUR website!
.
Gorimapa Nigeria plc has just received an order for it's bathroom cabinet which is made up of two kinds
that is standard and deluxe. The order is for at least 200 bathroom cabinets of either varieties and including
at least 60 of the deluxe kind.
The standard model takes 4 hours of the assembling time and has a valuable cost of #4000 whereas
the deluxe model takes 5 hours of assembling time and has a valuable cost of #6000 .
There are 400 hours available for assembling time. The equipment can be used to assemble either kind
of cabinet in any combination. The company's manager engages you as an expert and wishes to minimize
the company's cost of this special order. You are required as an expert to formulate this problem
in a linear programming form and using the graphical method advise the manager on the best product
that will enable his firm to minimize it's cost
~~~~~~~~~~~~~~~~~~~~~~~~

The answer in the post by @CPhill (125 standard cabinets, 60 Deluxe cabinets) does not satisfy
the restriction of 400 hours: 125*4 + 60*5 = 800 hours, which greatly exceeds the restriction of 400 hours.

It also does not satisfy the requirement "at least 200 bathroom cabinets", since 125 + 60 = 185 is less than 200.

So, @CPhill solved DIFFERENT problem from what is posed in the post.

He made it even without explicit announcement/declaration about changing the problem,
which is inappropriate practice and can confuse a reader.

For the correct treatment of the problem, see the post by @ikleyn at this spot,
where it was shown that the problem, as posed in the post, has no solution, at all.

Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
To help Gorimapa Nigeria PLC minimize its costs while meeting the order requirements, we can use **Linear Programming (LP)**. This mathematical approach allows us to find the most efficient combination of standard and deluxe cabinets within the given constraints.
### 1. Problem Formulation
First, we define our variables and the mathematical relationships between them.
**Decision Variables:**
* Let = Number of **Standard** cabinets to produce.
* Let = Number of **Deluxe** cabinets to produce.
**Objective Function:**
The goal is to minimize the total variable cost ():

**Constraints:**
1. **Total Order Quantity:** At least 200 cabinets in total.
*

2. **Deluxe Requirement:** At least 60 deluxe cabinets must be made.
*

3. **Assembly Time:** Total time cannot exceed 800 hours.
*

4. **Non-negativity:** Production cannot be negative.
*

---
### 2. Graphical Method Analysis
To solve this graphically, we treat the inequalities as equations to find the boundary lines.
* **Line 1 (Total Order):** .
* If ; if .

* **Line 2 (Deluxe Min):** .
* A horizontal line at .

* **Line 3 (Time Limit):** .
* If ; if .

#### Identifying the Feasible Region
The feasible region is the area that satisfies all three conditions simultaneously. Looking at the intercepts:
* The **Time Limit** and **Total Order** lines actually meet at the x-axis at .
* However, we must have at least 60 Deluxe cabinets ().
By solving the intersection of and the other constraints:
1. **Intersection of and :** . Point: **(140, 60)**.
2. **Intersection of and :** . Point: **(125, 60)**.
**Wait!** Note that at point (140, 60), the total hours needed would be hours. This exceeds our 800-hour limit.
### 3. Finding the Optimal Solution
The feasible region is a small triangle bounded by the points where these constraints overlap. To minimize cost, we test the corner points of the feasible region:
| Corner Point | Cost Calculation () | Total Cost |
| --- | --- | --- |
| **(125, 60)** | | **#860,000** |
| **(0, 160)** | | #960,000 |
| **(0, 200)** | *Not feasible (Exceeds 800 hours)* | - |
*Note: The point (200, 0) is also not feasible because must be at least 60.*
---
### 4. Expert Advice to the Manager
Based on the linear programming analysis, the best product mix to minimize costs while staying within the 800-hour limit and meeting the order requirements is:
* **Standard Cabinets:** 125 units
* **Deluxe Cabinets:** 60 units
* **Total Production:** 185 units

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
Gorimapa Nigeria plc has just received an order for it's bathroom cabinet which is made up of two kinds
that is standard and deluxe. The order is for at least 200 bathroom cabinets of either varieties and including
at least 60 of the deluxe kind.
The standard model takes 4 hours of the assembling time and has a valuable cost of #4000 whereas
the deluxe model takes 5 hours of assembling time and has a valuable cost of #6000 .
There are 400 hours available for assembling time. The equipment can be used to assemble either kind
of cabinet in any combination. The company's manager engages you as an expert and wishes to minimize
the company's cost of this special order. You are required as an expert to formulate this problem
in a linear programming form and using the graphical method advise the manager on the best product
that will enable his firm to minimize it's cost
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This problem, as it is presented in the post, has no solution, at all.

Indeed, to make 200 cabinets, it requires at least 4*200 = 800 hours of work,
counting the minimum required time of 4 hours for a standard cabinet.

It is just more than 400 hours available, so the problem does not have a solution.

You do not need to deploy the heavy artillery of the Linear Programming to get this conclusion.

Question 1210537: The curve surface area of a cylinder is 1.5 times the total surface area of a cone. If the diameter of the cylinder and cone are equal to 10cm and the vertical height of the cone is 12cm. Calculate the height of the cylinder
Found 2 solutions by ikleyn, CPhill:
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
The curve surface area of a cylinder is 1.5 times the total surface area of a cone.
If the diameter of the cylinder and cone are equal to 10cm and the vertical height of the cone is 12cm.
Calculate the height of the cylinder
~~~~~~~~~~~~~~~~~~~~~~~~~

The curve surface area of this cylinder is = = = . The slant height of the cone is L = = = = 13 cm. The total surface area of the cone is = + = + = = . Now write the equation = = Simplify it h = 1.5*9 = 13.5. ANSWER. The height of the cone is 13.5 cm.

Solved.

Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
To find the height of the cylinder, we first need to determine the total surface area of the cone and then use the relationship provided to solve for the cylinder's height.
### 1. Find the Dimensions of the Cone
We are given the diameter of the cone is **10 cm**, so the radius () is **5 cm**. The vertical height () is **12 cm**.
To find the Total Surface Area (TSA) of a cone, we first need the **slant height ()**:

### 2. Calculate the Total Surface Area (TSA) of the Cone
The formula for the TSA of a cone is :

---
### 3. Calculate the Curved Surface Area (CSA) of the Cylinder
The problem states that the CSA of the cylinder is **1.5 times** the TSA of the cone:

### 4. Solve for the Height of the Cylinder ()
The formula for the Curved Surface Area of a cylinder is . We know the diameter is 10 cm, so the radius () is **5 cm**:

Now, divide both sides by :

**The height of the cylinder is 13.5 cm.**
Would you like me to calculate the volume of either shape to see how they compare?

Question 1210538: Find the values of x in the range 0°≤x≤360°, which satisfy the equation below
i) sin3xsinx = 2cos2x + 1
ii) 3cotx + tanx - 4 = 0

Found 2 solutions by ikleyn, CPhill:
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

Dear @CPhill,

Your posts are doing more harm than good to students' learning.

Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
To solve these trigonometric equations, we will use fundamental identities to simplify them into a solvable form.
---
## i)
To solve this, we can use the **Product-to-Sum formula**:

**Step 1: Apply the formula to the left side.**
Let and :

**Step 2: Simplify and rearrange.**
Multiply the entire equation by :

**Step 3: Use the Double Angle formula for .**
Recall that . Here, let :

**Step 4: Factor the quadratic.**
Let :

So, or .
**Step 5: Solve for in the range .**
Since , then .
* **Case 1: **

* **Case 2: **

**Solution (i):**
---
## ii)
**Step 1: Rewrite in terms of .**
Since , the equation becomes:

**Step 2: Form a quadratic equation.**
Multiply through by (noting that ):

**Step 3: Factor the quadratic.**

So, or .
**Step 4: Solve for in the range .**
* **Case 1: **

* **Case 2: **

**Solution (ii):**
---
Would you like me to show you how to verify these solutions using their respective graphs?

Question 1205029: A heavy cube of side 8cm is placed vertically in a cylindrical tank of radius 7cm which contains water.
Calculate the rise in the water level if the original depth of water was:
a) 10 cm
b) 2 cm
Answer by n2(79)   (Show Source):
You can put this solution on YOUR website!
.
A heavy cube of side 8 cm is placed vertically in a cylindrical tank of radius 7 cm which contains water.
Calculate the rise in the water level if the original depth of water was:
    (a)   10 cm
    (b)   2 cm
~~~~~~~~~~~~~~~~~~~~~~~~~~~

(a) In this case, the entire cube is wholly submerged into the water in the tank. The water level rises over the entire base area of the cylindrical tank. The raised water represents the volume of the displaced water in the tank by the solid cube. Use the law of the volume of water conservation. To find the rise for question (a), we should divide the volume of the cube, cm^3 by the area of the base of the cylinder the rise = = = 3.326 cm. ANSWER to question (a). The rise of the water level is 3.326 cm. (b) In this case, the cube is only partly submerged into the water in the tank. The water level rises over the part of the base area of the cylindrical tank. This part of the area where the water rises is the entire area of the base of the tank minus the area of the base of the cube, which is only partially submerged. Use the law of the volume of water conservation. To find the new level of water for question (b), we should divide the volume of the water in the tank, which is cm^3, by the (area of the tank base MINUS area of the cube base) the new level = = 3.4232 cm (rounded). Thus the raise of the water level is 3.4232 - 2 = 1.4232 centimeters. The new level is still lower than the height of the cube, so our calculations make sense. ANSWER to question (b). The rise of the water level is 3.4232 cm.

Solved.

Question 426442: Solve log base 6 of x + log base 6 of (x+16)=2
Answer by MathTherapy(10806)   (Show Source):
You can put this solution on YOUR website!

Solve log base 6 of x + log base 6 of (x+16)=2 The other person is WRONG!! The solution set is NOT . Correct x-values are 2 and - 18, of which - 18 should be IGNORED, since it's an EXTRANEOUS solution!! Equation to be solved: , which factors to (x - 2)(x + 18) = 0 You can do the check to CONFIRM!

Question 1206821: The minute-hand of a clock is 6 cm long. How far does the end of the hand travel in 35 minutes?
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
The minute-hand of a clock is 6 cm long. How far does the end of the hand travel in 35 minutes?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        This my post is written in opposite to the solution by @mananth in his post.

        My goal was to present a short, clear and straightforward solution to the given problem.

A solution is to take = of the circumference of a circle having the radius of 6 cm travel distance is = = 21.99113 cm. Round reasonably to 22 cm. ANSWER

Solved.

Question 1188445: Simplify

Answer by MathTherapy(10806)   (Show Source):
You can put this solution on YOUR website!

Question 446383: If Jill fills a pool with water in 30 min, jack fills a pool with water in 45 min and John fills a pool with water in 1 hour and 30 min. How long will it take if they work together.
Found 3 solutions by math_tutor2020, greenestamps, ikleyn:
Answer by math_tutor2020(3835)   (Show Source):
You can put this solution on YOUR website!

Answer: 15 minutes

Reasoning
1 hr + 30 min = 60 min + 30 min = 90 min

When working alone we have these time durations in minutes only
Jill = 30 min
Jack = 45 min
John = 90 min
The LCM of that set is 90.

Consider a pool with a capacity of 9000 gallons. This hypoethical value can be changed to anything else and the final answer will be the same at the end.
I'm tacking a few zeros onto 90 to get some large capacity, so when we divide it later on we get integer results.

When working alone, Jill fills 9000 gallons in 30 min. Her rate is 9000/30 = 300 gallons per min.
rate = amountDone/time

When working alone, Jack does the same job in 45 min. His rate is 9000/45 = 200 gallons per min.

And when working alone, John's rate is 100 gallons per minute because 9000/90 = 100

The unit rates for each person are then added up. The assumption is that each person doesn't hinder the other when working together.
Eg: John doesn't slow down Jack
Jill + Jack + John = 300+200+100 = 600
They combine to a rate of 600 gallons per minute.

When working together, the pool gets filled in 15 minutes because (9000 gallons)/(600 gallons per min) = 15

Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

Here is an alternative to the standard algebraic method shown by the other tutor. This method works especially well in this particular problem because the numbers are "nice".

The three times for the three people to fill the pool, in minutes, are 30, 45, and 90.

Consider the least common multiple of those times, which is 90 minutes. In 90 minutes, ...
Jill could fill the pool 90/30 = 3 times;
Jack could fill the pool 90/45 = 2 times; and
John could fill the pools 90/90 = 1 time

Therefore, in 90 minutes, the three of them could fill the pool 3+2+1 = 6 times.

And if they can fill the pool 6 times in 90 minutes, the time it takes them to fill the pool once is 90/6 = 15 minutes.

ANSWER: 15 minutes

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
If Jill fills a pool with water in 30 min, Jack fills a pool with water in 45 min and John fills a pool
with water in 1 hour and 30 min. How long will it take if they work together ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        I will solve it in more understandable way than @mananth does it in his post.

Jill makes full job in 30 minutes - hence, Jill makes 1/30 of the job per minute.

Jake makes full job in 45 minutes - hence, Jake makes 1/45 of the job per minute.

John makes full job in 90 minutes - hence, John makes 1/90 of the job per minute.

Working together, they make

         + + = + + = =

of the job per minute.

Hence, it will take 15 minutes for the three participants to complete the job working together.

               Solved - hip-hip-hooray ( ! )

Question 437259: Solve. A boat moves 5 kilometers upstream in the same amount of time it moves 17 kilometers downstream. If the rate of the current is 8 kilometers per hour, find the rate of the boat in still water.
A. 7 1/12 kilometers per hour
B. 3 1/3 kilometers per hour
C. 8 kilometers per hour
D. 14 2/3 kilometers per hour
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

In his solution, @mananth mixed the notions/conceptions "wind" and "current",

so his post partially reminds a soup of words, but with this my notice,

the reader will be able to deal with these details.

But if somebody will integrate his solution into a greater composition, it should be fixed.

Question 437121: 15. sue can shovel snow from her driveway in 30mins. jim can do the same job in 35mins. how long would i ttake sue and ime to shovel teh driveway if they worked together?
16.simplify by removing factors of 1 (7y-14)/2-y
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
Assume each works on the same driveway.
Sue, 30 minutes for the job
Jim, 35 minutes the same job
The two of them working together, driveway per minutes

or looking at reciprocal
minutes per driveway
approximately 16 minutes

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
15. sue can shovel snow from her driveway in 30 mins. jim can do the same job in 35 mins. how long would i ttake sue and ime to shovel teh driveway if they worked together?
~~~~~~~~~~~~~~~~~~~~~~~~

        Calculations in the post by @mananth are incorrect and should be redone (polished).
        I came to make the job accurately and to present it an a way as it SHOULD be done.

Sue makes of the job per minute. Jim makes of the job per minute. Working together, they make + = = = = of the job per minute. So, they need = 16 minutes working together.

Solved correctly and presented properly.

Question 427596: when I open a book two page face me. the sum of the page number is 85. what are the page number ? if the sum is not give, but the product is gives to be 1806, how will your find the page number
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
(a) when I open a book two page face me, the sum of the page number is 85. what are the page number ?
(b) if the sum is not give, but the product is gives to be 1806, how will your find the page number
~~~~~~~~~~~~~~~~~~~~~~~

                Part (a)

Let the page numbers are n and (n+1), two consecutive integer numbers. The problem says that n + (n+1) = 85. Simplify and find 'n' 2n + 1 = 85, 2n = 85 - n1 = 84, n = 84/2 = 42. ANWER. The pages are '42' and '43'.

                Part (b)

This time, the equation is n*(n+1) = 1806. You can solve it by guessing n = 42 or by advanced guessing with reasoning. Alternatively, you can reduce it to quadratic equation n^2 + n - 1806 = 0 and solve it using the quadratic formula or factoring. Or you can evaluate the square root of 1806: = 42.497 approximately, which tells you that n = 42.

Thus, both parts, (a) and (b), are solved.

Question 428529: find the distance between the points (12,8) and (4,2)
1. 100 units
2. 14 units
3. 10 units
4. -10 units
Find the midpoint of the points (3,1) and (7,-5)
1. (1,6)
2. (2,1)
3. (2,-3)
4. (5, -2)

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.

Compare my solution with the solution of the other tutor and find differences

Distance between two points x1 y1 x2 y2 d= 12 8 4 2 d= d= d= d= 10 ........................ If the coordinates of A and B are ( x1, y1) and ( x2, y2) respectively, then the midpoint, M, of AB is given by the following formula M = ( 3 , 1 ) ( 7 , -5 ) M = x= ( 3 + 7 )/ 2 y= ( 1 - 5 )/ 2 x= 5 y= -2

Question 729458: A student scored 70 points on a test that had 4 seven- point fill in the blank questions and 24 three point multiple choice questions. If the student answered 1 fill in the blank question , how many multiple choice question did the student answer correctly?
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
A student scored 70 points on a test that had 4 seven- point fill in the blank questions
and 24 three point multiple choice questions. If the student answered 1 fill in the blank question,
how many multiple choice question did the student answer correctly?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

I will solve and answer in one line how many multiple choice questions did the student answer correctly = = = 21. ANSWER

Solved.

The formula is self-explanatory.

Question 1165403: Dunder Mifflin supplies paper to corporate clients. An important client needs 1500 reams of paper.
Dunder Mifflin can procure the paper from several suppliers. The characteristics of the suppliers
are as follows:
Cost per ream Delivery time Reams available
Supplier A 3.50 5 days 200
Supplier B 2.00 10 days 600
Supplier C 6.50 15 days 600
Supplier D 5.00 4 days 200
Supplier E 4.00 6 days 200
When procuring the paper, Dunder Mifflin needs to ensure the following conditions are satisfied:
(a) The client requires that at least 500 of the 1500 reams be delivered within the next 7 days.
(b) In order to maintain good relations with Supplier C, Dunder Mifflin must order at least 100
reams from them.
(c) Because of a rivalry between supplier B and D, Dunder Mifflin must order at least as many
reams from supplier D as they do from supplier B.
(d) Each supplier has a limited number of reams of paper available, as shown in the table.
Formulate a linear optimization model to help Dunder Mifflin procure the paper at the lowest cost.
(You don’t need to find the optimal solution to the model you formulate.)
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website!
This is a classic linear programming (LP) problem. We need to define the decision variables, the objective function (minimizing cost), and the constraints based on the client and supplier requirements.
## 1. Decision Variables
Let $x_i$ be the number of reams of paper ordered from **Supplier $i$**, where $i \in \{A, B, C, D, E\}$.
* $x_A$: Reams ordered from Supplier A
* $x_B$: Reams ordered from Supplier B
* $x_C$: Reams ordered from Supplier C
* $x_D$: Reams ordered from Supplier D
* $x_E$: Reams ordered from Supplier E
Since the number of reams must be non-negative:
$$x_A, x_B, x_C, x_D, x_E \ge 0$$
## 2. Objective Function (Minimize Cost)
The objective is to minimize the total procurement cost. This is the sum of (Cost per Ream $\times$ Reams Ordered) for each supplier.
| Supplier | Cost per Ream |
| :---: | :---: |
| A | 3.50 |
| B | 2.00 |
| C | 6.50 |
| D | 5.00 |
| E | 4.00 |
$$\text{Minimize } Z = 3.50x_A + 2.00x_B + 6.50x_C + 5.00x_D + 4.00x_E$$
## 3. Constraints
### a) Total Demand Constraint
The client requires a total of 1500 reams of paper.
$$x_A + x_B + x_C + x_D + x_E = 1500$$
### b) Delivery Time Constraint (Within 7 Days)
The client requires at least 500 reams to be delivered within 7 days. We must identify suppliers with a delivery time $\le 7$ days: Suppliers A (5 days), D (4 days), and E (6 days).
$$x_A + x_D + x_E \ge 500$$
### c) Supplier C Relationship Constraint
Dunder Mifflin must order at least 100 reams from Supplier C.
$$x_C \ge 100$$
### d) Rivalry Constraint (Supplier D vs. B)
The order from Supplier D must be at least as many reams as the order from Supplier B.
$$x_D \ge x_B$$
This is typically written as:
$$x_D - x_B \ge 0$$
### e) Availability Constraints
The order from each supplier cannot exceed the reams available.
| Supplier | Reams Available |
| :---: | :---: |
| A | 200 |
| B | 600 |
| C | 600 |
| D | 200 |
| E | 200 |
$$x_A \le 200$$
$$x_B \le 600$$
$$x_C \le 600$$
$$x_D \le 200$$
$$x_E \le 200$$
---
## Linear Optimization Model Summary
$$\text{Minimize } Z = 3.50x_A + 2.00x_B + 6.50x_C + 5.00x_D + 4.00x_E$$
$$\text{Subject to:}$$
1. $$x_A + x_B + x_C + x_D + x_E = 1500$$ (Total Demand)
2. $$x_A + x_D + x_E \ge 500$$ (7-Day Delivery)
3. $$x_C \ge 100$$ (Supplier C Minimum)
4. $$x_D - x_B \ge 0$$ (Rivalry)
5. $$x_A \le 200$$
6. $$x_B \le 600$$
7. $$x_C \le 600$$
8. $$x_D \le 200$$
9. $$x_E \le 200$$
10. $$x_A, x_B, x_C, x_D, x_E \ge 0$$ (Non-negativity)

Question 1210504: If A and B are positive 2 digest integers, how many solutions are there of the equation 2a+3b=100
Found 4 solutions by greenestamps, math_tutor2020, ikleyn, MathLover1:
Answer by greenestamps(13327)   (Show Source):
You can put this solution on YOUR website!

This is a linear Diophantine equation -- one equation with two variables, with a finite number of solutions because the solutions are positive integers. A common start on finding the set of solutions is to solve the equation for one variable.

Because the numbers 2 and 100 are both even, it is probably fastest to solve the given equation for b.

[1]

With the equation in the last form [1], we see that (50-a) must be divisible by 3. That means, among other things, that consecutive values of a that provide solutions will differ by 3.

(1) Solution with smallest value of a...

Remembering that the solutions must have both a and b positive 2-digit integers, we can see that the smallest 2-digit value for a that makes (50-a) divisible by 3 is 11.

That gives us
(2) Solution with the largest value of a...

The solution with the largest value of a is the one with the smallest value of b. Again since a and b must be 2-digit integers, equation [1] says that all values of b that give solutions will be even, so the smallest value of b that will give a solution is the smallest 2-digit even integer, which is 10.

From (1) and (2), the solutions are the ones with b having even values from 10 to 26 inclusive. The number of such solutions is

ANSWER: There are 9 solutions having both a and b 2-digit positive integers

Answer by math_tutor2020(3835)   (Show Source):
You can put this solution on YOUR website!

Answer: 9

-------------------------------------------------------------------------

Reasoning

It sounds like "digest" should be "digit".

I'm assuming the question is:
"If A and B are positive two-digit integers, how many solutions are there to the equation 2a+3b = 100?"

Let's look at the related equation 2a+3b = 1, since we can multiply both sides by 100 later.

Through fairly quick trial-and-error, you should find an integer solution is (a,b) = (-1,1)
Another method is to look at the graph of 2x+3y = 1 to find a lattice point, or you can use the Extended Euclidean Algorithm.

Since (-1,1) is a solution we can state that
2a+3b = 1
2*(-1)+3*(1) = 1

Multiply both sides by 100 to get,
2*(-1)+3*(1) = 1
100*( 2*(-1)+3*(1) ) = 100*1
100*2*(-1) + 100*3*(1) = 100
2*( 100*(-1) ) + 3*(100*1) = 100
2*( -100 ) + 3*(100) = 100
This demonstrates that (a,b) = (-100,100) is an integer solution to 2a+3b = 100.
Unfortunately a = -100 is not positive, and it's not a two-digit number either.
A similar problem shows up with b as well.
We must require that 10 <= a <= 99 and 10 <= b <= 99.
The symbol <= means "less than or equal to".

2x+3y = 100 solves to y = (-2/3)x+(100/3)
The slope -2/3 will tell us how to go from one integer point to another on this line.
Start at (-100,100).
Move down 2, right 3 to arrive at (-97,98)
Move down 2, right 3 to arrive at (-94,96)
etc

This "down 2, right 3" pattern can be expressed like this
a = -100+3t
b = 100-2t
where t is an integer.
If t = 0, then it leads to (a,b) = (-100,100)
If t = 1, then it leads to (a,b) = (-97,98)
If t = 2, then it leads to (a,b) = (-94,96)
and so on.
If t was negative, then you go the opposite direction "up 2, left 3".

Then,
10 <= a <= 99
10 <= -100+3t <= 99
10+100 <= -100+3t+100 <= 99+100
110 <= 3t <= 199
110/3 <= 3t/3 <= 199/3
36.67 <= t <= 66.33 approximately
Since t is an integer we round the lower bound up to the nearest integer; while the upper bound gets pulled down to the nearest integer.
Note how t = 36 is too small since it leads to a = 8; while t = 67 is too large because it leads to a = 101.

Anyways, we go from
36.67 <= t <= 66.33
to
37 <= t <= 66

Meanwhile,
10 <= b <= 99
10 <= 100-2t <= 99
10-100 <= 100-2t-100 <= 99-100
-90 <= -2t <= -1
-90/(-2) >= -2t/(-2) >= -1/(-2) .... inequality signs flip
45 >= t >= 0.5
0.5 <= t <= 45
1 <= t <= 45 .... rounding to nearest integer
The inequality signs flipped because we divided all sides by a negative number.

Overlap these two intervals
37 <= t <= 66
1 <= t <= 45
To get the interval 37 <= t <= 45
I recommend drawing out a number line.

This set of t values will make both 10 <= a <= 99 and 10 <= b <= 99 true simultaneously where a = -100+3t and b = 100-2t.

How many integers are in the set {37,38,...,45}?
We want the 37 to go to 1, the 38 to go to 2, etc.
Subtract 36 from each item and you'll get the set {1,2,...,9}

Or you could say 45-37+1 = 9
The formula I used is that there are n-m+1 items in the set {m,m+1,m+2,...,n-1,n} where m,n are integers and m < n.
Note the -37+1 portion combines to -36, which is what we subtracted from each item of the previous set.

We determined there are 9 integer solutions to the equation 2a+3b = 100 where a,b are positive two-digit integers.
Using a spreadsheet, or something like Python, the set of all solutions we're after is this
a = 11, b = 26
a = 14, b = 24
a = 17, b = 22
a = 20, b = 20
a = 23, b = 18
a = 26, b = 16
a = 29, b = 14
a = 32, b = 12
a = 35, b = 10
Each time 'a' goes up by 3, b drops by 2.
It seems like your teacher only cares about the number of solutions, rather than the solutions themselves. So this portion is likely optional.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
If 'a' and 'b' are positive 2-digit integers,
how many solutions are there of the equation 2a+3b=100.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The post by @MathLover1 isn't a solution to the problem because it doesn't answer the problem's question.
        The question "how many" remains unanswered, or she shifts the calculation to the reader.
        Therefore, her post is a talk about a solution, but not the solution itself.
        I came to deliver the solution as it should be.

In equation 2a + 3b = 100, the term '2a' is an even number and right side '100' is an even number, too. Hence, the term '3b' must be even number. It implies that number 'b' must be even number. Then '3b' is a multiple of 6. Thus, '3b' should be multiple of 6 and 'b' itself should be a two-digit positive integer number. The smallest such number '3b' is 30. So, the numbers '3b' form the set < 30, 36, 42, . . . >. We want to determine the maximum possible value of '3b'. It should allow 'a' in equation '2a = 100 - 3b' to be a 2-digit positive integer number. So, the greatest possible value of '3b' is not more than 80. Thus, our set for '3b' is the sequense < 30, 36, 42, . . . , 78 >. It contains = 9 numbers. So, equation 2a + 3b = 100 has 9 solutions in positive integer numbers (a,b), such that 'a' and 'b' are 2-digit positive integer numbers. ANSWER. There are 9 solutions under imposed conditions.

Solved.

Answer by MathLover1(20855)   (Show Source):
You can put this solution on YOUR website!

given:

and are positive digest integers

Rearrange the equation to express in terms of :

to ensure that is a positive integer which

Determine the range for :
since must be a positive integer, ()

determine
check the : note be divisible by
Count the valid integer values of (knowing that ) which satisfy both conditions to find the total number of solutions.

Question 1210503: if Tom gives maria 30 cents, they will have equal amounts of money. But if Maria then gives Tom 50 cents, he will how have twice as much money as she does. how much does each have now?
Found 2 solutions by ikleyn, josgarithmetic:
Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
If Tom gives Maria 30 cents, they will have equal amounts of money.
But if Maria then gives Tom 50 cents, he will how have twice as much money as she does.
How much does each have now?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let x be the amount which Tom and Maria each have equally after first exchange. It means that before first exchange, Tom had (x+30) cents, while Maria had (x-30) cents. Now we consider the second exchange. For it, we have this equation x + 50 = 2(x - 50) Simplify and find x x + 50 = 2x - 100, 50 + 100 = 2x - x x = 150. It implies that originally (or "now") Tom has 150+30 = 180 cents; Maria has 150-30 = 120 cents. ANSWER

Solved.

Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!

IF THEN IF Tom t t-30 t-30+50=t+20 Maria m m+30 m+30-50=m-20

Following the description given,

Solve for original t and m money cents quantities

ADD those.
and then

They each then have AFTER the second transfer
Tom, 200
Maria, 100

Question 730245: In a lake, there is a patch of lily pads. Every day the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?
Found 2 solutions by MathTherapy, ikleyn:
Answer by MathTherapy(10806)   (Show Source):
You can put this solution on YOUR website!

In a lake, there is a patch of lily pads. Every day the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake? I wholeheartedly agree with Tutor @Ikleyn that the patch will take 47 days to cover of the lake, considering that its growth DOUBLES every day. For this exponential growth scenario, we need to use the Simple Discrete Growth Formula: , or , where: f(x) = Growth percent/fraction at a particular time (max being 100, or 1, or ) x = time period during growth a = Initial/Beginning growth percent/fraction () b = growth rate <=== this can also be represented by 1 + r ---- Substituting 48 for x, and 2 for b ---- Substituting 1 (100%) for f(48) Initial percent/fraction of lake covered, or a = ----- Substituting for a, and 2 for b ----- Substituting for f(x) -- Converting to , and to - 1 = - 48 + x ----- BASES are equal, and so are the exponents Time it takes for the patch to cover of the lake, or x = - 1 + 48 = 47 days.

Answer by ikleyn(53748)   (Show Source):
You can put this solution on YOUR website!
.
In a lake, there is a patch of lily pads. Every day the patch doubles in size.
If it takes 48 days for the patch to cover the entire lake, how long would it take
for the patch to cover half of the lake?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

It is a classic puzzle for those who start learning exponential functions.

            The correct answer is 47 days.

The solution to this puzzle is extremely short:

        If the patch covers the entire lake in 48 days, it means that
        half of the lake is covered in the previous, 47-th day.

What is interesting: the correct answer and this short solution often shock the beginners.

But don't be discouraged - this is a normal reaction for a beginner.
That's why you're learning from experts to understand non-obvious truths.

/\/\/\/\/\//\/\/\/\/\/\/\/

The answer "24" in the post by @lynnlo is incorrect.

Question 730242: Let P(-3,6) and Q(10,1) be two points in the coordinate plane:
a. Find the distance between P and Q.
b. Find the slope intercept form of the equation of the line that contains P and Q.
c. Find an equation of the circle that contains P and Q and whose center is the midpoint of the segment PQ.
d. Find the equation for the line that contains P and that is perpendicular to the segment PQ.
e. Find an equation for the line that passes through the origin and that is parallel to the segment PQ.
Answer by josgarithmetic(39792)   (Show Source):
You can put this solution on YOUR website!
Question (c):

Circle containing P and Q with center at midpoint

Midpoint at ,

Diameter length

Radius is
Radius SQUARED is
Radius squared is

The asked-for circle is .
This was not rechecked.