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Jump to solutions: 0..29 , 30..59 , 60..89 , 90..119 , 120..149 , 150..179 , 180..209 , 210..239, >>NextVectors/1160091: 1. A plane is steering at S65°W at an air speed of 625 km/h. The wind is from the NW at 130 km/h. Find the ground speed and track of the plane. Include a vector diagram in your solution.
1 solutions
Answer 852205 by ikleyn(52788)   on 2025-09-14 19:41:16 (Show Source):
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Vectors/1159991: A plane is steering at S65°W at an air speed of 625 km/h. The wind is from the NW at 130 km/h. Find the ground speed and track of the plane. Include a vector diagram in your solution.
1 solutions
Answer 852204 by ikleyn(52788) on 2025-09-14 19:26:02 (Show Source):
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Matrices-and-determiminant/1159507: Find a non-zero 2×2 matrix such that:
[-9, -3] * [] = [0, 0]
[18, 6] [] [0, 0]
1 solutions
Answer 852202 by ikleyn(52788) on 2025-09-14 14:29:16 (Show Source):
You can put this solution on YOUR website! .
Find a non-zero 2×2 matrix such that:
[-9, -3] * [] = [0, 0]
[18, 6] [] [0, 0]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Looking at the first (2x2)-matrix, you may notice that its first column is three times the second column.
It tells you to take the first column of the second (2x2)-matrix as  .
Looking again at the first (2x2)-matrix, you may notice that its second column is one third times the first column.
It tells you to take the second column of the second (2x2)-matrix as  .
Thus, the second matrix can be taken as  .
Now you may check, making handle calculations, that
 * =  , QED.
This way, the problem is just solved, and one instance matrix is found.
But this solution is not a unique.
Every other matrix, c-times the found, where 'c' is any non-zero constant value, has the same property.
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Finance/1167311: tim did 10 lunges on day 1 and continued this doing 4 more lunges on each day.on day 7th he took a breakand did nt do any day day.what is the equation for this? express it in the form y=ax(x+b)+c, where a,b and c are constants.x= no of days and y=no of lunges timid
1 solutions
Answer 852201 by ikleyn(52788) on 2025-09-14 12:59:59 (Show Source):
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Finance/1167337: tim did 10 lunges on day 1 and continued this doing 4 more lunges on each day.on day 7th he took a breakand did nt do any day day.what is the equation for this? express it in the form y=ax(x+b)+c, where a,b and c are constants.x= no of days and y=no of lunges tim Did
1 solutions
Answer 852200 by ikleyn(52788) on 2025-09-14 12:56:46 (Show Source):
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Linear_Algebra/1167650: Show that the inner product ⟨u, v⟩ = 5u1v1 − u1v2 − u2v1 + 10u2v2 is the inner product on R2 generated by the matrix A=
| 2 1 |
| -1 3 |
1 solutions
Answer 852198 by Resolver123(6)  on 2025-09-14 08:37:50 (Show Source):
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Linear_Algebra/1167394: How does the rank of the following matrix depend on the value of t?
(1,1,t)
(1,t,1)
(t,1,1)
1 solutions
Answer 852197 by Resolver123(6) on 2025-09-14 07:46:14 (Show Source):
You can put this solution on YOUR website! We are given the following 3x3 matrix:
Compute the determinant det(A):
 .
Let  , or  .
Hence, det(A) = 0 if and only if t = 1 or t = -2.
Consider 3 cases:
Case 1: t ≠ 1 and t ≠ -2.
Then det(A) ≠ 0, and so the matrix is of full rank, that is, rank(A) = 3.
Case 2: t = 1
Then the matrix is:
All rows being identical means that there is only 1 linearly independent row. Hence, rank = 1.
Case 3: t = -2
Then we get the matrix:
Using the row operations  and  , we get the row equivalent matrix
Using the row operation  , we finally get
This gives 2 linearly independent rows, and therefore, rank = 2.
Thus, the rank of the matrix depends on  as follows:
* Rank = 3 if t ≠ 1 and t ≠ -2.
* Rank = 2 if t = -2, and
* Rank = 1 if t = 1.
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Miscellaneous_Word_Problems/1167598: Let
b0, b1, b2, be the sequence defined by the explicit formula
bn = C · 5^n + D(−4)^n for each integer n ≥ 0,
where C and D are real numbers.
(a)
Find C and D so that
b0 = 0
and
b1 = −9.
b) Find C and D so that
b0 = 4
and
b1 = 11.
c)What is
b2 in this case?
1 solutions
Answer 852196 by Resolver123(6) on 2025-09-14 05:33:50 (Show Source):
You can put this solution on YOUR website! (a)  for 
means that  , by hypothesis, after direct substitution.
Also, inserting  into the formula gives  .
Hence,  .
From c + d = 0, we get d = -c.
Substituting this into the equation 5c - 4d = -9 gives 5c - 4(-c) = 5c+4c = -9,
or 9c = -9, or c = -1.
Therefore, c = -1, and d = -(-1) = 1.
(b) Similarly, gives  , or c + d = 4, or d = 4 - c. Also,  , or 5c - 4d = 11.
Substituting, 5c - 4(4 - c) = 11, or 5c - 16 + 4c = 11.
This gives 9c = 27, or c = 3, which consequently gives d = 4 - 3 = 1.
(c) From part (b), we get  , so that 
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Inverses/1210395: What is the inverse of Y=(X+3)² ?
1 solutions
Answer 852194 by ikleyn(52788) on 2025-09-11 22:39:07 (Show Source):
You can put this solution on YOUR website! .
What is the inverse of Y=(X+3)² ?
~~~~~~~~~~~~~~~~~~~~~~~~
The direct function (which is the given function) does the following:
- it adds the number 3 to the argument and then squares the value.
Therefore, the range of the given function is the set of all non-negative real numbers.
It means that the domain of the inverse function is the set of all non-negative real numbers,
and at this set the inverse function makes the opposite operations in reverse order:
- first, it takes square root from the value, and after that subtract 3 from the square root.
So, we can write an expression for the inverse function
g(x) =  - 3.
But square root from a real number can be positive or negative.
Therefore, there are TWO possible expressions for "the" inverse function.
First expression is p(x) = - 3, using positive value of .
Second expression is q(x) = - - 3, using negative value of .
These two possible expressions define two possible inverse functions.
Both inverse functions are defined over the set of non-negative real numbers.
First inverse function has the range [ , ).
Second inverse function has the range (-,].
At this point, the problem is solved completely, with detailed explanations, so everybody can understand.
----------------------------
It is important to note that the question in the problem is not precisely correct/accurate.
Indeed, it asks " what is the inverse function of Y =  ?"
To be correct, in this situaion, the question should ask " what are possible inverse functions of Y = ?"
Now, after reading my post, you are armed to the teeth:
you know the right question and also know the right answer.
Come again to this forum soon to learn something new from the best source.
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Inverses/1210394: What is the inverse of Y=(X+3)² ?
2 solutions
Answer 852193 by ikleyn(52788) on 2025-09-11 22:07:14 (Show Source):
You can put this solution on YOUR website! .
What is the inverse of Y=(X+3)² ?
~~~~~~~~~~~~~~~~~~~~~~~~
The direct function (which is the given function) does the following:
- it adds the number 3 to the argument and then squares the value.
Therefore, the range of the given function is the set of all non-negative real numbers.
It means that the domain of the inverse function is the set of all non-negative real numbers,
and at this set the inverse function makes the opposite operations in reverse order:
- first, it takes square root from the value, and after that subtract 3 from the square root.
So, we can write an expression for the inverse function
g(x) = - 3.
But square root from a real number can be positive or negative.
Therefore, there are TWO possible expressions for "the" inverse function.
First expression is p(x) = - 3, using positive value of .
Second expression is q(x) = - - 3, using negative value of .
These two possible expressions define two possible inverse functions.
Both inverse functions are defined over the set of non-negative real numbers.
First inverse function has the range [,).
Second inverse function has the range (-,].
At this point, the problem is solved completely, with detailed explanations, so everybody can understand.
----------------------------
It is important to note that the question in the problem is not precisely correct/accurate.
Indeed, it asks " what is the inverse function of Y = ?"
To be correct, in this situaion, the question should ask " what are possible inverse functions of Y = ?"
Now, after reading my post, you are armed to the teeth:
you know the right question and also know the right answer.
Come again to this forum soon to learn something new from the best source.
Answer 852192 by josgarithmetic(39618) on 2025-09-11 19:55:32 (Show Source):
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Money_Word_Problems/1206600: Suppose you can afford to pay at most $2650 per month for a mortgage payment. If the maximum amortization period you can get is 20 years, and you must pay 6% interest per year compounded annually, what is the most expensive house you can buy? How much interest will you have paid to the lender at the end of the loan?
1 solutions
Answer 852188 by ikleyn(52788) on 2025-09-07 16:26:34 (Show Source):
You can put this solution on YOUR website! .
Suppose you can afford to pay at most $2650 per month for a mortgage payment. If the maximum amortization period
you can get is 20 years, and you must pay 6% interest per year compounded annually, what is the most expensive
house you can buy? How much interest will you have paid to the lender at the end of the loan?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This problem is tricky, since the payments are monthly, while compounding are annually.
So, the payments are desynchronized with compounding.
It means that monthly payments lie in the bank with no move and wait for the end of a year -
only then they are compounded, according to the problem.
Classic formulas for loan/mortgage are applicable for synchronized payments/compounding.
But we can modify the situation EQUIVALENTLY to get payments/compounding synchronized.
Indeed, we actually have annual payments of 12*2650 = 31800 dollars each, compounded annually.
Thus, it works as a classic loan for 20 years with annual payments of $31800
at the end of each year, compounded annually at the annual rate of 6%.
Now apply a standard loan formula
PMT = 
where PMT is the annual payment ($31800); L is the loaned amount; r = 0.06 is the percentage rate
of rounding and n = 20 years. Then the formula becomes
31800 =  .
From it, we get
L =  = 364743.50.
It means that the most expensive house you can buy under given conditions is for $364,743.50. <<<---=== ANSWER
You will pay for the loan 20*12*2650 = 636000 dollars.
It means that interest you have paid to the lender for the loan is
636,000 - 364,743.50 = 271256.50 dollars. <<<---=== ANSWER
At this point, the problem is solved completely.
//////////////////////////////////////////
The solution in the post by @Theo is inadequate to the problem.
It is because Theo introduces, considers and treats monthly compounding; but the bank does not perform
monthly compounding. According to the problem, the bank makes annual compounding, ONLY.
Theo introduces equivalent monthly rate; but it works as an equivalent scheme only under condition
when there are no intermediate compounding inside a year. When there are intermediate monthly compounding,
it immediately destroys equivalency.
Had the problem admit monthly compounding, the solution by @Theo would be correct.
But under the conditions, described in the post, @Theo' solution is inadequate.
It is why I called this problem "tricky".
It has a hidden underwater stone as a trap, and, therefore, should be treated carefully.
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Money_Word_Problems/1207430: A property worth $35 OOO is purchased for 10% down and semi-annual payments of $2100 for 12 years. What is the nominal annual rate of interest if interest is compounded quarterly?
1 solutions
Answer 852186 by ikleyn(52788) on 2025-09-06 18:25:03 (Show Source):
You can put this solution on YOUR website! .
A property worth $35,000 is purchased for 10% down and semi-annual payments of $2100 for 12 years.
What is the nominal annual rate of interest if interest is compounded quarterly?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The solution by tutor @Theo has an error (or a typo).
His answer 8.035524952% is incorrect.
I came to bring a correct solution, right numbers and proper answer.
Down payment is 10% of $35,000, i.e. 0.1*35000 = 3500 dollars.
Hence, the loan is the rest amount of $35,000 - $3,500 = $31,500.
+------------------------------------------------------------+
| Notice that, as it is given in the problem, semi-annual |
| payments are desynchronized with quarterly compounding. |
+------------------------------------------------------------+
Nevertheless, we can synchronize payments and compounding by considering an EQUIVALENT scheme
with semi-annual compounding with the effective growth coefficient 'r' semi-annually.
This coefficient 'r' is not known now, and we should find it from the problem.
We then have a loan of $31,500 with semi-annual payments of $2100 and semi-annual compounding
with the effective semi-annual rate of r.
Write the standard loan equation for such a loan
 = 
 = 
15 =
Solve this equation numerically to find 'r'.
I used online calculator https://www.wolframalpha.com/calculators/equation-solver-calculator/
It found the approximate real solution r = 0.0416015.
Thus, in the equivalent scheme, the semi-annual effective rate is 0.0416015.
// Notice that till point my solution coincides with that by @Theo.
Hence, the effective semi-annual growth factor is 1+r = 1.0416015.
It implies that in the basic scheme, the effective quarterly growth factor is the square root of that
 = 1.020589.
Hence, the effective quarterly rate is 0.020589.
Then the annual effective rate is four times this, or 4*0.020589 = 0.082388.
Thus the nominal annual compounding interest is 8.2388%. <<<---=== ANSWER
Solved.
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Probability-and-statistics/1207399: In a large population, about
45
%
of people prefer tea over coffee. A researcher takes a random sample of
13
people and surveys whether they prefer tea over coffee.
Use the binomial distribution to compute the probability that exactly
6
of the people in the sample prefer tea over coffee.
Identify the following information required to find the probability of people who prefer tea over coffee.
Provide your answer below:
n =
trials
x =
successes
p =
probability of those who prefer tea (as a decimal, not percent)
1 solutions
Answer 852185 by ikleyn(52788) on 2025-09-06 02:09:17 (Show Source):
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Money_Word_Problems/1207426: What payment made at the end of each year for 18 years will amount to $16,000 at 4.2% compounded monthly?
1 solutions
Answer 852184 by ikleyn(52788) on 2025-09-05 10:52:59 (Show Source):
You can put this solution on YOUR website! .
What payment made at the end of each year for 18 years will amount to $16,000 at 4.2% compounded monthly?
~~~~~~~~~~~~~~~~~~~~
In his post, tutor @Theo solved this problem using a calculator,
and described a methodology solving this problem using calculator.
It is good, but since this website is intended to teach mathematical methods,
I present here mathematical solution with all detailed explanations.
As it is given in the post, this annuity is not standard: the payments are made at the end of each year,
while compounding is made at the end of each month.
Analytic formulas exist only for coinciding schedules of payments and compounding.
But we can use an equivalent standard synchronized scheme, considering payments at the end of each year
and compounding at the end of each year with the  annual multiplicative growth rate
1+r =  = 1.042818007. (1)
Now we can use a standard formula for such ordinary annuity
FV =  . (2)
In this formula, FV is the future value in 18 years; P is the annual payment, the unknown value
which we should find.
We calculate the factor in the formula (2) separately
 =  = 26.31908947.
Then from formula (2) we find
P =  =  = 607.93 dollars.
Thus we found out the annual payment value. It is $607.93. ANSWER
Solved.
----------------------------------
My result is precisely consisted with the answer by @Theo.
Now you can solve similar problems mathematically and check them using calculator.
It is a good and reliable strategy.
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Linear_Algebra/1160838: Find an orthonormal basis of the plane x−4y−z=0.
1 solutions
Answer 852182 by ikleyn(52788) on 2025-09-04 00:57:17 (Show Source):
You can put this solution on YOUR website! .
Find an orthonormal basis of the plane x−4y−z=0.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider two vectors V1 = (1,0,1) and V2 = (2,1,-2).
You can manually check that both vectors V1 and V2 satisfy the given equation -
- so, they belong to the plane described by this equation.
Next, it is clear that vectors V1 and V2 are linearly independent - hence, they form
a basis in the plane described by the given equation.
The fact is that vectors V1 and V2 are orthogonal.
You may check it on your own.
To get vectors V1 and V2 orthonormal, we should divide each vector by its length.
Doing so, we get orthonormal vectors (  , ,) and ( , , ).
Thus the problem is solved completely.
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Triangles/1164146: Two sides of a triangle are AB=34cm and AC=25cm and their included angle measure 62°. Question is, Find the distance of the orthocenter to side AB.
1 solutions
Answer 852181 by ikleyn(52788) on 2025-09-03 10:07:07 (Show Source):
You can put this solution on YOUR website! .
Two sides of a triangle are AB=34cm and AC=25cm and their included angle measure 62°.
Find the distance of the orthocenter to side AB.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
In this problem, the given input data looks like is not related to the question,
so, it seems that it is unsolvable.
Perhaps, it is the reason why it remained unsolved about 5 years at this forum.
Nevertheless, the solution does exist and is quite beautiful, although not obvious, from the first glance.
So, we have a triangle ABC with the sides AB = 34 cm and AC = 25 cm.
Their included angle A is 62°, so we can find the length of the third side BC
opposite to angle A. Use the cosine law
BC =  =  = 31.3512096 cm.
Now, having the lengths of the three sides of triangle ABC, we can find its area,
using the Heron's formula. In order for do not bother with calculations, I will use
one of numerous online calculators,
https://www.omnicalculator.com/math/herons-formula
It gives the area of triangle ABC
area(ABC) = 375.253 cm^2.
Other online calculators
https://www.inchcalculator.com/herons-formula-calculator/
https://www.wolframalpha.com/widgets/view.jsp?id=7ac490665df1b278eb748160468147bc
give practically the same value.
Having the side lengths 'a', 'b' and 'c' of the triangle ABC, we can now
to determine the radius of the circumscribed circle around triangle ABC
R =  =  = 17.7537 cm (rounded).
Now the distance from the orthocenter to the side AB is the leg of the right angled triangle,
whose hypotenuse is R = 17.7537 cm and the other leg is half the length of the side AB.
So, we write
the distance from the orthocenter to the side AB =  = 5.118 cm (rounded).
ANSWER. The distance from the orthocenter to the side AB is 5.118 cm (rounded).
Thus, all the data was woven into one logical thread that led to a complete solution.
So, we can celebrate the victory.
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Miscellaneous_Word_Problems/1210391: The sum of two numbers is 21.five times the first number added to 2 times the second number is 66.find the numbers
6 solutions
Answer 852180 by greenestamps(13200)  on 2025-08-31 18:41:06 (Show Source):
You can put this solution on YOUR website!
Some of the tutors who have responded use a single variable to solve the problem. Others use two variables to make two equations and then use substitution to solve the problem.
Those are valid methods. But with the problem worded the way it is, forming two equations in two unknowns and solving by elimination would seem to be the most direct approach.
x = first number
y = second number
(1) x+y = 21 (the sum of the two numbers is 21)
(2) 5x+2y = 66 (5 times the first number plus 2 times the second is 66)
Multiply (1) by 2:
(3) 2x+2y = 42
Find the difference between (2) and (3) to eliminate y: 3x=24
Solve: x = 8
Substitute x = 8 into (1) to find y:
8+y = 21
y = 13
Note that this solution is exactly equivalent to the good "mental" solution provided by another of the tutors.
ANSWERS: 8 and 13
Answer 852178 by mccravyedwin(407)  on 2025-08-27 18:24:16 (Show Source):
You can put this solution on YOUR website!
We should not tutor on here as if the students who post here are gifted
students. The truth is that the majority of students who come on here are
STRUGGLING!!!
Edwin
Answer 852177 by ikleyn(52788) on 2025-08-27 17:01:43 (Show Source):
You can put this solution on YOUR website! .
The sum of two numbers is 21. Five times the first number added to 2 times the second number is 66. Find the numbers.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I will show here how to solve this problem MENTALLY, without using equations.
It is clear that we can re-formulate the problem  this way:
Two times the first number added to 2 times the second number is 42.
Five times the first number added to 2 times the second number is 66. Find the numbers
Looking in these two lines, your brain may conclude that three times (3 = 5-2) the first number is
the difference 66-42 = 24.
Hence, first number is 24/3 = 8.
Then the second number is 21-8 = 13.
ANSWER. First number is 8. Second number is 13.
Solved MENTALLY, without using equations.
Answer 852176 by timofer(105) on 2025-08-27 14:04:27 (Show Source):
Answer 852175 by MathLover1(20850)  on 2025-08-27 13:11:12 (Show Source):
Answer 852174 by Edwin McCravy(20056)  on 2025-08-27 13:01:36 (Show Source):
You can put this solution on YOUR website!
The sum of two numbers,
x = first number
y = second number
is 21.
So x + y = 21
five times the first number added to 2 times the second number is 66.
5x + 2y = 66
Solve the first equation for y: y = 21 - x.
Substitute in the second equation:
5x + 2(21 - x) = 66
5x + 42 - 2x = 66
3x = 66 - 42
3x = 24
x = 8
y = 21 - (8)
y = 13
Answer: The first number is 8 and the second number is 13.
Yes, I know it can be done with only one unknown. But I think it's
much easier when students can think of each single number mentioned
in the word problem as represented by one single letter, than it is
to have to think of the second number as the complicated combination
of a number, a sign, and an expression in the letter used for the
first number.
Edwin
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Miscellaneous_Word_Problems/1166337: A company has $14,830 available per month for advertising. Newspaper ads cost $190 each and can't run more than 24 times per month. Radio ads cost $590 each and can't run more than 32 times per month at this price.
Each newspaper ad reaches 5700 potential customers, and each radio ad reaches 6700 potential customers. The company wants to maximize the number of ad exposures to potential customers.
Use x for number of Newspaper advertisements and y for number of Radio advertisements.
1 solutions
Answer 852179 by ikleyn(52788) on 2025-08-30 19:51:01 (Show Source):
You can put this solution on YOUR website! .
A company has $14,830 available per month for advertising. Newspaper ads cost $190 each and can't run
more than 24 times per month. Radio ads cost $590 each and can't run more than 32 times per month at this price.
Each newspaper ad reaches 5700 potential customers, and each radio ad reaches 6700 potential customers.
The company wants to maximize the number of ad exposures to potential customers.
Determine the most profitable / (effective) way to do it.
~~~~~~~~~~~~~~~~~~~~~~~~
To find the maximum number of ad exposures, let's formulate the problem in terms
of objective function and constraints.
Let x be the number of newspaper ads and y be the number of radio ads.
The objective function is
P = 5700*x + 6700*y. (1)
It is the number of possible expositions, and we want to maximize it.
The constraints are:
190*x + 590*y ≤ 14830 (the budget), (2)
x ≤ 24 (newspaper ad limit), y ≤ 32 (radio ad limit), (3)
x ≥ 0, y ≥ 0 (non-negativity). (4)
Now it is presented as a typical Linear Programming problem. But it can be easily solved MENTALLY
using "the most aggressive" logical strategy/methodology.
From expression (1) for the objective function, we see that the contribution of each single newspaper ad
(in terms of the number of potential expositions, 5700) is comparable with (or distinct insignificantly from)
the contribution of each single radio ad (6700).
But each newspaper ad is much cheaper ($190) than each radio ad ($590). So, it is clear that
the most profitable strategy is to make as many newspaper ads as possible (x=24), and then
to spend the rest of the budget for the radio ads.
Thus the most effective solution is to make 24 newspaper ads, spending 24*190 = 4560 dollars for it.
The rest of the budget is then $14830 - $4560 = $10270.
This amount can be / (should be) spent for radio ads.
It provides the number of radio ads y =  = 17.40678,
and we should round this decimal number to the closest lesser integer number, which is y = 17.
So, the answer to the problem's question is THIS:
24 newspaper ads and 17 radio ads provide the greatest possible number of expositions (~ potential customers),
which is then 5700*24 + 6700*17 = 250700.
Solved.
--------------------------------------
Post-solution note
In this concrete problem, the presented method has one important advantage comparing with the traditional form
geometric solution of Linear Programming problems.
Working in the frame of traditional Linear Programming geometric method, you will get the solution with non-integer decimals,
so, you will be forced to use other arguments to complete the traditional solution.
Working in the way, presented here in the solution above, you will get the answer in integer numbers without any complications.
In other words, this problem is for INTEGER Linear Programming - not for regular Linear Programming.
But integer Linear Programming problems require their special solution methodology
(which is not studied in the school Math) and requires special solvers.
Or, as it is done in my solution above - a special logical treatment.
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Numeric_Fractions/1210390: 100 coins are on a table showing 10 heads and 90 tails. You are blindfolded. Is it possible to divide the coins into 2 piles with an equal number of heads?
2 solutions
Answer 852173 by math_tutor2020(3817) on 2025-08-24 23:02:36 (Show Source):
You can put this solution on YOUR website!
Label the piles A and B.
Randomly select 10 coins to form pile A.
The remaining 90 coins are pile B.
Let x be a whole number in the set {0,1,2,...,9,10}
x is between 0 and 10
If x coins in pile A are heads, then 10-x coins in pile A are tails since pile A has 10 coins.
Further we know there are 10-x heads in pile B since there are 10 heads total.
Turn over all coins in pile A.
x heads ---> x tails
10-x tails ---> 10-x heads
We'll have 10-x heads in pile A and 10-x heads in pile B.
At this point we have guaranteed both piles have the same number of heads.
Unfortunately we won't be able to determine how many heads are in each pile.
--------------------------------------------------------------------------
Let's look at an example.
Let's say there are x = 3 heads in pile A and 10-x = 10-3 = 7 tails in pile A.
The remaining 10-x = 10-3 = 7 heads are in pile B.
Now let's turn over every coin in pile A.
3 heads ----> 3 tails
7 tails ----> 7 heads
We will have 7 heads in each pile.
I encourage you to try other values of x in the set {0,1,2,...,9,10} so you can get a sense of how this is working.
Note carefully I say "turn over" rather than "flip" because I want to avoid the idea of randomly flipping a coin.
When I say "turn over" I simply mean "any heads becomes tails or vice versa".
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Quick Recap: Select any 10 random coins. Turn those 10 coins over (leave the rest as they are).
Answer 852172 by ikleyn(52788) on 2025-08-24 21:36:21 (Show Source):
You can put this solution on YOUR website! .
100 coins are on a table showing 10 heads and 90 tails. You are blindfolded.
Is it possible to divide the coins into 2 piles with an equal number of heads?
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Yes, of course - if you ask another person, who is sighted, to do it for you :)
Another way for a blindfolded person to perform this assignment on his or on her own
is to select 5 head coins from the entire collection of 100 coins by touch.
In this case, you use your tactile senses instead of your sight.
To me, it remains unclear how this post and your question do relate to Math . . .
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Complex_Numbers/1210389: The function f(x) = (1\x) is decreasing on its domain ( true or false)
3 solutions
Answer 852171 by math_tutor2020(3817) on 2025-08-24 16:43:07 (Show Source):
You can put this solution on YOUR website!
Answer: true
Explanation
I think you meant to type
f(x) = 1/x
As x gets bigger, i.e. as we move to the right, what happens to y = 1/x?
We can look at a graph to note how it goes downhill throughout its entire domain.
Therefore the answer is true
As x gets bigger, y gets smaller.
As we move to the right the curve goes downhill.
A way to do this without a graph is to plug in a value like x = 2
1/x = 1/2 = 0.50
Then plug in x = 3
1/x = 1/3 = 0.33 approximately
Comparing 0.50 with 0.33 shows that y has gone down, which helps show the function decreases.
Keep in mind that this does not prove it always decreases over the entire domain.
You'll have to use the method mentioned in the next paragraph.
Let's do a formal proof.
Let a > 0 and b > 0.
Since both are positive this means a+b > 0.
Plug in x = a to arrive at 1/a which is also positive.
Plug in x = a+b to arrive at 1/(a+b) which is also positive.
Now to compare 1/a with 1/(a+b)
Assume for now they are equal,
1/a = 1/(a+b)
a+b = a .... cross multiply
b = 0
We arrive at a contradiction since we made b > 0 earlier.
To fix those errors, replace each equal sign with a greater than sign
1/a > 1/(a+b)
a+b > a .... cross multiply
b > 0
Or basically we move through those steps backwards like so
b > 0
a+b > a
1/a > 1/(a+b)
This basically shows that for some input x = a, if we nudge that input to the right a bit to arrive at x = a+b, then we have shown that 1/a is larger than 1/(a+b)
In short we have proven the function is always going downhill no matter where you're on it.
Note that we can follow very similar steps when a < 0 and b > 0, to effectively mirror things over the y axis.
Answer 852170 by ikleyn(52788) on 2025-08-24 16:29:50 (Show Source):
You can put this solution on YOUR website! .
The function f(x) = (1\x) is decreasing on its domain ( true or false)
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First of all, I read this function as f(x) = 1/x,
following to the standard rules commonly accepted in Math.
The domain is the set of all real numbers except of 0 (zero).
Function f(x) = 1/x has two branches: one over the set of positive 'x',
another over the set of negative 'x'.
For each branch separately, function f(x) = 1/x is monotonic in each sub-domain.
Or, more accurately, function f(x) = 1/x is monotonically decreases locally in each sib-domain.
To prove it, take the derivative: it is -  .
The denominator  is always positive for all 'x' of the domain; so, the derivative is always negative.
You also can convince yourself by making a plot of this function.
For it, go to website https:\\www.desmos.com/calculator/
Print the formula for the function y = 1/x.
You will get the plot instantly.
The plot clearly shows that the function monotonically decreases as 'x' increases in the domain.
So, locally for each branch, the function f(x) = 1/x monotonically decreases.
You have this answer proved formally and demonstrated / illustrated visually.
But we can not say that function f(x) = 1/x is globally decreasing: when 'x' moves
from negative values to positive values, the function 1/x jumps from negative values to positive,
breaking monotonicity.
So, locally this function is monotonically decreasing, but globally it is not monotonic.
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This problem is an elementary  .
As I explained in my post above, locally and piece-wisely, the function is monotonically decreasing.
But globally, it is NOT monotonic.
It is OBVIOUS: compare these values of the function
x -2 -1 1 2
1/x -0.5 -1 1 0.5
decrease decrease
increase
They show non-monotonic behavior.
Tutor @math_tutor2000 fell into this trap.
Answer 852168 by josgarithmetic(39618) on 2025-08-24 15:23:23 (Show Source):
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Complex_Numbers/1210388: The function f(x) = (1\x) is monotonic on its domain ( true or false) ?
1 solutions
Answer 852169 by ikleyn(52788) on 2025-08-24 16:02:29 (Show Source):
You can put this solution on YOUR website! .
The function f(x) = (1\x) is monotonic on its domain ( true or false) ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
First of all, I read this function as f(x) = 1/x,
following to the standard rules commonly accepted in Math.
The domain is the set of all real numbers except of 0 (zero).
Function f(x) = 1/x has two branches: one over the set of positive 'x',
another over the set of negative 'x'.
For each branch separately, function f(x) = 1/x is monotonic in each sub-domain.
Or, more accurately, function f(x) = 1/x is monotonically decreases locally in each sib-domain.
To prove it, take the derivative: it is - .
The denominator is always positive for all 'x' of the domain; so, the derivative is always negative.
You also can convince yourself by making a plot of this function.
For it, go to website https:\\www.desmos.com/calculator/
Print the formula for the function y = 1/x.
You will get the plot instantly.
The plot clearly shows that the function monotonically decreases as 'x' increases in the domain.
So, locally for each branch, the function f(x) = 1/x monotonically decreases.
You have this answer proved formally and demonstrated / illustrated visually.
But we can not say that function f(x) = 1/x is globally decreasing: when 'x' moves
from negative values to positive values, the function 1/x jumps from negative values to positive,
breaking monotonicity.
So, locally this function is monotonically decreasing, but globally it is not monotonic.
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After reading the post by @math_tutor2020
https://www.algebra.com/algebra/homework/complex/Complex_Numbers.faq.question.1210389.html
This problem is an elementary .
As I explained, locally and piece-wisely, the function is monotonically decreasing.
But globally, it is NOT monotonic.
It is OBVIOUS: compare these values
x -2 -1 1 2
1/x -0.5 -1 1 0.5
decease decrease
increase
They show non-monotonic behavior.
Tutor @math_tutor2000 fell into this trap.
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Circles/1210387: A television tower broadcasts its signal at a range of 18.1 miles from the tower to its further point. How many square miles are covered by the range of this tower? Round your answer to the nearest hundredth if necessary.
1 solutions
Answer 852167 by ikleyn(52788) on 2025-08-23 22:37:06 (Show Source):
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Money_Word_Problems/1164634: Determine the present value of a series of 60 monthly payments of $2,500 each which begins 1 month from today. Assume interest of 10 percent per year compounded quarterly.
1 solutions
Answer 852165 by ikleyn(52788) on 2025-08-23 17:22:09 (Show Source):
You can put this solution on YOUR website! .
Determine the present value of a series of 60 monthly payments of $2,500 each
which begins 1 month from today. Assume interest of 10 percent per year compounded quarterly.
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In this tricky problem, the monthly payments are not compounded.
Monthly payments lie in the bank and wait for the end of a quarter -
only then they are compounded, according to the problem.
So, we actually have quarterly payments of 3*2500 = 7500 dollars each, compounded quarterly.
Thus, it works as an Ordinary Annuity saving plan with quarterly payments of $7500
at the end of each quarter, compounded quarterly at the annual rate of 10%.
60 monthly payments of $2500 each mean 60/3 = 20 quarterly payments of $7500 each.
So, the future value of the account after 20 quarters will be
FV =  = 191584.93 dollars.
Now we want to find the present value X. It is the starting value of the account,
which, when compounded quarterly at 10% per year, will have the same future value in 20 quarters.
So, we write this equation
191584.93 =  .
It gives the solution
X =  = 116918.72 dollars.
ANSWER. The present value is $116918.72 dollars.
Solved.
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On Aug. 26, 2025, I posted this problem to Google AI.
Determine the present value of a series of 60 monthly payments of $2,500 each which begins 1 month from today.
Assume interest of 10 percent per year compounded quarterly.
The Google AI Overview solution is placed below.
The present value (PV) of the series of 60 monthly payments is approximately $127,055.64.
To calculate this, you must first convert the annual interest rate to a quarterly rate,
then find the effective monthly interest rate by converting the quarterly rate to an equivalent monthly rate.
Finally, use this effective monthly rate in the present value of annuity formula to solve for the present value.
1. Convert the Annual Rate to a Quarterly Rate
Annual Rate: 10% or 0.10
Quarters in a year: 4
Quarterly Rate (r/q): 0.10 / 4 = 0.025 or 2.5%
2. Find the Effective Monthly Interest Rate
Since payments are monthly but compounding is quarterly, we need to find the equivalent monthly interest rate.
Formula: (1 + r/q)^(1/3) - 1 = Effective Monthly Rate (i)
Calculation: (1 + 0.025)^(1/3) - 1 = 1.025^(0.3333) - 1 = 1.008264 - 1 = 0.008264 (approximately)
3. Calculate the Present Value of the Annuity
Now that we have the effective monthly interest rate, we can use the present value of an ordinary annuity formula:
Formula: PV = PMT * [1 - (1 + i)^(-n)] / i
PMT (Payment): $2,500
i (Effective Monthly Rate): 0.008264
n (Number of Payments): 60
Calculation:
PV = $2,500 * [1 - (1 + 0.008264)^(-60)] / 0.008264
PV = $2,500 * [1 - (1.008264)^(-60)] / 0.008264
PV = $2,500 * [1 - 0.61158] / 0.008264
PV = $2,500 * 0.38842 / 0.008264
PV = $2,500 * 47.00778
PV = $127,055.64
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My (@ikleyn) comments.
The solution above, produced by this AI, is INCORRECT.
It is CONCEPTUALLY incorrect and TECHNICALLY incorrect in both aspects: conceptual and technical.
It is conceptually incorrect, since it introduces monthly compounding and operates with them.
But, according to the problem's description, the bank does not make monthly compounding.
It makes the quarterly compounding, ONLY.
They say that they select the monthly effective interest rate to make
monthly compounding equivalent to quarterly compounding.
But this equivalence works ONLY if there are no monthly deposits.
It they are and participate in compounding, then the two models
(one with monthly compounding and another with quarterly compounding) are NOT equivalent.
So, when the solution assumes monthly compounding of monthly deposits, it (the solution)
goes against the way, described in the problem, on how the bank treats this account.
Technically, the computations of PV are incorrect.
The correct calculation is THIS
PV =  = 117,663.42 dollars (without making any intermediate rounding till the end).
Compare it with the number of 127,055.64 dollars in the solution by the AI.
The difference is HUGE - - - not acceptable for finance predictions.
But even $117,663.42 is not a correct answer, since it is obtained via wrong conceptual assumption.
The correct solution/answer, adequate to the given problem, is what is given in my solution/post above.
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