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A rectangular field is to be enclosed by a fence and Ithen divided into two
small plots by a fence parallel to one of the sides.
If there is 1800 m of fencing available, find the dimensions of the field that will give the maximum area.
:
2L + 3W = 1800
2L = 1800-3W
Divide both sides by 2
L = (900-1.5W)
:
Area = L * W
Replace L with (900-1.5W)
A = W(900-1.5W)
A = -1.5W^2 + 900W
Find the axis of symmetry, x=-b/(2a); in this equation: a=-1.5, b=900
W =
W =
W = +300 m is the width for max area
then
L = 900 - 1.5(300)
L = 450 m is the length
and
300 * 450 = 135,000 sq/m is max area

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Find a number which is decreased by 54 and is 5 times its opposite?
:
Let x = "a number
then
-x = it's opposite
:
x - 54 = 5(-x)
x - 54 = -5x
x + 5x = +54
6x = 54
x =
x = 9
;
;
Check solution in original statement
9 - 54 = 5(-9)
-45 = -45

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1n 2005, a father was 4 times as old as his son.
In the year 2015,thrice the son's age will exceed the father's age by ten
years, in which year was the son born?
:
In 2005, let; f = father's age, s = son's age:
f = 4s
:
In 2015:
f + 10 = 3(s+10) - 10
f + 10 = 3s + 30 - 10
f + 10 = 3s + 20
f = 3s + 20 - 10
f = 3s + 10
Replace f with 4s, find s:
4s = 3s + 10
4s - 3s = 10
s = 10 yrs old in 2005
Son was born in 1995
:
:
Check solution, fathers age in 2005, 4(10) = 40
In 10 yrs
40 + 10 = 3(10+10) - 10
50 = 3(20) - 10

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why is it that in diff places you do not reciprocate the values that you need to eliminate?for example 1x+2y=3
2x+1y=3 then if youre trying to eliminate y or get it it becomes 1(1x+2y=3)
2(2x+1y=3)then becomes x+2=3
4x+2y=6
:
I am not sure what you are trying to say here,
but if you want to solve the following by eliminating y:
:
1x + 2y = 3
2x + 1y = 3
Multiply the 2nd equation by 2 and you have
1x + 2y = 3
4x + 2y = 6
-------------subtracting eliminates y, find x
-3x = -3
x =
x = +1
:
Find y using the 1st equation
1 + 2y = 3
2y = 3 - 1
2y = 2
y =
y = +1
:
Check solution in the 2nd original equation
2(1) + 1 = 3

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The product of four different positive integers is 360.
what is the maximum possible sum of these four integers ?
:
Find the 4th root of 360
= 4.35.......
integers above and below this value
3 * 4 * 5 * 6 = 360
:
3 + 4 + 5 + 6 = 18 is max

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Ethan cuts half a rectangular lawn 40 m by 30 m by mowing a strip of equal width around the perimeter.
Julia cuts the small rectangle left.
How wide a strip does Ethan cut so that they share the work equally?
:
Let x = the width of the strip cut by E
:
Find the total area of the lawn; 40 * 30 = 1200 sq/m
Each will mow an area of 600 sq/m
:
Dimension of the small rectangle which is 600 sq/ft:
(40-2x) by (30-2x)
;
Write an area equation of this rectangle. find x, which is the width of the strip
(40-2x)(30-2x) = 600
FOIL
1200 - 80x - 60x + 4x^2 = 600
Arrange as a quadratic equation
4x^2 - 140x + 1200 - 600 = 0
4x^2 - 140x + 600 = 0
Simplify divide by 4
x^2 - 35x + 150 0 =
Factor
(x - 5)(x - 30) = 0
The reasonable solution
x = 5 ft is the width of the strip
:
:
Check solution by finding the area of the inside rectangle
(40-10)*(30-10) = 600

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triangle has a perimeter of 36 centimeters.
If the length of each side is increased by 50%
:
Assume it is an equilateral triangle, each side 36/3 = 12
Increase each side by 50%, 18 cm, new perimeter, 3*18 = 54 cm

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Phone was thrown from top of a building.
The height h(t) of the cell phone at time (t) is described by the function h(t) = -16tsquared +96t+112.
When will the phone be at a height of 192 feet and how many seconds will it take to hit the ground?
:
-16t^2 + 96t + 112 = h(t)
The phone will be at 192 ft when h(t) = 192
-16t^2 + 96t + 112 = 192
Subtract 192 from both sides
-16t^2 + 96t + 112 - 192
-16t^2 + 96t - 80 = 0; a quadratic equation
Simplify, and change the signs, divide each term by -16
t^2 - 6t + 5 = 0
Factors to
(t - 5)(t + 1) = 0
The positive solution
t = 5 sec , tel at 192 ft
:
Phone will hit the ground when h(t) = 0
-16t^2 + 96t + 112 = 0
Simplify again, divide by -16
t^2 - 6t - 7 = 0
Factors to
(t - 7)(t + 1) = 0
positive solution
t = 7 sec, phone strikes the ground

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Bob received $14,000 inheritance and divided it between a mutual fund that
yielded a yearly return of 5% and a 1-year CD that paid 3.5% annually.
If his total earnings for the year on this investment was $640, how much did
he invest in each of these two funds?
:
Let x = amt invested at 5%
The total was given as 14000, therefore
(14000-x) = amt invested at 3.5%
:
Write a total interest equation in decimal form
.05x + .035(14000-x) = 640
:
.05x + 490 - .035x = 640
:
.05x - .035x = 640 - 490
:
.015x = 150
x =
x = $10,000 amt invested at 5%
:
I'll let you find out the amt invested a 3.5%
:
Check solution by finding the total interest using the two values

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Write a quadratic equation in the variable x having the given numbers as solutions. Type the equation in standard form, ax^2 + bx + c = 0
Problem:
Solution: 2, (only solution)
:
A solution of 2 will come from factors: (x-2), (x-2)
FOIL (x-2)(x-2) = x^2 - 4x + 4 = 0

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you charge $1 per cup of lemonade you sell 60 cups.
But when you raise your price to $1.50 you only sell 30 cups
Write an equation for the number of cups you sell as a function of the price you charge.
Denote "C" for number of cups, and "P" for the price you charge
:
the slope of the equation: m =
Assign the given values as follows:
x1=1.00, y1=60
x2=1.50, y2=30
Find the slope
m = = = -60 is the slope
:
Use the point/slope formula: y - y1 - m(x - x1)
y - 60 = -60(x - 1)
y - 60 = -60x + 60
y = -60x + 60 + 60
y = -60x + 120
:
c = -60p + 120, is the equation

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Three consecutive whole numbers are such that the square of the middle number
is greater than the product of the other two by 1.
Find the middle number:
There is no unique solution
:
let x = the middle number
then
(x-1) = the 1st number
and
(x+1) = the 3rd number
:
x^2 = ((x-1)*(x+1)) + 1
FOIL
x^2 = x^2 - 1 + 1
:
x^2 = x^2
Which means this is true for any 3 consecutive numbers even, -1, 0 +1
:
Try a few like; 142, 143, 144: 143^2 = 20449, 142*144 = 20448

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how long does it take a train to travel 27 feet at 4 mph
:
t = 27
5.867t = 27
t =
t = 4.6 seconds

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x + y = 90
x - y = 10
------------Adding eliminates y, find x
2x = 100
x = 50, then y = 40

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The son is 23 years younger than the mother.
In seven years, the mother will be twice as old as the daughter.
How old is each now?
:
Not much information here, let's assume Mommy is 31
then
son is 8 yrs old
and
31 + 7 = 2(d + 7)
31 + 7 = 2d + 14
31 + 7 - 14 = 2d
31 - 7 = 2d
24 = 2d
d = 24/2
d = 12 yrs is the daughter
:
There are many possibilities that are reasonable, and will give an integer value for the daughter's age
Mommy's age could be 27, 29, 31, 33, all odd numbers up to 89 and even beyond
For example when Mom is 89, son will be 66, daughter will be 41
:

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1.A silversmith has two alloys, one containing 30% silver and the other, 55%.
How much of each should be melted and combined to obtain 100 grams containing 45% silver?
:
Let x = amt of 55% alloy
Total amt will be 100 gr, therefore
(100-x) = amt of 30% alloy required
A typical mixture equation, using decimals
.55x + .30(100-x) = .45(100)
.55x + 30 - .30x = 45
.55x - .30x = 45 - 30
.25x = 15
x =
x = 60 gr of 55% alloy
then
100 - 60 = 40 gr of 30% alloy
:
:
2) A cellular phone company charges customers a certain amount for the first
minute of a long distance call and an additional amount for each succeeding minute.
A customer makes two calls to the same city, a 33-minute call for Php 269
and a 19-minute for Php 157.
How much does the cellular phone company charge for the first minute? for each succeeding minute?
:
Let x = amt charged for 1st min
Let y = amt charged for each succeeding min
:
Call 1: x + 32y = 269
Call 2: x + 18y = 157
-----------------------Subtraction eliminates x, find y
" "0x + 14y = 112
y =
y = 8 cents for each succeeding min
I'll let you find the value of x, the first minute cost
:
:
3) A swimming pool can be filled by three water pipes, A,B and C.
Pipe C alone can fill the pool in 8 hours.
If pipes B and C are used together, the pool can be filled in 6 hours.
If pipes A and B are used together, it takes 10 hours.
How long does it take to fill the pool if all three pipes are used?
:
Let the completed job = 1 (a full pool)
:
"pipes B and C are used together, the pool can be filled in 6 hours."
+ = 1
It tells us pipe C can do it in 8 hrs, therefore:
+ = 1
multiply by 8B
8(6) + 6B = 8B
48 = 8B - 6B
48 = 2B
B =
B = 24 hrs for B alone
:
"If pipes A and B are used together, it takes 10 hours." we know B, therefore
+ = 1
Multiply by 24A
24(10) + 10A = 24A
:
240 = 24A - 10A
240 = 14A
A =
A = 17.143 hrs for A alone
:
:

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1. Mr. Honda and Mr. Toyota arrange to meet on the highway connecting their hometown.
Mr Honda drives at 45 kph and Mr. Toyoya at 35 kph.
They leave their homes w/c are 120 kilometers apart at the same time.
In how many hours will they meet?
:
Let t = travel time when they meet
when they meet, the total distance they travel will equal 120 km
write a dist equation, dist = speed * time
:
H's dist + T's dist = 120 km
45t + 35t = 120
80t = 120
t =
t = 1.5 hrs to mee
:
;
2. There are two trains.
The rabbit heads North on the expressway at 4 kph.
Exactly 2 hours after, the Panther follows at a steady speed of 54 kph.
How long does it tAke the panther overtake the rabbit?
:
Let t = the panther's travel time
then
(t+2) = the rabbit's travel time
:
When p overtakes r, they will have traveled the same distance
Write a distance equation, dist = speed * time
;
P's dist = r's dist
54t = 4(t+2)
:
54t = 4t + 8
:
54t - 4t = 8
:
50t = 8
t =
t = .16 hrs which is: .16*60 = 9.6 min to catch the unfortunate rabbit
;
But what about the train? Did the rabbit jump on the train and escape?

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5x^2-4x=33 ?
Subtract 33 from both sides
5x^2 - 4x - 33 = 0
Factor this by using the factors of 33 and adding the correct signs to satisfy the middle term,
(5x + 11)(x - 3) = 0
x = 3
and
5x = -11
x =

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A rectangular swimming pool has a diagonal of 40 meters and makes a 30 degree angle with the length of the pool.
What is the area of the pool?
:
Find the width using the sine of 30
sin(30) = W/40
:
Find the length using the cos of 30
cos(30) = L/40

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how many 6 digit numbers we have in all?:
:
How about 999,999 - 99,999 = 900,000?

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My exact problem is:
2x^3-9x^2y-12^y+17xy^2 divided by 2x-3y
:
Rearrange the number divided, then the long division procedure will work
:
.................______________________________
(2x-3y)|2x^3 - 9x^2y + 17xy^2 - 12y^3
:
:
I got a quotient of: x^2 - 3xy - y^2, remainder -15y^3

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.While waiting for a train, alvin takes a bus ride at 60 kph to a certain point
and then take walks back leisurely at 10 kph. The bus takes 1/4 hour less than his walking.
How far does he walk and how long.
:
Let t = time he walked
then
(t-.25) = time rode; (changed 1/4 to .25)
:
Write a distance equation; dist = speed * time
:
riding dist = walking dist
60(t-.25) = 10t
60t - 15 = 10t
60t - 10t = 15
50t = 15
t =
t = .3 hrs, which is .3*60 = 18, min is time he walked
:
Find the dist
10 * .3 = 3 km he walked, (and rode)
:
:
Confirm our solutions by finding the dist he rode
(.3-.25) * 60 =
.05 * 60 = 3 km also

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The estimated masses of a proton and an electron are 1.67 x 10-24 grams and 9.11 x 10-28 grams, respectively. Find the ratio to three significant digits of the mass of the proton to the mass of the electron.
:
= = .183315(10^4) = 1833.15
:
Proton mass is about 1833 times an electron mass

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Find the x-intercept of the line that has an equation of –3x + 6y = 9.
:
The x intercept occurs when y=0, therefore:
-3x + 6(0) = 9
-3x = 9
divide both sides by -3
x =
x = -3

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I have a tanker that is sitting 450 ft from shore, is leaking oil that is
advancing in a circle at 2.2 ft per hour.
If it takes 204.55 hrs for the oil to reach the shoreline, how long will it take to contaminate 6 miles of shoreline?
:
As you probably noticed, 204.55 * 2.2 = 450 ft, which would be the radius of
spill when it just touched the shoreline after 204.55 hrs.
:
Visualize a larger circle that has a 6 mi chord which is 450 ft from the center.
(The source of the spill)
The center 450' distance to the 6 mi chord bisects the chord, so we can find the radius by solving a right triangle, with 1 leg 450' and the other leg 3 mi
:
Convert 3 mi to ft: 3*5280 = 15,840 ft
:
radius would be the hypotenuse, so we have
r =
r = 15846.4 ft
:
How long would it take the oil to travel 15,846.4 ft at 2.2 ft/hr?
= 7,203 hrs (about 300 days)
:
Did this make sense to you? Let me know please (in the comment section).

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Solve by completing the square
m^2 - m =
:
m^2 - m + ____ =
:
Calculate the 3rd term to complete the square (25/4)^2 = 625/16
m^2 - m + = +
:
m^2 - m + = +
:
m^2 - m + =
:
(m - )^2 =
Find the square root of both sides
m - = +/-
Extract the fraction from the radical
m - = +/-
Two solutions
m = +
or
m =
and
m =

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Yes it can, think of it as working backwards from solving a quadratic by factoring
:
x=-10, from a factor (x+10)
x=-1, from a factor (x+1)
:
FOIL (x+10)*(x+1) = x^2 + x + 10x + 10
:
y = x^2 + 11x + 10; is the equation with solutions of -10, -1

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=
this can be written under a single radical on the left
=
Now we have a single radical on each side, square both sides and we're rid of em
= 2
Multiply both sides by (7x-25), and you have
:
x^2 - 2 = 2(7x - 25)
:
x^2 - 2 = 14x - 50
arrange as a quadratic equation on the left
x^2 - 14x - 2 + 50 = 0
:
x^2 - 14x + 48 = 0
Easily factors to
(x-8)(x-6) = 0
Two solutions
x = 8
and
x = 6
:
Check both these solutions in the original problem using a calc
x=8
=
= 1.414
= 1.414; confirms this solution
:
You should check x=6, the same way

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you are going to make a box out of cardboard.
the box needs to measure 10cm by5cm by 5cm;
10% of the finale product will be wasted in the cutting process.
how much cardboard do you need
:
Find the surface area of the closed box
S.A. = 2(10*5) + 2(10*5) + 2(5*5)
S.A. = 100 + 100 + 50
S.A. = 250 sq/cm for the box
:
Let a = amt of cardboard required
then
.1a = wasted amt
:
a = .1a + 250
a - .1a = 250
.9a = 250
a =
a = 277.78 sq/cm required

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I gave one half of one third of one quarter of my money to Mathilda.
I have $23 left.
How much money did I give away?
:
Find out how much money you had before Mathilda came on the scene.
x = original amt
x - x = 23
x - x = 23
Multiply equation by 24
24x - x = 24(23)
23x = 24(23)
Divide both sides by 23
x = $24 original amt
Therefore, you gave Mathilda
24 - 23 = $1; you really got away cheap

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Using the definition of logarithm to solve logrithms equations,
solve this: 2ln(8x)=8.
Divide both sides by 2, results
ln(8x) = 4
Write it in exponent form of logs
e^4 = 8x
Find the value of e^4 on a calc:
54.598 = 8x
= x
x = 6.82475
:
Check solution on a calc: enter 2*ln(8*6.82475), results: 7.99999 ~ 8
:
;
16 grams of carbon-14 present.
The decay model A = 16*e^-0.000121t describes the amount of carbon-14 present after t years.
How many grams of carbon-14 will be present in 9477 years?
A = 16*e^(-0.000121*9477)
A = 16*e^-1.146717
Using a calculator, find e^-1.146717
A = 16*.317678
A = 5.084 grams after 9477 yrs