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In a physics experiment, a ball is dropped off a 144-ft platform. The height of the ball above the ground is given by the equation h=-16t^2+144
where h is the ball's height in feet, and t is the time in seconds after the ball is dropped (t is > or = 0)
Find the time required for the ball to hit the ground.(Let h =0)
:
-16t^2 + 144 = 0
simplify and change the signs, divide by -16
t^2 - 9 = 0
t^2 = 9
t =
t = 3 sec to hit the ground
:
:
Check this
-16(3^2) + 144 =
-16(9) + 144 =
-144 + 144 = 0

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two motorist set out at the same time to go from A to B, a distance of 100 miles.
they both followed the same route and traveled at different, though uniform, speeds of an integral number of miles per hour.
the difference in their speeds was a prime number of miles per hour, and after they had been driving for two hours, the distance of the slower car from A was five times that of the faster car from B.
how fast did the two motorist drive?
:
Let x = speed of 1
Let p = difference between the speeds of the two cars
then
(x - p) = speed of car 2
:
" after they had been driving for two hours, the distance of the slower car from A was five times that of the faster car from B
:
After two hrs
2(x-p) = car 2 dist from A
100 - 2x = car 1 dist from B
:
the equation
5(2(x-p)) = 100 - 2x
10x - 10p = 100 - 2x
-10p = 100 - 2x - 10x
10p = 12x - 100; multiplied by -1
p = x - 10
p = 1.2x - 10
the only prime number that will be generated by this equation is 2, (x=10)
therefore
Car 1 speed is 10 mph
Car 2 speed is 8 mph
;
:
We can check this
Car 1: 100 - 2(10) 80 mi from B
Car 2: 2(8) = 16 mi from A
:
80 mi = 5(16) mi
,

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n times p equals 185; n is a two digit number p is a one digit number.
what numbers do and p represent
:
185 is a multiple of 5, so 5 could be the single digit number (p)
= 37 is the two digit number (n)

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A sister while growing up was 5 times her brothers age, then she was 4 times, then 3 times, now she is 2 times his age………
:
Here's what I get, but having trouble getting a formula to find this:
:
Sis|bro|mult
---|---|----
15 | 3 * 5
16 | 4 * 4
18 | 6 * 3
24 | 12 * 2; their ages now
:
If I come up with a formula for this I will edit this solution

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Translate the problem into a pair of linear equations in two variables. Solve the equations using either elimination or substitution. State your answer for both variables.
1.The perimeter of a rectangle is six times the width. If the length were increased by 4 inches and the width were increased by 7 inches, then the perimeter would be 160 inches. What are the length and the width?
:
L=original length, W=original width
The perimeter of a rectangle is six times the width.
:
perimeter:
2L + 2W = 6W
2L = 6W - 2W
2L = 4W
Simplify, divide both sides by 2
L = 2W
:
If the length were increased by 4 inches and the width were increased by 7 inches, then the perimeter would be 160 inches.
2(L+4) + 2(W+7) = 160
2L + 8 + 2W + 14 = 160
2L + 2W + 22 = 160
2L + 2W = 160 - 22
2L + 2W = 138
Simplify, divide by 2
L + W = 69
:
What are the length and the width?
From the 1st statement we know L=2W; replace L with 2W
2W + W = 69
3W = 69
W =
W = 23 is the width
then
L = 2(23)
L = 46 is the width
:
:
Check solution in the 1st statement:
"The perimeter of a rectangle is six times the width."
2(46) + 2(23) = 6(23)
92 + 46 = 138;
:
YOu can check the solution using the 2nd statement
"If the length were increased by 4 inches and the width were increased by 7 inches, then the perimeter would be 160 inches. "
:
:
2. 30,000 people attended a ballgame at a stadium that offers two kinds of seats: general admission and reserved. The day's receipts were $201,000. How many people paid $13.00 for reserved seats, and how many paid $4.00 for general admission?
:
Let g = no. of general admission tickets
Let r = no. of reserved
:
Total attendance equation
g + r = 30000
or
g = (30000-r)
:
Total receipts equation
4g + 13r = 201000
:
Substitute (30000-r) for g in the above equation
4(30000-r) + 13r = 201000
120000 - 4r + 13r = 201000
9r = 201000 - 120000
9r = 81000
fr =
r = 9000 reserved tickets sold
and
30000 - 9000 = 21000 general admission tickets
:
Check solution using the receipt equation
4(21000) + 13(9000) =
84000 + 117000 = 201000

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Diophantus spent 1/6 of his life in childhood, 1/12 as a youth, and 1/7 more as a bachelor. 5 years after his marriage was born a son who died 4 years before his father at 1/2 the age at which his father died. How old was Diophantus when he died?
:
Let x = his final age
:
a lengthy but simple equation
Final age = child + youth + bach + marriage + son's age + 4
x = x + x + x + 5 + x + 4
:
Common denominator of 12*7 = 84
x = x + x + x + 5 + x + 4
:
x = x + 9
:
x - x = 9
:
x = 9
:
x = 84 is his final age
:
Check:
14 + 7 + 12 + 5 + 42 + 4 = 84

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Write equations for each phrase
;
"Two numbers differ by 8,
x - y = 8
or
x = y + 8
:
the sum of their squares is 320.
x^2 + y^2 = 320
:
Replace x with (y+8)
(y+8)^2 + y^2 = 320
:
FOIL
y^2 + 16y + 64 + y^2 = 320
:
A quadratic equation
2y^2 + 16y + 64 - 320 = 0
2y^2 + 16y - 256 = 0
:
Simplify, divide by 2
y^2 + 8y - 128 = 0
Factor
(y+16)(y-8) = 0
:
Two solutions
y = 8, then x = 16
Or
y = -16 then x = -8

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A lidless box is made from a square piece of metal by cutting 3 cm squares from the corners and bending up the sides.
If the volume of the box is 60 cm^3 find the dimensions of the square piece of metal
:
If 3 cm squares are cut, the height of the box is 3 cm
:
Find the area:
= 20 sq/cm is the area of the bottom of the box
:
Since it is square:
= 4.472 cm is the dimension of the side of the square bottom
:
Cutting 3 cm squares subtracts 6 cm from the original square piece of metal dimensions
Therefore
4.472 + 6 = 10.472 cm is the side of the square piece of metal

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The perimeter of a rectangle is six times the width.
:
perimeter:
2L + 2W = 6W
2L = 6W - 2W
2L = 4W
Simplify, divide both sides by 2
L = 2W
:
If the length were increased by 4 inches and the width were increased by 7 inches, then the perimeter would be 160 inches.
2(L+4) + 2(W+7) = 160
2L + 8 + 2W + 14 = 160
2L + 2W + 22 = 160
2L + 2W = 160 - 22
2L + 2W = 138
Simplify, divide by 2
L + W = 69
:
What are the length and the width?
From the 1st statement we know L=2W; replace L with 2W
2W + W = 69
3W = 69
W =
W = 23 is the width
then
L = 2(23)
L = 46 is the width
:
:
Check solution in the 1st statement:
"The perimeter of a rectangle is six times the width."
2(46) + 2(23) = 6(23)
92 + 46 = 138;
:
YOu can check the solution using the 2nd statement
"If the length were increased by 4 inches and the width were increased by 7 inches, then the perimeter would be 160 inches. "

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A tank can be filled by one pipe in four hours and by a second pipe in six hours;and when it is full,the tank can be drained by a third pipe in three hours.
If the tank is empty and all three pipes are open,in how many hours will the tank be filled?
:
Just like a "shared work problem, except the drain fraction is subtracted
:
Let t = time to fill with all pipes open
:
Let the completed job (a full pool) = 1
:
+ - = 1
Multiply by 12 to clear the fractions
3t + 2t - 4t = 12
t = 12 hrs to fill the pool
:
:
Check solution on a calc
12/4 + 12/6 - 12/3 =
3 + 2 - 4 = 1

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Let s = no. of 6 volt batteries
Let t = no. of 12 volt batteries
:
Write an equation for each statement:
:
"A group of ten 6- and 12- volt batteries are wired in a series"
The no. of batteries equation
s + t = 10
Arrange for substitution
s = (10-t)
:
" The sum of their voltages is 84 volts."
the total volts equation
6s + 12t = 84
:
Substitute (10-t) for s in the above equation
6(10-t) + 12t = 84
60 - 6t + 12t = 84
-6t + 12t = 84 - 60
6t = 24
t =
t = 4 ea 12v batteries
:
then
10 - 4 = 6 ea 6v batteries
;
:
Check solution in the total volts equation
6(6) + 4(12) = 84

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"It took a faster runner 10 seconds longer to run a distance of 1500 feet than it took a slower runner to run a distance of 1000 feet.
Let t = time in sec for the faster runner to run 1500 ft
then
(t-10) = time of the slower runner to run 1000 ft
:
if the rate of the faster runner was 5 ft/s more than the rate of the slower runner, what was the rate of each?"
:
Write a rate equation: Rate = dist/time:
:
Fast rate - slow rate = 5 ft/sec
- = 5
:
Multiply equation by t(t-10), results
1500(t-10) - 1000t = 5t(t-10)
:
1500t - 15000 - 1000t = 5t^2 - 50t
:
500t - 15000 = 5t^2 - 50t
:
0 = 5t^2 - 50t - 500t + 15000
A quadratic equation
5t^2 - 550t + 15000 = 0
Simplify, divide by 5
t^2 - 110t + 3000 = 0
Factor to
(t - 50)(t - 60) = 0
two solutions
t = 50 sec
t = 60 sec
:
Assume t = 50
1500/50 = 30 ft/sec for the fast guy
and
1000/40 = 25 ft/sec for the slow guy (5 ft/sec slower)
:
You can check the other solution

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Write a quadratic function given the following:
(-1,6), (1,4), (2,9)
:
Using the form; ax^2 + bx + c = y; we will write three equations:
eq1: x=-1; y=6, substitute for x and y
-1^2a + -1b + c = 6
1a - 1b + c = 6
or just
a - b + c = 6
:
eq2: x=1; y=4
1^2a + 1b + c = 4
a + b + c = 4
:
eq3: x=2, y=9
2^2a + 2b + c = 9
4a + 2b + c = 9
:
We will solve these equations by elimination
Subtract eq 1 from eq 2, eliminate a and c
a + b + c = 4
a - b + c = 6
-----------------subtraction eliminate a and c
0 +2b + 0 = -2
b = -1
:
Substitute -1 for b in the 1st equation
a - (-1) + c = 6
a + 1 = c = 6
a + c = 6 - 1
a + c = 5
:
do the same in the 3rd equation
4a + 2(-1) + c = 9
4a - 2 + c = 9
4a + c = 9 + 2
4a + c = 11
:
Using these two equations:
4a + c = 11
a + c = 5
-------------Subtraction eliminates c, find a
3a = 6
a = 2
:
Find c using a + c = 5
2 + c = 5
c = 5 - 2
c = 3
:
Our equation: y = 2x^2 - x + 3
:
You can check our solution, substitute for x and ensure that it equals y for each given coordinate
:
Did you understand what we did here? Any questions?

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"all my digits are odd. the digit in the tens place is twice the digit in the ones place,"
an inconsistency here, any digit that is twice another digit, is even.

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5 men can build a house in 13 days.
If 2 of the men fell sick after the 4th day, how many days would it have taken to complete the house.
:
From the information given, we know that it takes; 5*13 = 65 man-days to complete the house
:
Let d = no. of additional days if 2 fell sick after the 4th day
:
5*4 + 3d = 65
20 + 3d = 65
3d = 65 - 20
3d = 45
d = 15 additional days
hence
4 + 15 = 19 days to complete the house

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Assume the problems are (x is not part of the denominator)
= 36
Multiply both sides by 3 to get rid of the denominator
2x = 3(36)
2x = 108
x =
x = 54
:
Check solution by substituting 54 for x in the original equation
= 36
3 cancels into 54,
2(18) = 36
;
:
=
Multiplying by 18 will clear the denominators here
2(5)x = 7
10x = 7
x =
:
:
Check solution
=
Cancel 5 into 10
=

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Two men together can do a piece of work in one hour and twenty minutes.
The first man can do the work alone in three hours. How long would it take the second man alone to do the work?
:
Because of the fractional hr do the problem in minutes
Working together 80 min
1st man alone 180 min
Let t = 2nd man alone
Let the completed job = 1
:
+ = 1
Multiply by 180t
80t + 180(80) = 180t
80t + 14400 = 180t
14400 = 180t - 80t
14400 = 100t
t = 144 min, 2hr 24 min

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you spend 3/4 of your money on clothes and have 21 dollars left how much did you have before you bought the clothes
:
Let x = original amt
:
A simple equation
x = .75x + 21
x - .75x = 21
.25x = 21
x =
x = $84
:
:
you can check this in the original equation

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A person runs 3 miles and walks 2 miles.
He runs twice as fast as he walks and finishes in exactly one hour.
How fast does he walk.
:
let r = walking speed
then
2r = running speed
:
Write a time equation time = dist/rate
Walk time + run time = 1 hr
+ = 1
Multiply by 2r
2(2) + 3 = 2r
4 + 3 = 2r
r =
r = 3.5 mph walking rate
:
:
Check solution
2/3.5 + 3/7 = 1

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Hannah has a jar filled with 150 coins made up of quarters and nickels. The total value of the coins is $28.30
If x represents the number of quarters in the jar, which equation can be used to find the number of quarters?
:
(n = no. of nickels)
x + n = 150
n = (150 - x)
and
.25x + .05n = 28.30
.05n = (28.30 - .25x)
Multiply eq by 20
n = (566 - 5x)
:
find x using this equation
150 - x = 566 - 5x
-x + 5x = 566 - 150
4x = 416
x = 104 quarters

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Two cars leave a city at noon. One car travels north and the other travels east.
Suppose the northbound car is traveling at 60 mph, and the eastbound is traveling at 50 mph.
:
This is a pythag problem, distance between the cars is the hypotenuse: a^2 + b^2 = c^2
n = no. of hrs traveled; a = 50n; b=60n
:
Make a table to show the distance between the 2 cars at 1 hour, 2 hours, 3 hours and "n" hours.
1 hr: = 78.1 mi
2 hr: = 156.2 mi
3 hr: = 234,3 mi
n hrs = mi
:
Then, suppose that the northbound car is traveling at 40 mph and after 2 hours, the two cars are 100 miles apart.
How fast is the other car going?
Let a = 40*2, c=100, find b
80^2 + (2b)^2 = 100^2
6400 + 4b^2 = 100^2
4b^2 = 10000 - 6400
4b^2 = 3600
b^2 =
b^2 = 900
b =
b = 30 mph is the 2nd car
;
:
Check on a calc: enter = 100 mi

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How many gallons of milk that is 3 1/2% butterfat must be added to 80 gal of 1% butterfat milk to make milk that is 2% butterfat?
:
Let x = amt of 3.5% milk required
:
The the decimal equiv equation of the amt of butterfat
.035x + .01(80) = .02(x+80)
:
.035x + .8 = .02x + 1.6
:
.035x - .02x = 1.6 - .8
:
.015x = .8
x =
:
x = 53 gal of 3.5%

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find the measure of an angle such that the difference between the measures of its supplement and three times its complement is 10 degrees
:
Let = the angle
then
(180-x) = it's supplement
and
(90-x) = it's complement
:
The equation for:
"the difference between the measures of its supplement and three times its complement is 10 degrees"
(180-x) - 3(90-x) = 10
:
180 - x - 270 + 3x = 10
:
Combine like terms:
-x + 3x + 180 - 270 = 10
:
2x - 90 = 10
:
2x = 10 + 90
:
2x = 100
x =
x = 50 degrees
:
:
Is this true?
(180-50) - 3(90-50) = 10
130 - 3(40) = 10; confirms our solution

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A red ball and a green ball are tossed into the air. The red ball is given an
initial velocity of 96 feet per second and its height "t" seconds after it is
tossed is -16t^2 + 96t feet.
The green ball is given an initial velocity of 80 feet per second and its height
"t" seconds after being tossed is -16t^2 +80t feet.
:
5.1. Write the algebraic expression, in simplest form, that represents the difference in heights between the two balls at any time, t.
D(t) = (-16t^2 + 96t) - (-16t^2 +80t)
:
Remove brackets
D(t) = -16t^2 + 96t + 16t^2 - 80t
:
combine like terms:
D(t) = 16t
:
5.2. How much higher is the red ball after 2 sec?
D(t) = 16(2)
D(t) = 32 ft difference after 2 seconds
:
Looks like this:

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I have to find all the real number solutions of:
x^3 - 216 = 0
:
This can be factored as the "difference of cubes" (-216 = -6^3)
(x - 6)(x^2 + 6x + 36)
the real solution: x = 6

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Three people run a mile. The first person runs the mile in seven minutes.
The second person takes 5/4 as long as the first person.
How many minutes did it take the third person to run the mile if the average for all three people is eight minutes?
:
Let t = time of the 3rd person
:
Change 5/4 to 1.25
:
= 8
:
= 8
:
= 8
:
Multiply both sides by 3
15.75 + t = 24
:
t = 24 - 15.75
:
t = 8.25 min is the 3rd person's time
:
:
Check solution
= 8

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A hollow spherical vessel has the internal and external radii of 6cm and 6.4 respectively.
Calculate the weight of the vessel if it is made of metal of density 10g/cm^3.
:
weight = overall volume - inside volume times 10 gr/cm^3
w = - * 10

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In 1980 231 million was the population and its growing at a rate of 1.03% per year.
Change 1.03% to a decimal: .0103, for the finding the total, it will be 1.0103
:
P = 231(1.0103^t); where t is the number of years from 1980, p in millions
:
Predict the pop. in the year 2025.
P = 231(1.0103^45)
enter 231(1.0103^45) in to your calc, your answer will be in millions (336.3)

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A piece of aluminum strip of length l m is used to make a window frame.
The window frame consists of six identical small rectangles and each rectangle is x m long and y m long wide.Let the total area of the window encompassed by the aluminum stripes be A m^2.(Ignore the width of the aluminum strip in this question.)
:
Assume they mean L meters, (not 1 meter)
a)Find the maximum value of A in terms of l.
Max area is an overall shape of a square
7 equal strips of aluminum required to have 6 window panes
A =
:
b)Find the value of x/y so that the area of the window is a maximum allowing a maximum amount of light entering the room.
:
The shape of the whole window is a square, however, the six panes are rectangular, x by y:
Ratio: =
(The square will be 3 windows long by 2 windows wide)

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A cylindrical end of a pencil is sharpened to produce a perfect cone at the end with no overall loss of length.
If the diameter of the pencil is 1cm, and the cone is of length 2cm, calculate the volume of the shavings.
:
radius of the pencil = .5cm
:
the vol of cylinder is
and
the vol of a cone is:
therefore
the vol of the shavings:
v =
v =
v =
v = 1.047 cu/cm

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In a chemistry course, you learn that a substance’s density is found by dividing its mass by its volume.
If a 300 g block of wood measures 70 mm by 50 mm by 30 mm, find the density of the wood in g/cm^3
:
They want the answer in cu/cm, convert the measurements to cm
= 2.857 ~ 2.86 gr/cm^3