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Jim can fill a pool carrying buckets of water in 30 minutes. Sue
can do the same job in 45 minutes. Tony can do the same job in
1 1/2 hours. How quickly can all three fill the pool together?
:
Let t = time required if they work together
Let 1 = the full pool
Convert 1.5 hrs to minutes: 90 min
A simple equation:

Multiply equation by 90, this gets rid of the denominators;
You then have
3t + 2t + t = 90
6t = 90
t = 90/6
t = 15 minutes working together:
:
Check solution:
15/30 + 15/45 + 15/90 =
1/2 + 1/3 + 1/6 =
3/6 + 2/6 + 1/6 = 1

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Martina leaves home at 9 am bicycling at a rate of 24 miles per hour, 2 hours later John leaves driving at the rate of 48 miles per hour. At what time will john catch up with Martina?
:
Let t = M's travel time
Then
(t-2) = J's travel time
:
One thing to remember on all these "catch-up" problems is that they will have traveled the same distance when that occurs. Write a distance equation from this fact.
:
Distance = speed * time
:
J's dist = M's dist
48(t-2) = 24t
48t - 96 = 24t
48t - 24t = +96
24t = 96
t = 96/24
t = 4 hrs is M's travel time
:
Find the time: 9:00 AM = 4:00 = 1 PM is when J catches up with M
:
Confirm our solution by finding the distances:
24*4 = 96 mi
48*(4-2)) = 96 mi
;
Did this make sense? Any questions?

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36x^2 = 1
:
36x^2 - 1 = 0; recognize this as the difference of squares,
:
which factor to:
(6x+1)(6x-1) = 0
6x = -1
x =
and
6x = +1`
x =

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A bus leaves a station at 1 pm traveling west at an average rate of 44 miles per hour. One hour later a second bus leaves the same station, traveling east at an rate of 48 miles per hour. At what time will the two buses be 274 miles apart?
:
Let t = travel time of the westbound bus when they are 274 mi apart
Then
(t-1) = travel time of the eastbound bus when they are 274 mi apart
:
Write a distance equation: Distance = speed * time
:
West bus dist + East bus dist = 274 mi
44t + 48(t-1) = 274
44t + 48t - 48 = 274
92t = 274 + 48
92t = 322
t = 322/92
t = 3.5 hrs is the travel time of the west bound bus
Then we know:
2.5 hrs is the travel time of the east bound bus
:
Find the time:
1:00 + 3:30 = 4:30 PM, time they will be 274 mi apart
:
Check solution by finding the sum of the distances:
44(3.5) + 48(2.5) =
154 + 120 = 274 mi as given
:
:
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()
:
Multiply both terms inside the brackets by

Same as:

:
Extract the sqrt of 9

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Simplify:
√[49x^12y^4z^8]
Factor it into perfect squares:
=

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A small firm produces both AM and AM/FM car radios. The AM radios take 15 h to produce, and the AM/FM radios take 20 h. The number of production hours is limited to 300 h per week. The plant's capacity is limited to a total of 18 radios per week, and existing orders require that at least 4 AM radios and at least 3 AM/FM radios be produced per week. Write a system of inequalities representing this situation. Then, draw a graph of the feasible region given these conditions, in which X is the number of AM radios and Y is the number of AM/FM radios.
:
Let x = number of AM radios; let y = number of AM/FM radios

The production hour constraint:
15x + 20y =< 300
Arrange in general form to plot the graph
20y =< 300 - 15x
y =< 300/20 - (15/20)x
y =< 15 - .75x, use this to plot the production hr graph (Purple line)
:
Plant's capacity constraint:
x + y =< 18
y =< 18 - x, plot this graph also (Green line)
:
Min AM radio production constraint:
x => 4
This will be a vertical line going thru the x axis at +4, (I can't draw this in)
:
Min AM/FM radio production constraint
y => 3
This will be a horizontal line going thru the y axis at +3, (Dark blue line)
:
Here is the graph as presented by these equations except for x => 4,
:

:
The feasibility region:
1. At or below the purple or green line, whichever is lowest
2. At or above the horizontal line
3. At or to the right of the vertical line which you have to draw in at x = +4
:
How about this, did it all make some sense to you?

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Two Russian mathematicians meet on a plane.
“If I remember correctly, you have three sons,” says Ivan. “What are their ages today?”
“The product of their ages is thirty-six,” says Igor, “and the sum of their ages is exactly today’s date.”
“I’m sorry, Igor,” Ivan says after a minute, “but that doesn’t tell me the ages of your boys.”
“Oh, I forgot to tell you, my youngest son has red hair.”
“Ah, now it’s clear,” Ivan says. “I now know exactly how old your three sons are.”
How did Ivan figure out the ages?
:
I am going guess that Ivan knew a couple things that they didn't tell us:
1. The day of the month is the 13th
2. Two of the 3 brothers are twins
:
That could make the boys ages (which add up to 13):
6, 6, 1
2, 2, 9
:
When he said the youngest had red hair. that made the older ones the twins and confined the answer to:
6, 6, 1
:
This is just a guess, there may be other solutions.

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Here's an easy method to these type of problems:
:
Write an equation for each statement:
:
A father is four times as old as his daughter is now.
F = 4D
:
"In 20 years," their age will be:
F+20
and
D+20
:
In 20 years, he will be only twice as old as his daughter.
F + 20 = 2(D+20)
F + 20 = 2D + 40
F = 2D + 40 - 20
F = 2D + 20
:
How old are the father and daughter now.
From the 1st statement: substitute 4D for F in the above equation:
4D = 2D + 20
4D - 2D = 20
2D = 20
D = 20/2
D = 10 yrs old is the daughter now.
Then
4(10) = 40 yrs old is the father now
:
:
Check our solution using the statement:
"In 20 years, he will be only twice as old as his daughter."
(40 + 20) = 2(10 + 20); confirms out solution
:
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Two designs for and oil storage tank are being considered: spherical and cylindrical. The two tanks would have the same capacity and would each have an inside diameter of 60 feet.
:
a)what would be the height of the cylindrical tank?
r = 30 ft
Cylinder Vol = Sphere Vol
=
=
Multiply both sides by 3:
=
Divide both sides by and you have:
3h = 4*30
h = 120/3
h = 40 ft is the height of the cylinder
:
You can check this by finding the volume of both, see if they equal 113,097.3 cu/ft
:
b) If one cubic foot holds 7.5 gallons of oil, what is the capacity of each tank in gallons?
113097.3 * 7.5 = 848,230 gal
:
c)Which of the two designs has the smallest surface area and would thus require less material in its construction?
Sphere would have the smallest surface area
Sphere:
= 11,309.7 sq ft
Cylinder:
+ = 13,194.7 sq ft
:
Did this make sense to you?

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An alloy of metals is 25% copper another alloy is 50% copper. How much of each alloy is used to make 1000 grams of an alloy with 45% copper.
:
Let x = amt of 25% copper required
Then
(1000-x) = amt of 50% copper required:
:
A simple equation:
.25x + .50(1000-x) = .45(1000)

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Let x = the 1st number
:
Two numbers have sum of 32.
If the 1st number = x, then the 2nd number = (32-x)
:
"the second is five less than four times the first,"
(32 - x) = 4x - 5
32 + 5 = 4x + x
37 = 5x
:
what are the two numbers?
x = 37/5
x = 7.4 is the 1st number
:
Find the 2nd number:
32 - 7.4 = 24.6 is the 2nd number
:
:
Check solution in the statement:
"the second is five less than four times the first,"
24.6 = 4(7.4) - 5
24.6 = 29.6 - 5; confirms our solution

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Product x: Each item sold produces $100 profit
Labor cost: is 3 hours for assembly, 1 hour in detail work, and 2 hours for QA & packing product.
:
Product y: Each item sold produces $150 profit
Labor cost: is 5 hours for assembly, 3 hour in detail work, and 2 hours for QA & packing product.
:
Company has these hours available for production for each accounting period:
3900 hours for assembly, 2100 hours for detail work, & 2200 hours for QA & packing.
:
Assembly:
3x + 5y =< 3900
5y =< -3x + 3900; arrange in the "y=" form for graphing
y =< -.6x + 780; divided equation by 5
:
Detail
1x + 3y =< 2100
3y =< -x + 2100
y =< .33x + 700; divided equation by 3
:
QA & Packing
2x + 2y =< 2200
2y =< -2x + 2200
y =< -x + 1100; divided equation by 2
:
Graph this and you should have:

Note that the area of feasibility is at or below the the lowest graphing line
:
Questions are:
1) How many products of both x & y should be produced to make a max & min profit?
:
Find the coordinates of the corners of the area of feasibility, use these to
make an objective profit function for each corner.
The x/y values of the corners will be:
0, 700
300, 600
800, 300
1100, 0
:
0,700:
100x + 150y =
100(0) + 150(700) = $105,000 profit
:
300,600
100(300) + 150(600) =
30000 + 90000 = $120,000 profit
:
800,300
100(800) + 150(300) =
80000 + 45000 = $125,000 profit
:
1100, 0
100(1100) + 150(0) = $110,000 profit
:
:
2) Now what if product y profit goes up to $170 each. How many products are needed to make a max & min profit?
:
I am pressed for time, here. Substitute 170 for 150 and do the same as
above. If you have any questions email me, I will respond tonight.
I know this is not an easy question, but do you think you could help with this question please? Thank you kindly.

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x^3-9x+x^2-9 I am supposed to factor this completely:
:
How about this:
:
Rearrange:
x^3 + x^2 - 9x - 9
:
Group factoring
(x^3 + x^2) - (9x + 9);sign changes on the 9
:
x^2(x + 1) - 9(x + 1)
:
Factor out (x+1) and you have:
(x + 1) (x^2 - 9)
:
x^2 - 9 is the difference of squares
:
(x + 1) (x - 3) (x + 3)
:
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I think you're right, except if the whole thing is enclosed in brackets, and has
)^5 exponent, it would be x^10, wouldn't it?

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Mike and Matthew are each responsible for mowing half of their 60 foot by 100 foot rectangular back yard. MIke starts mowing at an outside corner, gradually working his way toward the middle by mowing around the outside edges. If the mower cuts a 3-foot-wide path, during which trip around the yard should Mike stop and Matthew start mowing?
:
Find the total area: 60 * 100 = 6000 sq/ft
:
Half the area = 3000 sq/ft
:
Let x = no. of trips by a 3' mower to do 3000 sq/ft (half the lawn
:
Area of the unmowed area remaining (each round trip cuts 6' off the edge:
(60-6x) * (100 - 6x) = 3000
:
6000 - 960x + 36x^2 = 3000
Arrange as a quadratic equation:
36x^2 - 960x + 6000 - 3000 = 0
:
36x^2 - 960x + 3000 = 0
:
Simplify, divide equation by 12:
3x^2 - 80x + 250 = 0
:
Using the quadratic formula, the lower value solution:
x = 3.6 trips
:
Try to verify this:
Remember that each round he would chop 6' off the lawn dimensions
:
if he mowed exactly 3 times 3(6) = 18 ft
The remaining lawn would be:
(60 - 18) * (100 - 18) = 3444 sq/ft remaining, not quite a half done
:
If he mowed exactly 4 time; 4(6) = 24 ft
(60 - 24) * (100 - 24) = 2736 sq/ft remaining, over half done
:
Between the 3rd and 4th round would be the time for Matthew to take over.
:
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(3-x+8x^3-10x^2)/(1+2x)
:
Arrange these expression into the general descending form:
(8x^3 - 10x^2 - x + 3)/(2x + 1)
:
Use grade school long division (I'm using .........to keep things lined up):
Choose a quotient term that will equal the 1st term, when multiplied by the divisor
:
............................4x^2 - 7x + 3
.............----------------------
(2x+1) |8x^3 - 10x^2 - x + 3
...............8x^3 + 4x^2
.............--------------
............................14x^2 - x
............................14x^2 -7x
............................----------
.........................................6x + 3
.........................................6x + 3
.........................................------
.................................................0
:
You can check our solution by ordinary multiplication:
4x^2 - 7x + 3
Times 2x + 1
----------------
:
:
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The sum of two numbers is 82. The larger number is 17 less than twice the smaller number. What are the two numbers?
:
Just write what it says:
:
"The sum of two numbers is 82."
Let larger number = x;
then
(82-x) = the smaller number
:
"The larger number is 17 less than twice the smaller number"
x = 2(82-x) - 17
x = 164 - 2x - 17
x + 2x = 164 - 17
3x = 147
x = 147/3
x = 49
Then
82 - 49 = 33 = the smaller number
:
:
Check our solution using the statement:
"The larger number is 17 less than twice the smaller number."
49 = 2(33) - 17
49 = 66 - 17; confirms our solution

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For a recent job, a plumber earned $28 per hour, adn the plumber's apprentice earned $15 per hour. The plumber worked 3 hours more htan the apprentice. If together they were paid $213, how much did each earn?
:
All I know is that I need to find an expression first for the number of hours each worked on the job, which would be something like (28x+3)+ 15x = 213 ???
:
you're almost right (bracket), just solve for x to find the apprentice hrs
28(x+3) + 15x = 213
28x + 84 + 15x = 213
28x + 15x = 213 - 84
43x = 129
x = 129/43
x = 3 hrs
:
Find the wages:
Plumber: 6 * 28 = 168
Apprent: 3 * 15 = 45
-----------------------
our given total = 213,

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a) the length of rectangle is 1 cm less than 3 times the width, if the perimeter is 54 cm, find the dimensions of a rectangle.
:
Just write an equation for what it says:
"the length of rectangle is 1 cm less than 3 times the width,"
L = 3W - 1
:
"the perimeter is 54 cm,"
2L + 2W = 54
Substitute (3W-1) for L (from the 1st statement)
2(3W-1) + 2W = 54
Find W
6W - 2 + 2W = 54
8W = 54 + 2
8W = 56
W = 56/8
W = 7
I'll let you find L using the 1st statement equation
:
:
b) 400 tickets were sold for a school play, General admission tickets were $4 while student tickets were $3, if the total tickets sales were $1350, how many of each type of ticket were sold?
:
Let x = no. of general admission tickets; let y = no. of student tickets
:
Write an equation for each statement:
"400 tickets were sold for a school play,"
x + y = 400
y = 400 - x; we can use this for substitution
:
"General admission tickets were $4 while student tickets were $3, the total tickets sales were $1350,"
4x + 3y = $1350
Substitute (400-x) for y in the above equation, find x:
4x + 3(400-x) = 1350
4x + 1200 - 3x = 1350
4x - 3x = 1350 - 1200
x = 150 general admission tickets
Here again I will let you find y from the 1st statement equation
Check your solution in the $$ equation
:
:
c) How much pure alcohol must be added to 9 quarts of water to obtain a mixture of 40% alcohol?
:
Let x = amt of alcohol required
A simple equation will do this:
x = .40(x+9)
x = .4x + 3.6
x - .4x = 3.6
.6x = 3.6
x = 3.6/.6
x = 6 quarts of alcohol required
:
:
d) On Friday morning, Ali drove from his house to the beach in 4 hrs, in coming back on Sunday afternoon, heavy traffics slowed his speed by 10mi/hrs, and the trip took 5 hrs, what was his average speed in each direction?
:
Let s = his speed to the beach
then
(s-10) = his speed back
:
The distance there and back are the same, so write a distance equation
Distance = time * speed
:
return dist = to beach dist
5(s-10) = 4s
5s - 50 = 4s
5s - 4s = +50
s = 50 mph to the beach
:
I'll let you provide the speed on the return trip
:
Did this help you get a handle on this stuff? Hope so.

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Some students planned for a get-together. The budget for food was $500. Five of the students failed to come because of the distance and therefore the cost of food for each member increased by $5. How many students attended the get-together?
500/x + 5 = 500/(x-5)
:
I would do it this way:
Let x = actual number that attended:
Then
(x+5) = number that was anticipated to come
:
Anticipated cost + $5 = Actual cost
+ 5 =
:
Multiply equation by x(x+5)
500x + 5(x(x+5)) = 500(x+5)
500x + 5x^2 + 25x = 500x + 2500
:
Arrange as a quadratic on the left:
5x^2 + 500x - 500x + 25x - 2500 = 0
5x^2 + 25x - 2500 = 0
:
Simplify, divide by 5
x^2 + 5x - 500 = 0
:
Factors to:
(x+25)(x-20) = 0
:
x = +20 is the solution we want here
:
20 actually attended
:
Actual cost 500/20 = $25 each
:
If 25 attended:
500/25 = $20, $5 less

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A farmer plans to fence a rectangular grazing area along a river with 300 yards of fence. Write an expression for the area A of grazing land in terms of the width w of the rectangle. Also, what is the largest area he can enclose?
:
Since one side is the river, the rectangle's fence perimeter will be:
L + 2W = 300
L = 300 - 2W
:
Area = Length * Width
Substitute (300-2W) for L:
A = W(300 - 2W)
A = -2W^2 + 300W; this would be the expression
:
This is a quadratic equation, Find the axis of symmetry, x = -b/(2a)
In our equation it would be:
W = -300/(2*-2)
W = -300/-4
W = +75 is the width for max area:
:
Find the max area, substitute 75 for W
A = -2(75^2) + 300(75)
A = -2(5625) + 22500
A = -11250 + 22500
A = 11250 sq ft is the max area
:
If you plot this, y = area and x = width

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Carl Lewis, one of the great Olympian sprinters of this century can run 100 meters in 10 seconds. How fast is this in kilometers per hour?
:
Here's one way:
Find his speed in meters per second:
100/10 = 10 m/s
:
How many meters would he travel in one hour at that speed (3600 sec in 1 hr)
10 * 3600 = 36000 meters in one hr
:
Change that to km/hr
= 36 km/hr

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The edges of three cubes are consecutive odd integers. If the cubes are stacked as shown the total exposed surface area is 381 cm2. Find the lengths of the sides of the cubes.
:
The areas of the 3 cubes would be:
ax^2 + b(x+2)^2 + c(x+4)^2 = 381
where and a,b,c are the number of sides of that cube that are exposed
:
An example,
x cube had two sides exposed
(x+2) had two sides exposed
(x+4) had one side exposed
:
2x^2 + 2(x + 2)^2 + 1(x + 4)^2 = 381
:
2x^2 + 2(x^2 + 4x + 4) + (x^2 + 8x + 16) = 381
:
2x^2 + 2x^2 + 8x + 8 + x^2 + 8x + 16 = 381
:
2x^2 + 2x^2 + x^2 + 8x + 8x + 8 + 16 = 381
:
5x^2 + 16x + 24 - 381 = 0
:
5x^2 + 16x - 357 = 0
Factors to:
(5x + 51)(x - 7) = 0
Only the positive solution wanted here:
x = +7
:
Nos. 7, 9, 11 would satisfy this situation:
2(7^2) + 2(9^2) + 11^2 = 381
:
Your problem may be different, but this method should help

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An escalator in a department store moves down at the rate of sixty-four steps per minute. Wendy can walk up ordinary steps at the rate of ninety-six steps per minute. In the store, forty-eight steps go from one level to another. Wendy walks up the down escalator. How long will it take her to go up one level?
:
48 steps is the distance from one floor to the other.
:
Wendy's speed walking up: 96 steps/min
:
Escalator speed going down: 64 steps/min
:
Let t = time required to travel the 48 steps to the next floor
:
Dist = speed * time
:
Wendy's dist - escalator dist = 48 steps
96t - 64t = 48
32t = 48
t = 48/32
t = 1.5 min
:

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Red candle: 45 cm tall and burns 3cm per hour
White candle: 30 cm tall and burns 1 1/2 cm per hour.
:
How tall will each candle be after 1 hour?
Red: 45 - (1*3) = 42 cm
White: 30 - (1*1.5) = 28.5 cm
:
How tall will each candle be after 3 hours?
Red: 45 - (3*3) = 36 cm
White: 30 - (3*1.5) = 25.5 cm
:
How tall will each candle be after 12 hours?
Red: 45 - (12*3) = 9 cm
White: 30 - (12*1.5) = 12 cm
:
Can you generalize-a formula- a system for calculating the candle's height at any given time?
Let t = time in hrs and h = height in cm
Red: h = 45 - 3t
White: h = 30 - 1.5t
:
Which candle will last longer? How do you know?
The white one. The red burns twice as fast, but it is less than twice as tall
:
If you burn both candles continuously (same start time) will they ever be the same height at the same time?
:
Use the equations to see how long (t), they will be the same height
30 - 1.5t = 45 - 3t
+3t - 1.5t = 45 - 30
1.5t = 15
t = 15/1.5
t = 10 hrs they will be the same
:
Check the height when they are the same:
45 - (3*10) = 15"
30 - (1.5*10) = 15"

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h = -4.9t^2 + 10t + 1

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An object is shot upward from a height of 80 feet with an initial velocity of 64 feet per second. Its height (h) above the ground in feet after t seconds is given by the formula
h = - 16t2 + 64t + 80. At what times will the object be 128 feet above the ground?
:
use the formula and set the height (h) to 128 ft
-16t^2 + 64t + 80 = 128
:
-16t^2 + 64t + 80 - 128 = 0; subtract 128 from both sides
:
-16t^ + 64t - 48 = 0
:
Divide equation by -16, simplifies and changes the signs, easier to factor then
+t^2 - 4t + 3 = 0
Factors to:
(t - 3) (t - 1) = 0
Two solutions:
t = +1 sec, at 128 ft on the way up
and
t = +3 sec, at 128 ft on the way down

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On a river a boat travels 240 miles downstream in the same amount of time that it takes to travel 160 miles upstream. The speed of the boat without current is 25 mph. Find the speed of the river.
:
Let x = speed of the river
Then
(25-x) = speed upstream
and
(25+x) = speed downstream
:
It tells us the times are equal, make a simple time equation from this fact:
Time = distance/speed
:
Time up = time down
=
:
Cross multiply and you have:
160(25+x) = 240(25-x)
:
4000 + 160x = 6000 - 240x
:
160x + 240x = 6000 - 4000
:
400x = 2000
:
x = 2000/400
:
x = 5 mph is the current:
:
:
Check solution by seeing if, indeed, the times are the same:
Upstream: 160/20 = 8 hrs
Downstream: 240/30 = 8 hrs, confirms our solution
:
Make sense to you? Any questions?

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A river has a steady current. A boat travels 30 mph over still water. If a trip 85 miles downstream and back takes 10 hours, then what is the speed of the current?
:
Let x = speed of the current
then
(30+x) speed downstream
and
(30-x) speed up-stream
:
Write a time equation: Time = Distance/speed
:
Time upstream + Time downstream = 10 hrs
+ = 10
:
Multiply equation by (30-x)(30+x) to get rid of the denominators:
(30-x)(30+x)* + (30-x)(30+x)* = 10(30-x)(30+x)
:
Cancel out the denominators, FOIL (30-x)(30+x) and you have;
85(30-x) + 85(30+x) = 10(900-x^2)
:
2550 - 85x + 2550 + 85x = -10x^2 + 9000
:
The 85x's cancel leaving us with:
5100 = -10x^2 + 9000
Or
10x^2 = 9000 - 5100
:
10x^2 = 3900
:
x^2 = 3900/10
:
x^2 = 390
:
x =
:
x = 19.75 mph is the current, that seems high so let's check it in the equation
:
+ = 10
+ = 10
8.29 + 1.71 = 10 hrs, confirms our solution
:
Could you follow the logic here OK? Any questions?

You can put this solution on YOUR website!
Driving down a mountain, Tom finds that he has descended 1800 ft in elevation by the time he is 3.25 mi horizontally away from the top of the mountain. Find the slope of his descent to the nearest hundredth.
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One of the definitions of slope is "Rise/run"
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Convert miles to ft like the given elevation: 3.25 * 5280 = 17160 ft
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He is descending so the 1800 ft will be negative:
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Slope = = -.104895 or -.10