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How many cubic feet of sand will I need to fill a pipe 1.5" I.D. which is 10 feet long?
:
Use the volume of a cylinder: V =
Diameter of 1.5, a radius of .75 inches
Change length to inches also; 12(10) = 120 inches
V =
V ~ 212 cu/in
:
Change to cubic ft, there are 1728 cu/in in 1 cu/ft
= .123; roughly cu/ft

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A person is in a rowboat 3 miles from the closest point on a straight shoreline, as illustrated in the figure.
The person would like to reach a cabin that is 8 miles down the shoreline. The person can row at 4 miles per hour and jog at 7 miles per hour.
:
A.)How long will it take to reach the cabin if the person rows straight toward shore at point A and then jogs to the cabin?
A time equation: time = dist/speed
total time (t)= row time + walk time
t = +
t = .75 + 1.143
t 1.893 hrs or about 1 hr + .893(60) = 53.6 min
:
B.) How long will it take to reach the cabin if the person rows straight to the cabin and does no jogging?
Solve this as a right triangle, direct to the cabin will be the hypotenuse
h =
h = 8.544 miles
Rowing at 4 mph: = 2.136 hrs or about 2 hrs 8 min
:
C.)Find the minimum time to reach the cabin.
This is a little more complicated. He will row to a point on the shore line, then jog from that point to the cabin.
The position of the landing point has to be determined to find the rowing distance and the jogging distance that will yield the least total travel time>
:
Let x = distance from the closest point on the shoreline to the landing point
Then
(8-x) = distance from the landing point to the cabin (the jogging distance)
The rowing distance will be the hypotenuse of the triangle formed by x and 3
:
total time = rowing time + jogging time
t(x) = +
:
Graphically

Minimum on the graph, x=2.1 mi from the closest point on shoreline, then
Minimum time is when landing point is: 8-2.1 = 5.9 mi from the cabin
Then it will take about 1.76 hrs to get to the cabin
:
Check this
find the rowing distance: = 3.66 mi
then 3.66/4 = .915 hrs rowing
find the jogging distance: 8 - 2.1 = 5.9 mi
then 5.9/7 = .843 hrs jogging
Total time: .915 + .843 = 1.76 hrs min time to the cabin

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a farmer was asked how many cows he had on his property.
he replied that he was unsure, but he knew that when he counted them by twos, threes, fours, fives, or sixes, he always had one left over.
the only way he could avoid this was to count by sevens; he then had none left over.
what is the smallest number of cows the farmer could possibly have owned?
:
We know it is a multiple of 7, that it is an odd number, has to end in 1
multiply by numbers ending in 3 to get a 1
13 * 7 = 91, no
23 * 7 = 161, no
33 * 7 = 231, no
43 * 7 = 301, looks like a winner

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A motorist noticed that as he got 10 km closer to a mountain, the angle of elevation of the top of the mountain changed from 10 degrees to 70 degrees.
Approximate the height of the mountain.
:
Draw this out, consider the triangle formed the by 10 km and the slant ranges from the two given points.
The interior angles: 10, 180-70=110, and 60 degrees
Find the slant range(s) to the top of the mountain from initial position using the law of sines.
=
cross multiply
.866s = .9397*10
s =
s = 10.85 km to the top of the mountain
:
A right triangle where the side opposite is the height(h) of the mountain
sin(10) =
h = 10.85 * .17365
h = .32125 km or 321.25 meters high

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The Healthy Favorites Juice Company sells apple juice for 8.3 cents an ounce and raspberry for 9.3 cents and ounce.
The company wishes to market and sell 8 ounce cans of apple raspberry juice for 8.7 cents an ounce.
How many ounces of each should be mixed?
:
Let x = amt of raspberry required
then result is to be an 8 oz can therefore:
(8-x) = amt of apple required
:
A typical mixture equation
9.3x + 8.3(8-x) = 8.7(8)
9.3x + 66.4 - 8.3x = 69.6
9.3x - 8.3x = 69.6 - 66.4
x = 3.2 oz of raspberry required
then
8 - 3.2 = 4.8 oz of apple
:\
:
Check these values in the original mixture equation

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Just remember a simple rule:
"Any term, variable or number, with an exponent of 0 is = to 1

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A carpenter can complete a job in 5 hours.
After working on the job for 2 hours, an assistant helped finish the job.
Together they completed the job in 1 hour.
How long might it take the assistant, working alone, to complete a job similar to this one?
:
let a = time required by the assistant working alone
let the completed job = 1
:
The carpenter worked a total 3 hrs, the assistant for 1 hr,
a shared work equation
;
+ = 1
multiply by 5a, results:
3a + 5(1) = 5a
5 = 5a - 3a
5 = 2a
a = 5/2
a = 2.5 hrs
:
:
Check
+ = 1

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invested $8000 for 1 year, part at 3% and part at 5% simple interest.
How much was invested in each account if the same amount of interest was received from each account?
:
let x = amt invested at 5%
then
(8000-x) = amt invested at 3%
:
5% int = 3% int
.05x = .03(8000-x)
.05x = 240 - .03x
.05x + .03x = 240
.08x = 240
x = 240/.08
x = $3000 invested at 5%
I'll let you find the amt invested at 3%
:
Check this by finding the actual interest from each investment

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1. a man cover a distance of 6 km at the rate of 4km/h & cover a distance of 4 km at rate of 3km/h .
what is his average speed?
The total distance traveled: 10km
let a = the average speed for the trip
Write a time equation, time = distance/speed
+ =
multiply equation by 12a to clear the denominators, resulting in:
3a(6) + 4a(4) = 12(10)
18a + 16a = 120
34a = 120
a = 120/34
a ~ 3.53 km/hr is his average speed
:
2. the average marks obtained by 120 candidate is 35.
if average marks of pass candidate is 39 and that of fail candidate is 15, what is the number of candidate who passed in examination?
let p = number that passed
then
(120-p) = number that failed
:
= 35
multiply both sides by 120
39p + 15(120-p) = 120(35)
39p + 1800 - 15p = 4200
39p - 15p = 4200 - 1800
24p = 2400
p = 100 passed
:
:
You should check these solutions in the original equations.

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you are an electrician working in an industrial plant.
A circuit contains 8 incandescent lamps connected in series across 480 volts.
One lamp has burned out, and you must determine which one is the defective.
You have available a voltmeter, ammeter, and ohmmeter.
Which meter would you use to determine which lamp is defective in the shorter possible time? Explain how you use this meter and why
:
I think the best and safest way is to use an ohmmeter
:
Turn off the power, Check continuity between lamps 4 & 5 to one end of the line of lights.
If continuity is shown, the defective lamp is among the other 4 lamps.
Check from the that middle point across two lamps on the open side.
Then across the two remaining lamps to determine which one is open.

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a^3 - 2ab - ac
Factor out a
a(a^2 - 2b - c)

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To determine the appropriate landing speed of an airplane, the formula D=.1x−3x+22 is used, where x is the initial landing speed in feet per second and D is the distance needed in feet.
:
I think this should be D = .1x^2 - 3x + 22
:
If the landing speed is too fast, the pilot may run out of runway; if the speed is too slow, the
plane may stall. What is the appropriate landing speed if the runway is 800 feet long?
:
.1x^2 - 3x + 22 = 800
.1x^2 - 3x + 22 - 800 = 0
.1x^2 - 3x - 778 = 0
solve this using the quadratic formula

this equation:

You can do the math, I got x = 104.47 mph

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Take your calculator, press "log" , then 5, then "enter,
should get .69897...

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6 times the volume of a cube is equal to the sum of its surface area and the total length of its 12 edges. Find the dimensions of the cube.
:
let x = one side of the cube
then
6x^2 = the surface area
and
x^3 = the volume
and
12x = total length of the 12 edges
:
" 6 times the volume of a cube is equal to the sum of its surface area and the total length of its 12 edges. "
6x^3 = 6x^2 + 12x
simplify, divide by 6x
x^2 = x + 2
x^2 - x - 2 = 0
factors to
(x-2)(x+1) = 0
the positive solution
x = 2 cm is the side of the cube
:
:
You can confirm this in the original equation

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let the two number be a & b
:
Write an equation for each statement
:
"the sum of twice a first number and five times a second number is 99?"
2a + 5b = 99
:
"if the second number is subtracted from five the first number the result is 45.
Perhaps you mean:
"if the second number is subtracted from five times the first number the result is 45."
5a - b = 45
or
b = 5a - 45
:
In the 1st equation, replace b with (5a-45)
2a + 5(5a-45) = 99
2a + 25a - 225 = 99
27a = 99 + 225
27a = 324
a = 324/27
a = 12
:
Find b using b = 5a - 45
b = 5(12) - 45
b = 60 - 45
b = 15
:
Find the numbers. 12 & 15
;
;
Check this in the statement:
"the sum of twice a first number and five times a second number is 99?"
2(12) + 5(15) =
24 + 75 = 99

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let a = the 10's digit
let b = the units
then
10a+b = "the number"
:
Write an equation for each statement
:
"The tens digit of a two-digit number is 5 more than the units digit."
a = b + 5
:
"If "the number" is divided by the sum of its digits, the partial quotient is 7 and the remainder is 6."
subtract the remainder (6) from the number to get an even 7
= 7
multiply both sides by (a+b)
10a + b - 6 = 7(a+b)
10a + b - 6 = 7a + 7b
10a - 7a = 7b - b + 6
3a = 6b + 6
simplify, divide by 3
a = 2b + 2
Replace a with (b+5)
b + 5 = 2b + 2
5 - 2 = 2b - b
b = 3,
then
a = 3 + 5
a = 8
:
83 is the number
:
:
Check this, divide 83 by 11, 7 a remainder of 6

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Here we are, exactly seven minutes late. This time I averaged 30 miles per hour.
Last time we did the trip, we were five minutes early.
We started at the same time. This time I averaged 36 miles per hour.
What was the distance?
:
Since we are using mph:
Change 5 min to 5/60 hr
Change 7 min to 7/60 hr
:
let t = normal "on-time" time for the trip
:
Write a distance equation, dist = speed * time,
36(t-) = 30(t+)
divide both sides by 6
6(t-) = 5(t+)
6t - = 5t +
reduce the fractions
6t - = 5t +
6t - 5t = +
t = hr is the normal travel time
:
Find the distance when 7min late, 67 min
d = 30 * [ + ]
d = 30 * [ + ]
d = 30 *
d =
d = 36 miles
:
Check the distance when 5 min early
d = 36 * [ - ]
d = 36 *
d = 36 miles,

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A diver is standing on a platform 24 feet above the pool.
He jumps from the platform with an initial upward velocity of 8ft/s.
How long will it take for him to hit the water?
I know the formula is h=-16t^2+vt+s, but I don't understand the difference between h and s.
:
The Formula h = -16t^2 + vt + s; where
h = height after t seconds
t = time in seconds
v = upward velocity
s = initial height (t=0)
also note that
-16t^2 represents the downward force of gravity for t seconds
+vt = the upward force for t seconds
:
the height when he hits the water = 0 therefore:
0 = -16t^2 + 8t + 24
usually written
-16t^2 + 8t + 24 = 0
we can simplify this, divide by -8, then we have
2t^2 - t - 3 = 0
this will factor to
(2t - 3)(t + 1) = 0
the positive solution
2t = 3
t = 3/2
t = 1.5 seconds to hit the water
-

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Kerry rides a 60km route on his road bike and can average 30km per hour if there is no head or tail wind.
For every 5km of wind Kerry's average speed is affected by 1 km/hr.
Write a rational function that calculates the time, T, it would take Kerry to ride the 60km route with a wind speed of s.
:
Time = dist/speed
speed = 30 - s
or
speed = 30-.2s
:
T(s) = ; Time as a function of wind speed
:
How long would it take Kerry to ride the 60km if he was riding into a 30km/hr headwind?
T(s) =
T(s) =
T(s) =
T(s) = 2.5 hrs to ride 60 km with 30km/hr headwind
:
Calculate the speed of the wind if it took Kerry 1hour 42mins to ride the 60km route.
Wait a minute, if there was no wind, it would take 60/30, 2hrs to make the trip
Therefore:
Find the speed of the tail wind, assume 5 km wind adds 1 km/hr
1.7 =
Multiply both sides by (30+.2s)
1.7(30+.2s) = 60
51 + .34s = 60
.34s = 60 - 51
.34s = 9
s = 26.47 km/hr tail wind to help him ride 60 km in 1 hr 42 min
i

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The mean of 15 scores is 82. If the mean of 7 of those scores is 78, what is the mean of the remaining 8 scores?
:
Let x = mean of remaining 8 scores
:
= 82
multiply both sides by 15
7(78) + 8x = 15(82)
546 + 8x = 1230
8x = 1230 - 546
8x = 684
x = 684/8
x = 85.5 is the mean of the remaining 8 scores

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convert the improper fraction into a mixed number.35x11
:
Using old fashioned 3rd grade division
:
... . ___
11 |35
11 goes into 35 three times, with a remainder of 2, so it's 3

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Madam Ling was 28 years old when her son was born.
If the product of their ages now is 480, how old is her son now?
:
Let s = son's age now
we know
s + 28 = Ling's age now
then the product of their ages:
s(s+28) = 480
s^2 + 28s - 480 = 0
Factors to
(s+40)(s-12) = 0
the positive solution
s = 12 yrs old is the son's present age
:
:
Check this yourself, find Ling's age and multiply by 12

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Let b = B's present age
Let j = J's present age
;
Write an equation for each statement:
:
Eighteen years ago, Benjamin was eight more than one-third as old as Jessica.
b - 18 = (j - 18) + 8
multiply both sides by 3
3(b-18) = j - 18 + 3(8)
3b - 54 = j - 18 + 24
3b = j - 18 + 24 + 54
3b = j + 60
:
Today, Jessica is twenty less than two times the age of Benjamin.
j = 2b - 20
:
replace j with (2b-20) in the 1st simplified equation
3b = (2b-20) + 60
3b - 2b = 60 - 20
b = 40 yrs old is B
:
find j
j = 2(40) - 20
j = 60 yrs is Jessica's age
:
:
:
Check this in the 1st original equation
40 - 18 = (60 - 18) + 8
22 = (42) + 8
22 = 14 + 8

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a man travels 10km in 50mins if he runs for 8km and walks for 2km.
if he runs 4km and walks 6km, his time is 1h 15mins.
find his running and walking speed.
:
Change 50 min to 5/6 hr
Change 1 hr 15 min to 5/4 hr
:
Let r = his running speed in km/hr
Let w = his walking speed
:
Write a time equation for each scenario; time = dist/speed
:
+ =
and
+ =
:
Multiply the above by 2 and subtract the 1st equation
+ =
+ =
--------------------------------- subtraction eliminates r, find w
0 + = -
multiply by 12w, resulting in
12(10) = 3w(10) - 2w(5)
120 = 30w - 10w
120 = 20w
w = 120/20
w = 6 km/hr walking
:
Replace w with 6 in the 1st equation
+ =
= -
=
cross multiply
3r = 8 * 6
3r = 489
r = 48/3
r = 16 km/hr running
:
;
Check solution in the 2nd equation
+ =

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if a running track is circular, 140 yards across the middle and eight feet wide, how much further do you go in running four laps if you run on the outside edge rather than the inside edge?
:
running inside edge, diameter = 140 yds
running outside edge diameter = 156 yrs, add 2*8
:
Find circumferences
*4 = 1960.35
*4 = 1759.29
--------------------------
yds further = 201.6 yds

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4. Determine the equations of:
(a) the horizontal line that passes through the point (3, 7)
y = 7
(b) the vertical line that passes through the point (-5, 0)
x = -5

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A bricklayer does one day of work building a wall.
On the second day, he is joined by a second worker.
These two workers are joined by a third worker on the third day, and they finished building the wall at the end of the fourth day.
If all three workers worked at the same rate, how long would it have taken all three workers to build the wall if they worked together from the start?
:
first man works 4 days, 4 man-days
second man works 3 days, 3 man-days
third man works 2 days, 2 man-days
---------------------------------
Total man-days for wall: 9 man-days
:
"how long would it have taken all three workers to build the wall if they worked together from the start"
9/3 = 3 days if all worked together

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Rewrite each as an equivalent exponential equation.
1)
The base of 3, raised to the power of y, equals 5

:
2

:
Solve


The reciprocal gets rid of the neg exponent


:


square both sides


or just
x = 81

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+
Multiply inside the brackets by 1/2, gets rid of the 2
+
+
an exponent of 1/2 is the square root
+
can be written

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In 1862, two companies were granted the rights to build the transcontinental railroad from Omaha, Nebraska, to Sacramento, California.
The Central Pacific railroad began in 1863 from Sacramento heading east.
The Union Pacific began 24 months later, leaving from Omaha and heading west.
The Central Pacific averaged 8.75 miles of track each month, while the Union Pacific averaged 20 miles of track per month.
The two companies met at Promontory, Utah, after completing 1590 miles of track. How much track did each build?
:
First find out how many months to complete the task
:
m = no. of months of work by the Union Pacific
then
(m+24) = no. of months of work of the Central Pacific
:
20m + 8.75(m+24) = 1590
20m + 8.75m + 210 = 1590
28.75m = 1590 - 210
28.75m = 1380
m = 1380/28.75
m = 48 month of work by the UP
then
48(20) 960 mi of track completed by the UP
:
and
8.75(48+24) = 630 mi of track completed by the CP
;
:
See if that adds up: 960 + 630 = 1590 total miles

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the average age of a family of eight is 30 years.
the average age of six children in the family is 19 years.
If the mother is four years younger than the the father.
Calculate the age of the father?
:
let f = fathers age
then
(f-4) = mothers age
:
= 30
2f - 4 + 114 = 8(30)
2f + 110 = 240
2f = 240 - 110
2f = 130
f = 65 yrs is the age of the father
:
:
You can confirm this solution in the original equation