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15390..15419 , 15420..15449 , 15450..15479 , 15480..15509 , 15510..15539 , 15540..15569 , 15570..15599 , 15600..15629, >>NextLinear_Equations_And_Systems_Word_Problems/173499: Sam buys 3 kinds of fruit for 40 dollars, 10 dollars, and 1 dollar each. He pays 259 dollars for 100 pieces of fruit. How many of the cheapest did he buy?
1 solutions
Answer 128392 by ankor@dixie-net.com(15638)   on 2008-12-15 19:18:55 (Show Source):
You can put this solution on YOUR website!: Sam buys 3 kinds of fruit for 40 dollars, 10 dollars, and 1 dollar each. He pays 259 dollars for 100 pieces of fruit. How many of the cheapest did he buy?
:
Three kinds of fruit x, y, z
:
x + y + z = 100
and
40x + 10y + z = 259
:
Subtract the 1st equation from the 2nd
40x + 10y + z = 259
x + y + z = 100
----------------------
39x + 9y = 159
Simplify, divide by 3
13x + 3y = 53
3y = 53-13x
y = 
We know there can't be many $40 fruits (x) and y has to be an integer
try x = 2
y = 
y = 
y = 9 ea $10 fruit, this is the only positive integer solution
:
Find z
2 + 9 + z = 100
z = 100 - 11
z = 89 ea $1 animals, (the cheapest fruit)
:
Check in the Cost equation
40(2) + 10(27) + 1(89)
80 + 90 + 89 = 259; confirms our solutions
|
Percentage-and-ratio-word-problems/173487: Mya played in two softball games one afternoon. The first game lasted 42 minutes. The second game lasted 1 2/3 times longer that the first game. How long did Mya's second game last?
1 solutions
Answer 128349 by ankor@dixie-net.com(15638) on 2008-12-15 16:51:04 (Show Source):
You can put this solution on YOUR website!Mya played in two softball games one afternoon. The first game lasted 42 minutes. The second game lasted 1 2/3 times longer that the first game. How long did Mya's second game last?
:
Convert 1 2/3 to an improper fraction, namely 
;
* 42 =  = 70 minutes
|
Miscellaneous_Word_Problems/173430: This question is from textbook introduction to college mathematics
The decay rate of krypton-85 is 6.3% per day. What is the half-life?
Using the exponential decay function P(t) =Po^e^-kt
1 solutions
Answer 128347 by ankor@dixie-net.com(15638) on 2008-12-15 16:44:28 (Show Source):
You can put this solution on YOUR website!The decay rate of krypton-85 is 6.3% per day. What is the half-life?
Using the exponential decay function P(t) =Po^e^-kt
:
The exponential decay formula that I am familiar with:
A = Ao[2^(-t/h)]
where
Ao is the initial amt
A = final amt
t = time
h = half-life of the substance
:
Assume Ao = 100, Assume t = 1 day, find h (half life in days)
After 1 day, A = 100 - 6.3 = 93.7
:
100 * 2^(-1/h) = 93.7
:
2^(-1/h) = 
:
2^(-1/h) = .937
:
ln[2^(-1/h)) = ln(.937)
:
 *.693 = -.065
:
 = -.065
:
h = 
h = 10.66 days is the half life of krypton-85 according to this information
:
In actual fact I think they mean 6.3% per year, hence, 10.66 yrs half life
|
Volume/173460: Jeremy has a fish tank that has 40-cm by 70-cm rectangular base. The water is 25-cm deep. When he drops rocks into the tank, the water goes up by 2-cm. What is the volume in liters of the rocks?
1 solutions
Answer 128334 by ankor@dixie-net.com(15638) on 2008-12-15 15:45:28 (Show Source):
You can put this solution on YOUR website!Jeremy has a fish tank that has 40-cm by 70-cm rectangular base. The water is
25-cm deep. When he drops rocks into the tank, the water goes up by 2-cm.
What is the volume in liters of the rocks?
:
Find the volume of water displaced in cu/cm
:
40 * 70 * 2 = 5600 cu/cm
;
Convert cu/cm to liters:
 = 5.6 liters
|
Volume/173211: A rectangular piece of metal 32cm by 22cm, has a square of side X cm removed from each corner in order to form a rectangular box. If the volume of the box is to be a maximum what is the value of X?
1 solutions
Answer 128222 by ankor@dixie-net.com(15638) on 2008-12-14 21:57:55 (Show Source):
You can put this solution on YOUR website!A rectangular piece of metal 32cm by 22cm, has a square of side X cm removed
from each corner in order to form a rectangular box.
If the volume of the box is to be a maximum what is the value of X?
:
From the given information we know the dimensions (L,W,H) of the box is:
(32-2x) by (22-2x) by x
:
Area = length * width * height
A = (32-2x) * (22 - 2x) * x
FOIL
A = x(704 - 64x - 44x + 4x^2)
A = x(704 - 108x + 4x^2)
A = 704x - 108x^2 + 4x^3
or the standard arrangement is:
y = 4x^3 - 108x^2 + 704x
:
Plot this equation, we only are interested in the positive values x,y values
:
Using my trusty Ti83, max volume occurs when x = 4.274 cm about 1348 cu/cm
|
Geometry_Word_Problems/173274: The lenght of a rectangle is 4cm more than its width and 4cm less than the length of a diagonal of the rectangle. What is the area of the rectangle?
1 solutions
Answer 128203 by ankor@dixie-net.com(15638) on 2008-12-14 20:46:51 (Show Source):
You can put this solution on YOUR website!The length of a rectangle is 4cm more than its width and 4cm less than the length of a diagonal of the rectangle. What is the area of the rectangle?
;
Keep in mind the right triangle formula a^2 + b^2 = c^2
:
Let x = the length of the rectangle (a)
then
(x-4) = width of the rectangle (b)
and
(x+4) = diagonal of the rectangle (c)
:
x^2 + (x-4)^2 = (x+4)^2
FOIL
x^2 + x^2 - 8x + 16 = x^2 + 8x + 16
Combine like terms on the left
x^2 + x^2 - x^2 - 8x - 8x + 16 - 16 = 0
Results
x^2 - 16x = 0
Factor
x(x - 16) = 0
Two solutions;
x = 0
and
x = 16 is the length of the rectangle
then
12 = width of the rectangle
and
20 = diagonal of the rectangle
;
Area = 16 * 12 = 192 sq/cm
:
Check it:
16^2 + 12^2 = 20^2
256 + 144 = 400; confirms our solution for length & width
|
Miscellaneous_Word_Problems/173079: At a well-known unversity,1/4 of the undergraduate student commute,and 1/3 of the gradute student commute.One-tenth of the undergradute students drive more than 40 miles daily,and 1/6 of the graduate students drive more daily than 40 miles daily.If there are twice as many undergradute students as there are graduate students,then what fraction of the commuters drive more than 40 miles daily.
1 solutions
Answer 127946 by ankor@dixie-net.com(15638) on 2008-12-13 17:05:03 (Show Source):
You can put this solution on YOUR website!Commuting students. At a well-known university, 1/4 of the undergraduates commute; 1/3 of the graduates commute. 1/10 of the undergraduates drive more than 40 miles daily and 1/6 of the graduates drive more than 40 miles daily. If there are twice as many undergrads as there are grad students, then what fraction of the commuters drive more than 40 miles daily?
:
Let's use a convenient number for the total students: 720 has a lot of factors
:
It says,"there are twice as many undergrads as there are grad students,"
That would give us us 480 undergrads, and 240 grads
:
1/4 of the undergraduates commute;
 * 480 = 120
:
1/3 of the graduates commute.
 * 240 = 80
:
1/10 of the undergraduates drive more than 40 miles daily
 * 480 = 48
:
1/6 of the graduates drive more than 40 miles daily.
 * 240 = 40
:
what fraction of the commuters drive more than 40 miles daily?
Total commuters: 120 + 80 = 200
Total commuters drive more than 40 mi: 48 + 40 = 88
:
Then:  =  of the commuters commute over 40 mi
|
Functions/172882: You want to enclose a rectangular lot with fencing. One side of the lot runs along the highway. The fencing along the highway costs $10 per foot and the rest of the fencing costs $7 a foot. The total money available for fencing is $6,000. The cost equation for the fencing is 7(2W+L)+10L = 6000, where L is the length of the lot and W is the width of the lot. The area of the lot, A, is given by the formula A=LW.
Can you step me through the following:
1. By writing the width as a function of the length, find the width of the lot, if its length is 100 feet (Width = ).
2. Write the area of the lot as a function of the length (A(L) = ).
3. What will be the area of the lot if the length is 100 feet (Area = )?
1 solutions
Answer 127904 by ankor@dixie-net.com(15638) on 2008-12-13 10:11:33 (Show Source):
You can put this solution on YOUR website!You want to enclose a rectangular lot with fencing. One side of the lot runs along the highway. The fencing along the highway costs $10 per foot and the rest of the fencing costs $7 a foot. The total money available for fencing is $6,000. The cost equation for the fencing is 7(2W+L)+10L = 6000, where L is the length of the lot and W is the width of the lot. The area of the lot, A, is given by the formula A=LW.
:
Can you step me through the following:
;
1. By writing the width as a function of the length, find the width of the lot, if its length is 100 feet (Width = ).
:
Take the original equation, rearrange to find W:
7(2W+L)+10L = 6000,
14W + 7L + 10L = 6000
14W + 17L = 6000
14W = 6000 - 17L
W = 
:
L = 100'
W = 
W = 
W = 
W = 307.14'
:
:
2. Write the area of the lot as a function of the length (A(L) = ).
:
from the above we have W as the function L;
Substitute, for W in the A=L*W
A = L*( )
:
:
3. What will be the area of the lot if the length is 100 feet (Area = )?
:
Substitute 100 for L in the above equation:
A = L*( )
A = 100*( )
A = 10*( )
A = 100 * 307.14
A = 30714 sq ft
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Miscellaneous_Word_Problems/173014: How many solutions does 11x + 7y = 292 have if x and y are positive integers? This question is not from a textbook.
1 solutions
Answer 127890 by ankor@dixie-net.com(15638) on 2008-12-12 21:24:27 (Show Source):
You can put this solution on YOUR website!How many solutions does 11x + 7y = 292 have if x and y are positive integers?
:
7y = 292 - 11x
:
y = 
:
Put this into a graphing calc like Ti83, look at the table for positive integer x & y:
I got 4:
x | y
------
3 |37
10 |26
17 |15
24|| 4
|
Expressions-with-variables/173015: My question is how to do substitution. I am extremely confused because with it because my teacher will not explain it well no matter what.
As an example, questions like these:
y=5-4*x
3*x-2*y=12
Please help and possibly explain in detail how to solve these problems.
1 solutions
Answer 127887 by ankor@dixie-net.com(15638) on 2008-12-12 20:59:45 (Show Source):
You can put this solution on YOUR website!y=5-4*x
3*x-2*y=12
:
You can write it like this
y = 5 - 4x
3x - 2y = 12
;
Look at the 1st equation; that means that any place you see y, you can substitute
(5-4x) for it
:
In the 2nd equation (3x - 2y = 12), we can substitute (5-4x) for y, so we have
3x - 2(5-4x) = 12
That gives just one unknown (x), which is easy to solve with basic algebra:
:
multiply what's inside the brackets by -2 and you have:
:
3x - 10 + 8x = 12; note that a minus times a minus is plus
:
Perform the necessary operations to get the variables on the left, numbers on the right
3x + 8x = 12 + 10; added 10 to both sides
:
11x = 22
x =  ; divided both sides by 11
x = 2
:
Now substitute 2 for x in the 1st original equation and find y:
y = 5 - 4x
y = 5 - 4(2)
y = 5 - 8
y = -3
;
Check by substituting both solution for x and y in the 2nd equation
3x - 2y = 12
3(2) - 2(-3) = 12
6 + 6 = 12; which is true. (remember minus times a minus is a plus)
:
How about this, did it make sense to you, now?
|
Graphs/172917: The sum of number is 12. If two times the greater number is substracted from 5 times the lesser number. The result is 25 what is the product of the two original numbes if the greater one is doubled and the lesser one is cut in half?
1 solutions
Answer 127788 by ankor@dixie-net.com(15638) on 2008-12-12 08:31:30 (Show Source):
You can put this solution on YOUR website!Two numbers, x & y
:
Write an equation for each statement:
:
"The sum of number is 12."
x + y = 12
x = (12-y)
:
"If two times the greater number is subtracted from 5 times the lesser number, the result is 25"
5y - 2x = 25
Substitute (12-y) for x, find y
5y - 2(12-y) = 25
5y - 24 + 2y = 25
5y + 2y = 25 + 24
7y = 49
y = 
y = 7
then
x = 12 - 7 = 5
:
However, this make y the greater number, anyway charging ahead we have:
:
what is the product of the two original numbers if the greater one is doubled and the lesser one is cut in half?"
:
Here again, if you double one number, & halve the other they are no longer original, are they? Anyway do what they say:
:
2(7) * .5(5) =
14 * 2.5 = 35
but if you reverse it
2(5) * .5(7)
10 * 3.5 = 35, the same solution
:
A strange problem.
:
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Travel_Word_Problems/172884: A man leaves Paris driving at a constant speed. After a while he passes a kilometer stone that displays a 2 digit number. An hour later he passes another kilometer stone displaying the same 2 digit number but in reverse order. Another hour after that, he passes a 3rd kilomer stone marker with the same 2 digits(in any order seperated by a zero. What is the speed of the man's car?
1 solutions
Answer 127747 by ankor@dixie-net.com(15638) on 2008-12-11 21:53:27 (Show Source):
You can put this solution on YOUR website!A man leaves Paris driving at a constant speed. After a while he passes a kilometer stone that displays a 2 digit number. An hour later he passes another kilometer stone displaying the same 2 digit number but in reverse order. Another hour after that, he passes a 3rd kilometer stone marker with the same 2 digits(in any order separated by a zero. What is the speed of the man's car?
:
two digit x and y
;
Let (10x + y) = 1st km marker
then: (10y + x) = 2nd km marker
and: (100x + y) = 3rd km marker
:
2nd marker - 1st marker = 3rd marker - 2nd marker
(10y + x) - (10x + y) = (100x + y) - (10y + x)
10y + x - 10x - y = 100x + y - 10y - x
9y - 9x = 99x - 9y
Simplify, divide equation by 9
y - x = 11x - y
y + y = 11x + x
2y = 12x
:
In order for y to equal a single digit, x has to = 1
2y = 12
y = 6
:
1st marker = 16
2nd marker = 61
3rd marker = 106
:
106 - 61 = 45 km in one hr therefore speed = 45 km/hr
;
Check:
61 - 16 = 45 km
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Equations/172554: Given the rational expression:
x^2-8x+12
__________ (divided by)
x^2-7x+6
state the restrictions on the variable x
1 solutions
Answer 127711 by ankor@dixie-net.com(15638) on 2008-12-11 19:18:22 (Show Source):
You can put this solution on YOUR website!Given the rational expression:
x^2-8x+12
__________ (divided by)
x^2-7x+6
state the restrictions on the variable x
:
Find the value of x that will give you 0 in the denominator
x^2 - 7x + 6 = 0
Factors to:
(x-6)(x-1) = 0
x = +6
and
x = +1
:
x cannot equal +1 or +6, any other value is OK
(if you substitute these values for x you have a 0 denominator, a big no-no in algebra)
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Signed-numbers/172819: Divide and if possible simplify: (5p-5/p)/(6p-6/5p^2)
I know that i have to flip the second fraction and multiply, but I'm not seeing how to get the right answer.This is a study problem for an Accuplacer test that I'm taking tomorrow, It really helps me to understand if I can See each step to solve this problem written out like in a math book. Thank you.
1 solutions
Answer 127701 by ankor@dixie-net.com(15638) on 2008-12-11 18:56:58 (Show Source):
You can put this solution on YOUR website!Divide and if possible simplify:
 / 
Invert the dividing fraction and multiply (like you said)
* 
:
Cancel p into 5p^2
5p - 5 * 
;
Factor out 5 and 6
5(p-1) * 
;
Cancel out (p-1)'s
5 *  = 
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Travel_Word_Problems/172651: A mover's van left Middleton at 8:00 AM and drove to Bayside at an average speed of 70km/h. At 9:00 AM, the Riley's left Middleton following the same route as the van. If both arrived in Bayside at 4:00 PM, what was the Rileys' average speed?
1 solutions
Answer 127610 by ankor@dixie-net.com(15638) on 2008-12-11 13:57:57 (Show Source):
You can put this solution on YOUR website!A mover's van left Middleton at 8:00 AM and drove to Bayside at an average speed
of 70km/h. At 9:00 AM, the Riley's left Middleton following the same route as
the van. If both arrived in Bayside at 4:00 PM, what was the Rileys' average speed?
:
Let s = Riley's average speed
:
From the information given we know that:
Mover's travel time = 8 hrs
Riley's travel time = 7 hrs
:
They both travel the same distance, write a distance equation: Dist = speed * time
:
Riley's dist = Mover's dist
7s = 8*70
:
7s = 560
s = 
s = 80 km/hr is Riley's speed
:
You can confirm this: 7 * 80 = 560 km also
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Quadratic_Equations/172694: A manufacturer charges $24 for stereo headphones and has been selling about 1000 a week. He estimates that for every $1 price reduction, 100 more headphones can be sold per week. (For example, he could sell 1100 headphones at $23 each and 1200 headphones at $22 each.)
a. Let 24-x be the reduced price per set of headphones. write a quadratic function that gives the total revenue recieved by the manufacturer in a week.
My work- would it be 24-1 I really don't understand a & b
b. What price will maxamize the total revenue?
Please explain
I would really appreciate any help
Thank you
1 solutions
Answer 127602 by ankor@dixie-net.com(15638) on 2008-12-11 13:40:13 (Show Source):
You can put this solution on YOUR website!A manufacturer charges $24 for stereo headphones and has been selling about 1000 a week.
He estimates that for every $1 price reduction, 100 more headphones can be sold per week.
(For example, he could sell 1100 headphones at $23 each and 1200 headphones at $22 each.)
:
a. Let 24-x be the reduced price per set of headphones. write a quadratic function that gives the total revenue received by the manufacturer in a week.
"
:
x = no. of dollar reductions in price
and
x = additional no. of 100's of units sold
:
Price = (24-x)
Units sold = 1000+100x
:
Revenue = price * units sold
:
R = (24-x) (1000+100x)
FOIL
R = 24000 + 2400x - 1000x - 100x^2
:
R = -100x^2 + 1400x + 24000; the quadratic equation for revenue
:
:
b. What price will maximize the total revenue?
:
Since this is a quadratic equation, the axis of symmetry will give a value
for x that will make (24-x) be the price for maximum revenue
:
Find the axis of symmetry using the formula x = -b/(2a)
In this equation a=-100; b=1400
x = 
x = 
x = +7; a $7 reduction
:
Price = 24 - 7 = $17 for max revenue
:
:
If you plotted this it would look like this:
Where x = dollar reduction from $24, and y is the total revenue
You can see max revenue occurs when x = 7 which is a unit price of $17
:
If you want to know exactly what the max revenue is, substitute 7 for x in the
original equation.
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Miscellaneous_Word_Problems/172670: This question is from textbook Algebra and Trigonometry Structure and Method book 2
I have been struggling to figure out this math problem and I was wondering if someone could help me? Please and Thank You!! I would deeply appreicate it!!
Frational Equations
6t^2-t-1 over 3(t^2+1)= 2
I was also wondering if you could include the domain restriction and the LCD PLEASE!!
1 solutions
Answer 127539 by ankor@dixie-net.com(15638) on 2008-12-10 21:46:20 (Show Source):
You can put this solution on YOUR website! = 2
;
Don't worry about the LCD. Just multiply both sides by 3(t^2+1) to get rid of the denominator:
 = 
:
 = 
:
Variables on the left, numbers on the right
 = 6 + 1
-t = 7
t = -7
:
There are no restriction on the domain. Denominator can never = 0
:
:
Check this solution by substitution in the original equation
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Numbers_Word_Problems/172625: This question is from textbook Algebra Structure and Method Book 1
The sum of three digits is 12. The tens digit exceeds the hundreds digit by the same amount that the units digit exceeds the tens digit. If the digits are reversed, the new number exceeds the original number of 198. Find the original number.
1 solutions
Answer 127533 by ankor@dixie-net.com(15638) on 2008-12-10 21:20:06 (Show Source):
You can put this solution on YOUR website!The sum of three digits is 12. The tens digit exceeds the hundreds digit by the same amount that the units digit exceeds the tens digit. If the digits are reversed, the new number exceeds the original number of 198. Find the original number.
:
x = 100's digit
y = 10's digit
z = units
:
The number 100x + 10y + z
;
Write an equation for each statement:
:
"The sum of three digits is 12. "
x + y + z = 12
:
"The tens digit exceeds the hundreds digit by the same amount that the units digit exceeds the tens digit."
y - x = z - y
y + y - x - z = 0
2y - x - z = 0
:
We can use these first two equations to find y using elimination
x + y + z = 12
-x +2y - z = 0
------------------addition eliminates x and z
0 + 3y + 0 = 12
y = 
y = 4
:
"If the digits are reversed, the new number exceeds the original number of 198.
100z + 10y + x = 100x + 10y + z + 198
Simplify
100z - z + 10y - 10y + x - 100x = 198
99z - 99x = 198
Simplify divide equation by 99, results:
z - x = 2
z = x + 2
:
Find the original number.
:
In the 1st equation, substitute 4 for y, and (x+2) for z; find x
:
x + 4 + (x+2) = 12
2x + 6 = 12
2x = 12 - 6
x = 
x = 3
:
An easy step to find z:
z = 3 + 2
z = 5
:
The number 345
:
:
Check the solution using the statement:
"If the digits are reversed, the new number exceeds the original number of 198.
543 = 345 + 198
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Age_Word_Problems/172631: A father, being asked his age and that of his son, said, "If you add 4 to my age and divide the sum by 4, you will have my son's age. But 6 years ago I was 7 1/2 times as old as my son." find their ages.
1 solutions
Answer 127514 by ankor@dixie-net.com(15638) on 2008-12-10 20:07:53 (Show Source):
You can put this solution on YOUR website!A father, being asked his age and that of his son, said,
"If you add 4 to my age and divide the sum by 4, you will have my son's age.
But 6 years ago I was 7 1/2 times as old as my son."
find their ages.
:
Let f = father's age now; s = son's age now
:
"If you add 4 to my age and divide the sum by 4, you will have my son's age."
 = s
Multiply equation by 4
f + 4 = 4s
f = 4s - 4
:
"But 6 years ago I was 7 1/2 times as old as my son."
f - 6 = 7.5(s - 6)
f - 6 = 7.5s - 45
f = 7.5s - 45 + 6
f = 7.5s - 39
Substitute (4s-4) for f in the above equation
4s - 4 = 7.5s - 39
-4 + 39 = 7.5s - 4s
35 = 3.5s
s = 
s = 10 yrs old is the son
then
f = 4(10) - 4
f = 36 yrs old is the father:
:
:
Check solution in the statement:
"If you add 4 to my age and divide the sum by 4, you will have my son's age."
 = 10; confirms our solution
:
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Money_Word_Problems/172635: This question is from textbook
Can someone please help me I came up with three crazy answer. thank you
A rectangular garden has dimensions of 18 feet by 13 feet. A gravel path of uniform width is to be built around the garden. How wide can the path be if there is enough gravel for 516 square feet?
1 solutions
Answer 127510 by ankor@dixie-net.com(15638) on 2008-12-10 19:47:53 (Show Source):
You can put this solution on YOUR website!A rectangular garden has dimensions of 18 feet by 13 feet. A gravel path of
uniform width is to be built around the garden.
How wide can the path be, if there is enough gravel for 516 square feet?
:
Let x = the width of the path required to utilize 516 sq/ft of gravel
then
The overall dimensions (including the path) are: (2x+18) by (2x+13)
FOIL this and we have:
4x^2 + 62x + 234 = the area of the garden and the path
:
18 * 13 = 234, the area of the garden
:
Overall area - garden area = path area (516 sq/ft)
:
(4x^2 + 62x + 234) - 234 = 516
;
4x^2 + 62x - 516 = 0
Simplify divide by 2:
2x^2 + 31x - 258 = 0
This will factor:
(2x + 43)(x - 6) = 0
Positive solution
x = + 6 ft is the width of the path
:
:
Check our solution:
overall dimensions would be 30 by 25 = 750 sq/ft (add 12 to length & width)
750 - 234 = 516, the area of the path
:
Did this clear up some of the difficulty you were having here?
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Linear_Equations_And_Systems_Word_Problems/172429: A stream Flows at a rate of 5 mph. A boat travels 75 mi upstream and returns in a total of 8 hours. What is the speed of the boat in still water?
1 solutions
Answer 127448 by ankor@dixie-net.com(15638) on 2008-12-10 16:06:53 (Show Source):
You can put this solution on YOUR website!A stream Flows at a rate of 5 mph. A boat travels 75 mi upstream and returns
in a total of 8 hours. What is the speed of the boat in still water?
:
Let s = boat speed in still water
then
(s-5) = speed upstream
and
(s+5) = speed down stream
:
Write a time equation: Time= 
:
Time upstream + time downstream = 8 hours
 +  = 8
:
Multiply equation by (s-5)(s+5)
(s-5)(s+5)* + (s-5)(s+5)* = 8(s-5)(s+5)
results
75(s+5) + 75(s-5) = 8(s^2 - 25)
:
75s + 375 + 75s - 375 = 8s^2 - 200
:
150s = 8s^2 - 200
:
Arrange as a quadratic equation
8s^2 - 150s - 200 = 0
:
Simplify divide equation by 2:
4s^2 - 75s - 100 = 0
Factors to:
(4s + 5)(s - 20) = 0
Positive solution:
s = +20 speed of boat in still water
:
:
Check the solution by finding the times of up and down streams
75/15 = 5 hrs
75/25 = 3 hrs
-------------
total = 8 hrs
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Travel_Word_Problems/172465: Steve traveled 200 miles at a certain speed. Had he gone 10mph faster, the trip would have taken 1 hour less. Find the speed of his vehicle.
1 solutions
Answer 127381 by ankor@dixie-net.com(15638) on 2008-12-10 10:14:10 (Show Source):
You can put this solution on YOUR website!Steve traveled 200 miles at a certain speed. Had he gone 10mph faster, the trip
would have taken 1 hour less. Find the speed of his vehicle.
:
Let s = original speed of the vehicle
then
(s+10) = the faster speed
:
Write a time equation, time =
:
original time = faster time + 1 hour
 =  + 1
:
Multiply equation by s(s+10)
s(s+10)* = s(s+10)* + s(s+10)
Results:
200(s+10) = 200s + s^2 + 10s
:
200s + 2000 = 200s + s^2 + 10s
:
Arrange as quadratic equation:
s^2 + 10s + 200s - 200s - 2000 = 0
:
s^2 + 10s - 2000 = 0
Factors to:
(s + 50)(x - 40) = 0
:
s = + 40 mph is the original speed
:
:
Check solution by finding the times
200/40 = 5 hrs
200/50 = 4 hrs
|
Age_Word_Problems/172442: Amme and Becky have the same birthday. Amme is twice as old as Becky was when Amme was as old as Becky is now. If Amme is now 24 years old, how old is Becky??
1 solutions
Answer 127377 by ankor@dixie-net.com(15638) on 2008-12-10 09:44:20 (Show Source):
You can put this solution on YOUR website!Amme and Becky have the same birthday. Amme is twice as old as Becky was when Amme was as old as Becky is now. If Amme is now 24 years old, how old is Becky??
:
Let b = Becky's age; Let a = Anne's age
:
Write an equation for the statement:
"Amme is twice as old as Becky was when Amme was as old as Becky is now."
a = 2(b - (a-b))
a = 2(b - a + b)
a = 2(2b - a)
a = 4b - 2a
a + 2a = 4b
3a = 4b
:
"If Amme is now 24 years old,"; substitute 24 for a:
3(24) = 4b
72 = 4b
b = 
b = 18 is Becky's age now
:
:
Check solution in the statement(the difference in their ages is 6):
" Amme is twice as old as Becky was when Amme was as old as Becky is now."
24 = 2(18 - 6)
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Systems-of-equations/172376: a cyclist, riding at 21 miles per hour, leaves town. Four hours later, another cyclist leaves from the same starting point, traveling in the same direction at an average of 33 miles per hour. how long did it take the second cyclist to catch up to the first cyclist?
1 solutions
Answer 127302 by ankor@dixie-net.com(15638) on 2008-12-09 19:43:16 (Show Source):
You can put this solution on YOUR website!a cyclist, riding at 21 miles per hour, leaves town. Four hours later,
another cyclist leaves from the same starting point, traveling in the
same direction at an average of 33 miles per hour.
:
how long did it take the second cyclist to catch up to the first cyclist?
;
Let t = 2nd cyclist travel time
then
(t+4) = 1st cyclist travel time
:
When the 2nd cyclists catches the 1st, they will have traveled the same distance
Write distance equation from this fact: Dist = speed * time
:
2nd cyclist dist = 1st cyclist dist
;
33t = 21(t+4)
33t = 21t + 84
33t - 21t = 84
12t = 84
t = 
t = 7 hrs, for the 2nd cyclist to catch the 1st
;
:
Check solution by finding the dist traveled by each:
11*21 = 231 mi
7*33 = 231 mi; confirms our solution
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Money_Word_Problems/172168: a bank teller is counting his money and notices that he has an equal number of tens and twenties and $147 in other bills. if the total value of the bills he has is 1,167, how many tens and twenties does he have
1 solutions
Answer 127191 by ankor@dixie-net.com(15638) on 2008-12-08 21:13:54 (Show Source):
You can put this solution on YOUR website!A bank teller is counting his money and notices that he has an equal number
of tens and twenties and $147 in other bills. if the total value of the bills
he has is 1,167, how many tens and twenties does he have?
:
Let x = no. of 10's and number of 20's
:
10x + 20x + 147 = 1167
:
You should be able to do it now
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Quadratic_Equations/171929: suppose that an object is thrown into the air with an intial upward velocity of vo meters per second from a height ho meters above the ground. Then t seconds later, its height h(t) meters above the ground is modeled by the function h(t)= -4.9 t^2 + vot + ho. (this model dosent' account for resistance)
if a stone is thrown with an upward velocity of 14 m/s from a cliff 30 meters high
I answered part a which asked:
a. find its height above the ground t seconds later.
h(t)= -4.9t^2 + 14 t + 30
I don't understand part b or c
b. when will the stone reach it's highest elevation
c.when will the stone hit the ground?
I would really appreciate any help I can recieve.
Thank you very much
please help soon!
1 solutions
Answer 127186 by ankor@dixie-net.com(15638) on 2008-12-08 21:04:41 (Show Source):
You can put this solution on YOUR website!suppose that an object is thrown into the air with an intial upward velocity of vo meters per second from a height ho meters above the ground. Then t seconds later, its height h(t) meters above the ground is modeled by the function h(t)= -4.9 t^2 + vot + ho. (this model dosent' account for resistance)
if a stone is thrown with an upward velocity of 14 m/s from a cliff 30 meters high
I answered part a which asked:
a. find its height above the ground t seconds later.
h(t)= -4.9t^2 + 14t + 30
:
I don't understand part b or c
b. when will the stone reach it's highest elevation
:
It will reach the highest point (max) on the axis of symmetry
Find the axis of symmetry using the formula: x = -b/(2a)
:
In this problem x=t, a=-4.9, b=14, we have:
t = 
t = 
t = +1.43 seconds, it will be at it's highest point
:
:
c.when will the stone hit the ground?
;
When it hits the ground h(t) = 0, find t: we have;
-4.9t^2 + 14t + 30 = 0
:
Solve for t using the quadratic formula: a=-4.9, b=14, c=30
:
You should get a positive solution of about 4.28 seconds
:
:
A graph of this:
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Linear-equations/172003: divide 184 into two parts such that one third of one part may exceed one seventh of the other part by 8
1 solutions
Answer 127174 by ankor@dixie-net.com(15638) on 2008-12-08 19:44:03 (Show Source):
You can put this solution on YOUR website!divide 184 into two parts such that one third of one part may exceed one seventh of the other part by 8
:
Let one part = x
and
the other part = (184-x)
:
"one third of one part may exceed one seventh of the other part by 8"
x =  (184-x) + 8
:
Multiply equation by 21 to get rid of the denominators; results:
7x = 3(184-x) + 21(8)
:
7x = 552 - 3x + 168
:
7x + 3x = 552 + 168
:
10x = 720
x = 
x = 72 is one part
and
184 - 72 = 112 is the other part
;
:
Check these values in the statement
"one third of one part may exceed one seventh of the other part by 8"
(72) = (112) + 8
24 = 16 + 8
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Polynomials-and-rational-expressions/172005: (x-1) (x4+x3+x2+1)
answer: x^5-x^2+ x-1
any math mathician can come up with a full break down of how my math book came up with x^5-x^2+x-1 as an answer
1 solutions
Answer 127170 by ankor@dixie-net.com(15638) on 2008-12-08 19:16:01 (Show Source):
You can put this solution on YOUR website!(x-1) (x4+x3+x2+1)
:
Each degree has to be represented, so write it (x^4 + x^3 + x^2 + 0x + 1)
:
Multiply it the old fashioned way:
x^4 + x^3 + x^2 + 0x + 1
....................*..............x - 1
-----------------------------
answer: x^5-x^2+ x-1
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