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Quadratic_Equations/267481: x/(.900-2x)^2 = 4 How do I solve for x?
1 solutions
Answer 196433 by ankor@dixie-net.com(15649)   on 2010-02-09 19:28:15 (Show Source):
You can put this solution on YOUR website!x/(.900-2x)^2 = 4 How do I solve for x?
:
Assume the problem is:
 = 4
FOIL
 = 4
:
 = 4
:
x = 4(4x^2 - 3.6x + .81)
:
x = 16x^2 - 14.4x + 3.24
:
0 = 16x^2 - 14.4x - x + 3.24
:
16x^2 - 15.4x + 3.24 = 0
:
Solve this using the quadratic formula
Two solutions: x~.652 and x~.311
;
Check both of these in the original equation
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Linear_Algebra/267630: Tanya, who is a long distance runner, runs at an average velocity of 8mph. two hours after Tanya leaves your house, you leave in your Honda and follow the same route. If your average velocity of 40mph, how long will it be before you catch up with Tanya? How far will you be from your home?
1 solutions
Answer 196418 by ankor@dixie-net.com(15649) on 2010-02-09 19:02:41 (Show Source):
You can put this solution on YOUR website! runs at an average velocity of 8mph.
two hours after Tanya leaves your house, you leave in your Honda and follow the same route.
If your average velocity of 40mph, how long will it be before you catch up with Tanya?
:
Find the time required to catch up with the runner first
when that happens both will run the same distance, write a distance equation: Dist = speed*time
:
Let t = Honda's travel time
then
(t+2) = Runner's travel time
:
honda's dist = runner's dist
40t = 8(t+2)
40t = 8t + 16
40t - 8t = 16
32t = 16
t = 
t = .5 hrs, Honda's travel time (2.5 hrs = runner's travel time)
:
How far will you be from your home?
:
Find the dist using the Honda time:
.5(40) = 20 mi
Check the dist using the runner's time
2.5(8) = 20 mi also
|
Travel_Word_Problems/267440: Going an average of 65 mph, Bob can travel 500 miles per day. How far can he travel in a day if he increases his average speed to 75 mph? Show (explain) how you can calculate your answer without calculating the time Bob spent on the road.
1 solutions
Answer 196398 by ankor@dixie-net.com(15649) on 2010-02-09 17:58:37 (Show Source):
You can put this solution on YOUR website!Going an average of 65 mph, Bob can travel 500 miles per day.
How far can he travel in a day if he increases his average speed to 75 mph?
Show (explain) how you can calculate your answer without calculating the time Bob spent on the road
:
Use a ratio equation, d = dist at 75
 = 
Cross multiply
65d = 75 * 500
65d = 37500
d = 
d = 576.92 mi in a day
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Linear-systems/267425: i dont understand how to do this:
3x+2y=-6
x=y+3
1 solutions
Answer 196396 by ankor@dixie-net.com(15649) on 2010-02-09 17:51:29 (Show Source):
You can put this solution on YOUR website!i don't understand how to do this:
3x + 2y =-6
x = y+3
:
Since x = (y+3), why not substitute (y+3) for x in the 1st equation so you have:
3(y+3) + 2y = -6
:
Multiply what's inside the brackets by 3
3y + 9 + 2y = -6
;
Subtract 9 from both sides, add the y's
3y + 2y = -6 - 9
5y = -15
:
Divide both sides by 5
y = 
y = -3
:
Remember that x = y + 3; substitute -3 for y
x = (-3) + 3
x = 0
:
:
Check the solution by substituting for x and y in the 1st equation
3x + 2y =-6
3(0) + 2(-3) = -6
-6 = -6; confirms the solutions
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Linear_Equations_And_Systems_Word_Problems/267414: You inherit $13,000 and decide to invest the money in two different investments, one paying 10% and the other paying 14%. After 6 months your investments are worth $13,741. How much did you invest in each account?
1 solutions
Answer 196382 by ankor@dixie-net.com(15649) on 2010-02-09 16:54:06 (Show Source):
You can put this solution on YOUR website!You inherit $13,000 and decide to invest the money in two different investments, one paying 10% and the other paying 14%.
After 6 months your investments are worth $13,741.
How much did you invest in each account?
:
We know the interest earned = $741; write a total interest equation
Let x = amt invested at 14%
then
(13000-x) = amt invested at 10%
:
We assume the investment return % were per year, but this total is for 6 months
We have to assume that after 1 yr the amt earned would be 2*741 = $1482
Using that assumption:
:
.14x + .10(13000-x) = 1482
.14x + 1300 - .10x = 1482
.14x - .10x = 1482 - 1300
.04x = 182
x = 
x = $4,550 invested at 14%
and
13000 - 4550 = $8,450 invested at 10%
:
:
Check: .14(4550) + .10(8450) =
637 + 845 = 1482 per year
then
781 for half a year
|
Numbers_Word_Problems/267461: How to solve:
The differences between the square of a positive number and ten times the number 11. Find the positive number.
How to solve:
The height of a parallelogram is 2cm longer than its base. Its area is 120cm2. Find the base and height of the parallelogram.
1 solutions
Answer 196379 by ankor@dixie-net.com(15649) on 2010-02-09 16:32:51 (Show Source):
You can put this solution on YOUR website!The differences between the square of a positive number and ten times the number is 11.
Find the positive number.
x^2 - 10x = 11
x^2 - 10x - 11 = 0; our old friend, the quadratic equation!
Factors to
(x-11)(x+1) = 0
The positive solution
x = 11
Check: 11^2 - 10(11) = 11
:
How to solve:
The height of a parallelogram is 2cm longer than its base.
Its area is 120cm2. Find the base and height of the parallelogram.
:
Area of a parallelogram;
b * h = 120
"height is 2cm longer than its base."
h = (b+2)
Replace h with (b+2) in the area equation
b * (b+2) = 120
b^2 + 2b = 120
b^2 + 2b - 120 = 0; what luck! Another quadratic equation!
Factors to
(b-10)(b+12) = 0
Positive solution
b = 10 cm
then
h = 10+2 = 12 cm
;
Check 10 * 12 = 120 sq/cm
|
percentage/267426: I need help finding the regular price of an item before a tax was applied.
"After a discount of 37%, the sale price for a book was $32.09. What was the regular price.
I've tried many different equations...I'm missing something here.
x+.37x=32.09 ...I get 23.42. Please help...ty :)
1 solutions
Answer 196378 by ankor@dixie-net.com(15649) on 2010-02-09 16:13:34 (Show Source):
You can put this solution on YOUR website!"After a discount of 37%, the sale price for a book was $32.09. What was the regular price.
I've tried many different equations...I'm missing something here.
x+.37x=32.09 ...I get 23.42
:
The discount has to be subtracted from the regular price
x - .37x = 32.09
:
.63x = 32.09
x = 
x = $50.94 is the regular price
|
Travel_Word_Problems/267665: Lucas is 1/3 mile away from home bicycling at 20 miles per hour when his brother takes off on his bike to catch up. How fast must Lucas's brother go to catch him a mile from home.
1 solutions
Answer 196372 by ankor@dixie-net.com(15649) on 2010-02-09 15:49:48 (Show Source):
You can put this solution on YOUR website!Lucas is 1/3 mile away from home bicycling at 20 miles per hour when his brother takes off on his bike to catch up.
How fast must Lucas's brother go to catch him a mile from home.
:
Briefly, bro has to travel 1 mi in the same time that L travels 2/3 of a mile
:
Let s = bro's speed to accomplish difficult feat
:
Write a time equation: Time = dist/speed
:
bro's time = L's time
 = 
:
= 
:
= 
Cross multiply
s = 30 mph bro has pedal to catch L 1 mi from home
;
:
Check this by finding the times
1/30 = .033 hrs
.667/20 = .033 hrs (about 2 min)
|
Exponential-and-logarithmic-functions/267560: At 2pm the number of bacteria in a colony was 100, by 4pm it was 4000. Write an exponential function to model the population y of bacteria x hours after 2pm.
Please help
1 solutions
Answer 196361 by ankor@dixie-net.com(15649) on 2010-02-09 15:16:35 (Show Source):
You can put this solution on YOUR website!At 2pm the number of bacteria in a colony was 100, by 4pm it was 4000.
Write an exponential function to model the population y of bacteria x hours after 2pm.
:
Ao*k^x = A
where
Ao is the initial amt
k = constant factor
x = time in hrs
A = resulting amt
;
Write an equation from the 1st statement, x = 2 hrs, find k
100 * k^2 = 4000
k^2 = 
k^2 = 40
k = 
k = 6.3246
:
The equation
y = 100(6.3246^x)
|
Mixture_Word_Problems/267615: How much water should be added to 1 gallon of pure antifreeze to obtain a solution that is 50% antifreeze?
1 solutions
Answer 196354 by ankor@dixie-net.com(15649) on 2010-02-09 15:04:53 (Show Source):
You can put this solution on YOUR website!How much water should be added to 1 gallon of pure antifreeze to obtain a solution that is 50% antifreeze?
:
1 gal of water, seems pretty obvious, but it you want an equation where x = amt of water
:
x = .5(x+1)
x = .5x + .5
x - .5x = .5
.5x = .5
x = 1 gal
|
Geometry_Word_Problems/267524: Need help with the following:
A box is being constructed by cutting 3-inch squares from the corners of a rectangualr sheet of metal that is 4 inches longer than it is wide. If the box is to have a volume of 135 cubic inches, find the dimensions of the metal sheet.
1 solutions
Answer 196347 by ankor@dixie-net.com(15649) on 2010-02-09 14:48:29 (Show Source):
You can put this solution on YOUR website!A box is being constructed by cutting 3-inch squares from the corners of a rectangular sheet of metal that is 4 inches longer than it is wide.
If the box is to have a volume of 135 cubic inches, find the dimensions of the metal sheet.
:
Assume they mean an open box.
:
Cutting 3" squares from each corner make the box base dimensions 6" less than the metal sheet
We know the height of the box = 3" therefore:
Area of the base =  = 45 sq/in
:
Let x = width of the metal
then we can say:
(x+4) = length of the metal
:
The base of the box dimensions: (x-6) by (x+4-6)
The area
(x-6)*(x-2) = 45
FOIL
x^2 - 2x - 6x + 12 = 45
x^2 - 8x + 12 - 45 = 0
x^2 - 8x - 33 = 0
Factors to
(x-11)(x+3) = 0
positive solution
x = 11" is the width of the metal sheet
then
11+4 = 15" is the length
:
:
See if that's true
(11-6) * (15-6) * 3 =
5 * 9 * 3 = 135 cu/in
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Travel_Word_Problems/267516: How far can a man drive out into the country at an average rate of 30 mph and return over the SAME road at the average of 50 mph, if he travels total time of 8 hours?
1 solutions
Answer 196339 by ankor@dixie-net.com(15649) on 2010-02-09 14:25:15 (Show Source):
You can put this solution on YOUR website!How far can a man drive out into the country at an average rate of 30 mph and return over the SAME road at the average of 50 mph, if he travels total time of 8 hours?
:
Let t = time driving at 50 mph
then
(8-t) = time driving at 30 mph
:
Write a distance equation, Dist = speed * time
:
Dist at 50 mph = dist at 30 mph
50t = 30(8-t)
50t = 240 - 30t
50t + 30t = 240
80t = 240
t = 
t = 3 hrs at 50 mph
then
8 - 3 = 5 hrs at 30 mph
:
Find the dist at 50 mph
3 * 50 = 150 mile
Check the dist at 30 mph, (should be the same)
5 * 30 = 150 miles
:
He can go 150 mi and satisfy the 8 hr requirement.
|
Rectangles/267515: a rectangle has an area that is numerically twice its perimeter. if the length is twice the width, what are the dimensions?
1 solutions
Answer 196325 by ankor@dixie-net.com(15649) on 2010-02-09 13:28:04 (Show Source):
You can put this solution on YOUR website!a rectangle has an area that is numerically twice its perimeter.
:
L*W = 2(2L+2W)
:
If the length is twice the width, what are the dimensions?
:
Replace L with 2W
(2W)*W = 2[2(2W) + 2W)]
2W^2 = 2(4W + 2W)
2W^2 = 2(6W)
;
divide both sides by 2
W^2 = 6W
W^2 - 6W = 0
:
factor out W
W(W - 6) = 0
:
Two solutions
W = 0, meaningless
and
W = 6 units is the width
then
2(6) = 12 units is the length
:
:
Is this true?
12*6 = 2(2(12) + 2(6))
72 = 2(24 + 12),
|
Travel_Word_Problems/267467: Two planes leave an airport 30 minutes apart. The first planes flies east and the second west. After 75 minutes the planes are 700 miles apart. How fast is each flying?
I know that S X T = D. I know that the East plane flys for 1.25hrs at x speed so its distance is 1.25x. I know that the West plane flys for .75hrs at y speed so its distance is .75x. I know that if you add the two distances together you will get 700 miles. 1.25x + .75y = 700. I just don't know where to go next. Can you help?
1 solutions
Answer 196324 by ankor@dixie-net.com(15649) on 2010-02-09 13:11:56 (Show Source):
You can put this solution on YOUR website!Two planes leave an airport 30 minutes apart. The first planes flies east and the second west.
After 75 minutes the planes are 700 miles apart.
How fast is each flying?
I know that S X T = D. I know that the East plane flys for 1.25hrs at x speed so its distance is 1.25x. I know that the West plane flys for .75hrs at y speed so its distance is .75x. I know that if you add the two distances together you will get 700 miles. 1.25x + .75y = 700.
:
The problem is there are many combinations of x & y that will work here
Using your equation:
.75y = -1.25x + 700
divide by .75
y = -1.67x + 933.33
you can choose a value for x, say 300 mph, and find y = 432.33
or: x = 500, y = 98.33
an integer solution: x=399, y=267
an infinite number of solutions
|
Expressions-with-variables/267422: Here is a confusing question for me: The ratio of Aldo's cycling speed to Jose's cycling speed is 6:5. Jose leaves school at 3 P.M. and Aldo leaves at 3:10 P.M. By 3:30, Aldo is only 2 km behind Jose. How fast is each cycling?
1 solutions
Answer 196322 by ankor@dixie-net.com(15649) on 2010-02-09 12:42:42 (Show Source):
You can put this solution on YOUR website!The ratio of Aldo's cycling speed to Jose's cycling speed is 6:5.
Let m = the multiplier
then
6m = A's speed
and
5m = J's speed
:
Jose leaves school at 3 P.M. and Aldo leaves at 3:10 P.M. By 3:30,
J's travel time 30 min;  hr
A's travel time 20 min;  hr
:
By 3:30, Aldo is only 2 km behind Jose. How fast is each cycling?
;
Write a distance equation. Dist = time * speed
J's dist - A's dist = 2 km
(5m) - (6m) = 2
Multiply by 6 to get rid of the denominators, results
3(5m) - 2(6m) = 6(2)
15m - 12m = 12
3m = 12
m = 4 is the multiplier
then
4(6) = 24 km/hr is A's speed
and
4(5) = 20 km/hr is J's speed
:
:
Check solution by finding the dist traveled by each
J: .5(20) = 10 km
A: .33(24)= 8 km
---------------------
difference = 2 km
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Rectangles/267275: A box is 720 cubic centimeters. The height of the box is five centimeters and width is w centimeters. If the length of the box is four times the width, what will be the width in cnetimeters of the box?
1 solutions
Answer 196320 by ankor@dixie-net.com(15649) on 2010-02-09 12:23:33 (Show Source):
You can put this solution on YOUR website!A box is 720 cubic centimeters.
The height of the box is five centimeters and width is w centimeters.
If the length of the box is four times the width,
what will be the width in centimeters of the box?
:
The volume equation
L * W * H = 750
:
L * W * 5 = 750
:
5LW = 750
LW = 
LW = 150
:
"the length of the box is four times the width"
L = 4W
:
Replace L with 4W
W*4W = 150
4W^2 = 150
W^2 = 
W^2 = 37.5
W = 
W = 6.1237 cm is the width
:
:
See if that is true, check on a calc:
4(6.1237) * 6.1237 * 5 = 749.997 ~ 750
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Miscellaneous_Word_Problems/267398: A car traveling on a road that is perpendicular to a railroad track. When the car is 30 meters from the crossng, the car's new collision detector warns the driver that there is a train 50 meters from the car and heading toward the same crossing. How far is the train from the crossing?
1 solutions
Answer 196318 by ankor@dixie-net.com(15649) on 2010-02-09 12:12:46 (Show Source):
You can put this solution on YOUR website!A car traveling on a road that is perpendicular to a railroad track.
When the car is 30 meters from the crossing, the car's new collision detector warns the driver that there is a train 50 meters from the car and heading toward the same crossing.
How far is the train from the crossing?
:
This is a pythag problem: a^2 + b^2 = b^2
Where:
c = 50,
a = 30
b = dist the train is from the crossing
:
30^2 + b^2 = 50^2
900 + b^2 = 2500
b^2 = 2500 - 900
b^2 = 1600
b = 
b = 40 meters, is the train from the crossing
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Graphs/267219: what does it mean to draw a line with a slope of 5
1 solutions
Answer 196288 by ankor@dixie-net.com(15649) on 2010-02-09 09:50:37 (Show Source):
You can put this solution on YOUR website!what does it mean to draw a line with a slope of 5
:
Remember the definition of slope is "rise over run"
:
A slope of 5 is actually  where rise = 5 and run = 1
:
We can say for every 1 unit horizontally, we have 5 units vertially
:
an equation y = 5x would look like this
:
note when x (run) = 1, y (rise) = 5
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Exponents/267263: A polynomial in x has degree 3. The coefficient of x^2 is 3 less than the coefficient of x^3. The coefficient of x is three times the coefficient of x^2. The remaining coefficient is 2 more than the coefficient of x^3. The sum of the coefficients is -4. Find the polynomial.
1 solutions
Answer 196282 by ankor@dixie-net.com(15649) on 2010-02-09 09:43:02 (Show Source):
You can put this solution on YOUR website!We will try to get every variable in terms of a
:
A polynomial in x has degree 3.
ax^3 + bx^2 + cx + d
:
The coefficient of x^2 is 3 less than the coefficient of x^3.
b = a - 3
:
The coefficient of x is three times the coefficient of x^2.
c = 3b
Replace b with (a-3)
c = 3(a-3)
c = 3a - 9
:
The remaining coefficient is 2 more than the coefficient of x^3.
d = a + 2
:
The sum of the coefficients is -4. Find the polynomial.
a + b + c + d = -4
Substitute, solve for a
a + (a-3) + (3a-9) + (a+2) = -4
6a - 10 = -4
6a = -4 + 10
6a = 6
a = 1
then
b = 1 - 3
b = -2
and
c = 3(1) - 9
c = -6
:
d = 1 + 2
d = 3
:
x^3 - 2x^2 - 6x + 3; is the polynomial
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Human-and-algebraic-language/267214: Translate the following sentence into an algebraic equation:"6 more than triple a number is 3 times the difference of a number & 2."
1 solutions
Answer 196278 by ankor@dixie-net.com(15649) on 2010-02-09 09:25:07 (Show Source):
You can put this solution on YOUR website!Translate the following sentence into an algebraic equation:
"6 more than triple a number is 3 times the difference of a number & 2."
:
Just write what it says, the word "is" usually mean equal "="
6 + 3x = 3*(x - 2)
but we usually write this as
3x + 6 = 3(x-2)
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Travel_Word_Problems/267173: A woman rides a bicycle at 12 miles per hour. What is her speed in
yards per minute?
If light can travel 186,000 miles per second, how many miles can it
travel in one year? (Assume one year has 365.24 days.)
1 solutions
Answer 196270 by ankor@dixie-net.com(15649) on 2010-02-09 08:23:21 (Show Source):
You can put this solution on YOUR website!a bicycle at 12 miles per hour.
What is her speed in yards per minute?
:
There are 1760 yds in 1 mile
:
12 mph is:  = 352 yds/min
:
If light can travel 186,000 miles per second, how many miles can it
travel in one year? (Assume one year has 365.24 days.)
:
60 * 60 * 24 * 365.24 * 186,000 ~ 5.87 (10^12)
which is
5,869,552,896,000 miles on my calc
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Mixture_Word_Problems/267282: A chemist wishes to mix some pure acid with some water to produce 16L of a solution that is 30% acid. How much pure acid and how much water should be mixed
1 solutions
Answer 196262 by ankor@dixie-net.com(15649) on 2010-02-08 22:31:40 (Show Source):
You can put this solution on YOUR website!A chemist wishes to mix some pure acid with some water to produce 16L of a solution that is 30% acid.
How much pure acid and how much water should be mixed
:
let x = amt of pure acid
:
x = .30(16)
x = 4.8 liters of pure acid
then
16-4.8 = 11.2 liters of water
:
:
Check:  = .3 or 30% acid
and:  = .7 or 70% water
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Graphs/267223: what does it mean to draw a line with a slope of -1 over 4
1 solutions
Answer 196259 by ankor@dixie-net.com(15649) on 2010-02-08 21:55:11 (Show Source):
You can put this solution on YOUR website!what does it mean to draw a line with a slope of -1 over 4
:
One definition of slope is "Rise over Run" which means vertical over horizontal
This slope is  , so it drops 1 unit vertically for every 4 horizontal units
If you graph x, it would look like this
note that when x=4, y=-1
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Numbers_Word_Problems/267378: how do i solve and write two equations for the sum of two numbers is 30. the sum of the greater number and 3 times the lesser number is 54. Find both numbers?
1 solutions
Answer 196254 by ankor@dixie-net.com(15649) on 2010-02-08 21:39:47 (Show Source):
You can put this solution on YOUR website!how do i solve and write two equations:
for the sum of two numbers is 30.
x + y = 30
:
the sum of the greater number and 3 times the lesser number is 54.
x + 3y = 54
x + y = 30; subtract the 1st equation from the 2nd equation, eliminate x
------------
2y = 24
y = 12
then
x = 30 - 12 = 18
;
:
Check solution in the statement
"the sum of the greater number and 3 times the lesser number is 54."
18 + 3(12) = 54
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