SOLUTION: can anyone solve this problem? thank you for that someone.The circumference of a hind wheel of a wagon is 8 decimeters longer than that of the front wheel. In traveling one kilomet
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Question 356738: can anyone solve this problem? thank you for that someone.The circumference of a hind wheel of a wagon is 8 decimeters longer than that of the front wheel. In traveling one kilometer, the front wheel makes 62 ½ more revolutions than the hind wheel. Find the circumference of each wheel.
Found 2 solutions by Theo, Alan3354:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
It took a few tries because my mathematics was sloppy, even with a calculator, but I think I found the solution.
Here's how:
The main equation to work with is c*x = d
d = distance
c = circumference
x = number of revolutions
First thing you want to do is convert everything to a common measure.
I used meters.
1 kilometer = 1000 meters
8 decimeters = .8 meters
your formulas to work with are:
h * x = 1000
h is the circumference of the hind wheel.
f * (x + 62.5) = 1000
f is the circumference of the front wheel.
since they are both equal to 1000, then they are both equal to each other and you get:
h * x = f * (x + 62.5)
simplify this to get:
h * x = (f * x) + (62.5 * f)
you are given that h = f + .8 so you can substitute in this equation to get:
(f + .8) * x = (f * x) + (62.5 * f)
remove parentheses in the first equation to get:
(f * x) + (.8 * x) = (f * x) + (62.5 * f)
subtract (f * x) from both sides of the equation to get:
.8 * x = 62.5 * f
divide both sides of the equation by .8 to get:
x = 62.5 * f / .8 which becomes:
x = 78.125 * f
substitute for x in the equation of f * (x + 62.5) = 1000 to get:
f * ( 78.125 * f + 62.5) = 1000
simplify by removing parentheses to get:
78.125 * f^2 + 62.5 * f = 1000
subtract 1000 from both sides of the equation to get:
78.125 * f^2 + 62.5 * f - 1000 = 0
since this is the standard form of a quadratic equation, we get:
a = 78.125
b = 62.5
c = -1000
from this, we get:
2a = 156.25
-b = -62.5
b^2 = 3906.25
4ac = -312500
b^2 - 4ac = 3906.25 - (-312500) = 316406.25
sqrt(b^2-4ac) = sqrt(316406.25) = 562.5
f will be equal to ( -b +/- sqrt(b^2-4ac) ) / (2a)
this becomes f = (-62.5 +/- 562.5 ) / 156.25
you get:
f = 3.2 or f = -4
since f can't be negative, f must be equal to 3.2.
since h = f + .8, h must be equal to 4.
h * x = 1000 becomes 4 * x = 1000 which yields x = 1000 / 4 = 250
Let y = the number of revolutions of the smaller wheel.
You get f * y = 1000 which becomes 3.2 * y = 1000 which becomes y = 1000 / 3.2 = 312.5
y minus x yields 312.5 - 250 which equals 62.5.
y is equal to x + 62.5 which confirms the solution is good.
your equations are:
h * x = 1000 which becomes 4 * 250 = 1000 which becomes 1000 = 1000 which is true.
f * (x + 62.5) = 1000 which becomes 3.2 * (250 + 62.5) = 1000 which becomes 3.2 * 312.5 = 1000 which becomes 1000 = 1000 which is also true.
that's your answer.
the circumference of the front wheel is 3.2 meters.
the circumference of the hind wheel is 4 meters.
the number of revolutions of the front wheel is 312.
the number of revolutions of the hind wheel is 250.
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
The circumference of a hind wheel of a wagon is 8 decimeters longer than that of the front wheel. In traveling one kilometer, the front wheel makes 62 ½ more revolutions than the hind wheel. Find the circumference of each wheel.
-------------------
f = circumference of front
b = of rear
-----------
f = b+8
1 km = 10000 decimeters
---------
10000/f = revs of front
10000/b = revs of back
---------
10000/f = 10000/b - 62.5
10000/(b+8) = 10000/b - 62.5
10000b = 10000(b+8) - 62.5b(b+8)
6.25b^2 + 500b -10000b - 80000 = -10000b
125b^2 + 1000b - 20000b - 160000 = -20000b
125b^2 + 1000b - 160000 = 0
b^2 + 8b - 1280 = 0
(b - 32)*(b+40) = 0
b = 32 dm
f = 40 dm
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