Question 1050565: The circumference of the rear wheel of a unusual bicycle is 3 feet more than the circumference of the front wheel. The rear wheel makes 100 fewer rotations than the front wheel in traveling 6000 feet. Find the circumference of the rear wheel.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let f = circumference of the front wheel.
let r = circumference of the rear wheel.
you are given that the circumference of the rear wheel is 3 feet more than the circumference of the front wheel.
this means that r = f + 3.
when the bicycle travels 6000 feet, the number of rotations of the front wheel is equal to 6000 / f and the number of rotations of the rear wheel is equal to 6000 / r.
you are given that the number of rotations of the rear wheel is 100 less than the number of rotations of the front wheel.
this means that (6000 / r) = (6000 / f) - 100.
since you already know that r = f + 3, then this equation becomes:
(6000 / (f + 3) = 6000 / f - 100
if we multiply both sides of this equation by f * (f + 3), we get:
6000 * f = 6000 * (f + 3) - 100 * f * (f + 3).
we simplify this equation to get:
6000 * f = 6000 * f + 18000 - 100 * f^2 - 300 * f
if we subtract 6000 * f from both sides of the equation, we get:
0 = 18000 - 100 * f^2 - 300 * f
if we divide both sides of the equation by 100, we get:
0 = 180 - f^2 - 3 * f
if we multiply both sides of the equation by -1, we get:
0 = -180 + f^2 + 3f
if we reorder the terms of the equation in descending order of degree, we get:
0 = f^2 + 3f - 180.
this is the same as f^2 + 3f - 180 = 0
this is a quadratic equation.
we can factor this quadratic equation to get:
(f + 15) * (f - 12) = 0
we solve for x to get:
x = -15 or x = 12.
x can't be negative, so x = 12 should be our solution.
x = 12 is the circumference of the front wheel.
x = 15 is the circumference of the rear wheel.
6000 / 12 = 500
6000 / 15 = 400
the difference is 100 with the rear wheel rotating 100 less times than the front wheel.
the solution looks good.
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