<PRE><B>Question 1116861</B>
let the circumference of the front wheel be f


since the circumference of the front wheel is 2 less than the circumference of the rear wheel, then the circumference of the rear wheel is f + 2.


let the number of revolutions that the rear wheel makes in order to travel 3500 feet be equal to x.


since the front wheel makes 200 more revolutions than the rear wheel to travel the same distance, then the number of revolutions that the front wheel makes in order to travel 3500 feet is equal to x + 200.


circumference * revolutions = distance traveled.


for the front wheel, the equation becomes 3500 = f * (x + 200)


for the rear wheel, the equation becomes 3500 = (f + 2) * x.


simplify both equations to get:


3500 = fx + 200f for the front wheel.


3500 = fx + 2x for the rear wheel.


subtract the rear wheel equation from the front wheel equation to get:


0 = 200f - 2x


solve for x to get x = 100f.


in the equation for the rear wheel, replace x with 100f to get:


3500 = fx + 2x becomes 3500 = f * 100f + 200f


this becomes 3500 = 100f^2 + 200f


subtract 3500 from both sides of the equation and flip sides to get:


100f^2 + 200f - 3500 = 0


divide both sides of this equation by 100 to get:


f^2 + 2f - 35 = 0


factor this quadratic equation to get:


(f-5) * (f+7) = 0


solve for f to get f = -7 or f = 5


f can&#39;t be negative, therefore f = 5 has to be the solution, if at all.


this means the circumference of the front wheel is 5 and the circumference of the real wheel is 7.


go back to your two original equations, which are:


3500 = f * (x + 200) for the front wheel.
3500 = (f + 2) * x for the rear wheel.


replace f with 5 in each equation to get:


3500 = 5 * (x + 200) for the front wheel.
3500 = 7 * x for the rear wheel.


solve for x in both equations to get x = 500 for both equations.


your original equation are, once again:


3500 = f * (x + 200) for the front wheel.
3500 = (f + 2) * x for the rear wheel.


when f = 5 and x = 500, these equations become:


3500 = 5 * 700 for the front wheel.
3500 = 7 * 500 for the rear wheel.


since both original equations are true, the solution looks good.


the solution is:


the circumference of the front wheel is 5 feet.
the circumference of the rear wheel is 7 feet.








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