Composite function bike problem

Question

When an 18-speed touring bicycle is in sixth gear, the gear ratio is 3:2, which means that the wheels of the bicycle revolve 3 times for every 2 revolutions of the pedals. This relationship can be expressed as w = 3/2 p

where w and p represent wheel and pedal revolutions, respectively. Since the wheels of a touring bicycle have a diameter of 27 inches, w revolutions of the wheels move the bicycle a distance d, in inches, given by: d=27πw

The composite function d=81/2 πp will give the distance as a function of pedal revolutions.

How many wheel revolutions are made if the pedals make 100 revolutions = Answer

What distance is travelled when the wheels make 100 revolutions? (round answer to the nearest tenth) = Answer

How many revolutions do the pedals make when the bicycle travels 1 mile? (round to nearest whole revolution) = Answer

Answer 1

Hello, Mike,

Using the formulas a noted, 100 revolutions of the pedals will produce 150 revolutions of the wheels. (w=(3/2)p)

The distance for 100 wheel revolutions is 25,604.0 inches.(distance = (81.5)(pi)*100)

To calculate how many pedal revolutions are required for the bike to travel one mile, we can rearrange one equation and use a conversion factor.

Since distance, d, (in inches) = (81.5)(pi)(w), we can state:

w(revolutions) = d/(81.5*pi), where d is in inches

But we have miles, not inches, for the distance ridden. 1 mile. Convert that into inches:

1 mile = 5280 feet

1 foot = 12 inches

Therefore, we have the conversion factors: 12 in/foot and 5280 feet/mile.

Then: 1 mile * 5280 feet/mile * 12 in/foot = the distance in inches. See how the units cancel. The first step removes the mile and leaves just feet. The second step removes the feet and leaves just inches.

Thus, 1 mile = 63,360 inches.

To find the number of wheel revolutions in 1 mile, take the earlier equation:

w(revolutions) = d/(81.5*pi), where d is in inches, and use 63,360 inches for d:

w = 63,360 in/(81.5*pi) = 247 revolutions of the wheel (247.46153, exactly)

To get revolutions of the pedals, w = (3/2)p, so rearrange:

p = (2/3)w

p = (2/3)*(247) = 165 revolutions of the pedals.

I hope this helps,

Bob