Precalculus Help!

Question

Hi! I'm struggling greatly with these last two questions for my homework assignment. I've already attempted it like 5 times and I keep getting lost. Any help is welcome please!

1.    The rear wheels of DeMarius’ car complete 4/5 of a rotation for every full rotation of a front wheel. What is the radius, in feet, of a rear wheel on the car? Write your answer as a simplified fraction.

 (Score for Question 6: ___ of 3 points)

2.    DeMarius’ original design for his car used rear wheels with a radius of 12 inches. What is the measure of the central angle of this rear wheel such that the arc length is equivalent to that of a full rotation of the rear wheel that is actually used on DeMarius’ car?

Answer:

Answer 1

These questions are written in a... misleading way?

First of all, to clarify, arc length and radians are the same thing. First, of course, Pi is the ratio between a perfect circle's circumference and its diameter. So, a circle with Radius 1 has a circumference of 2*Pi, since its diameter is then 2. Second, a radian is really just a measure of this circumference, related to the amount of a full rotation taken along the circle. In a unit circle, with a radius of 1, taking any particular section of a pie piece from this circle results in a similar ratio of its circumference.

All that is a bunch of definitions, but basically, by taking the arc angle of a piece of pie, and dividing that by a full rotation gives you a ratio, which is the ratio of the circumference of the pie piece versus the circumference of the circle. In simplest terms, the radians are just a measure of a circle's perimeter.

The first question is swaying that one full rotation of the rear wheels are worth 4/5 of a rotation of the front wheel. In the comments, the front wheel's radius was given as 8, and I'll take 8 as generic units rather than inches or feet, for example.

this would make the circumference of a front wheel 16pi. The circumference of the rear wheel is then 4/5 of this, or 12.8pi units. Dividing by 2pi gives the radius of the rear wheels as approximately 2.037 units.

For the second question, it is in a rather confusing way asking for the equivalent arc angle of a pie piece of a wheel 12 units in radius, where that angle's arc length is the same as the perimeter or circumference of the rear wheel we just calculated for.

So, we can take the circumference of the wheel 12 units in radius, which is 2pi * 12 or 24pi, and then divide 12.8pi by this to find the amount of a full rotation of the 12 unit wheel a 2.037 unit wheel is worth.

That calculation provides 12.8/24, so a little over on half at 0.5333

This multiplied by 360 degrees gives us 192 degrees of the planned wheel.

So, aside from the actual math, a really quick intuitive explanation of the second question is, if you had the actual rear wheels with radius 2.037 units, and the planned wheel with a radius of 12 units, and you rolled them against each other, how far along the 12 unit wheel would the 2.037 unit wheel roll along the larger wheel. this proportion gives you exactly just how far that 12 unit wheel would have to roll on the actual ground to go the same distance.