Solving Tire Nail Height & Distance Problems: Grade 11

[eleventhxhour]

8) The diameter of a car's tire is 52cm. While the car is being driven, the tire picks up a nail.
a) Draw a graph of the height of the nail above the ground in terms of the distance the car has traveled since the tire picked up the nail.

For this part, I drew a graph with the max 52, the mid 26 and the min 0. However, the graph in the textbook has the max as 26, the mid as 0 and the min as -26. Why is this?

b) How high above the ground will the nail be after the car has traveled 0.1km?
c) How far will the car have traveled when the nail reaches a height of 20cm above the ground for a fifth time?

And then I'm not really sure how to do b and c. I got that there would be 61.23 full rotations (10 000 / 163.28 = 61.23) for b, but then I'm not sure what to do after that.

Could someone explain in relatively simple terms? (this is just grade 11).

Thanks! (:

 

[Evgeny.Makarov]

For this part, I drew a graph with the max 52, the mid 26 and the min 0. However, the graph in the textbook has the max as 26, the mid as 0 and the min as -26. Why is this?

This must be a mistake on the textbook's part.

b) How high above the ground will the nail be after the car has traveled 0.1km?
...
And then I'm not really sure how to do b and c.

I recommend writing explicitly the function whose graph you drew for part a). Note that the angle by which the wheel turned since the fateful moment is $\varphi=x/(\pi d)$ where $x$ is the traveled distance in centimeters. Turn $\varphi$ into height $h(x)$. Once you have a formula for $h(x)$, just compute $h(10^4)$.

And, of course, the diameter of the wheel changed after it was punctured, so the rest of the problem does not make sense. (Smile)