MHB Solving Tire Nail Height & Distance Problems: Grade 11

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The discussion revolves around solving a problem related to the height of a nail picked up by a car tire with a diameter of 52 cm. There is confusion regarding the graph of the nail's height, as the user's graph shows maximum height at 52 cm while the textbook indicates a maximum of 26 cm, leading to questions about potential errors in the textbook. For part b, the user calculates the number of tire rotations after traveling 0.1 km but struggles with the subsequent calculations. A suggestion is made to derive a function for the nail's height based on the distance traveled and the tire's rotation, emphasizing the need to consider the tire's diameter change after the puncture. The overall consensus is that the problem's parameters need clarification for accurate resolution.
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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! (:
 
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eleventhxhour said:
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.

eleventhxhour said:
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)
 
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