SOLUTION: You work at a pioneer historical site. On this site you have handcarts. One cart has a handle that connects to the center of the wheel. We have to maintain the handle of the cart a

Question 1119202: You work at a pioneer historical site. On this site you have handcarts. One cart has a handle that connects to the center of the wheel. We have to maintain the handle of the cart at an angle of no more than 20� with the ground so the contents do not spill out. The distance from where the handle rests on the ground to the point where the wheel is sitting on the ground is 45 inches. The distance of the center of the wheel to the end of the handle is approximately 48 inches.
picture: https://api.agilixbuzz.com/Resz/~PW4LEAAAAAQsZBc1R9ciiB.CLm56DbZ6T1uAwIk1KSyJA/48780739,5F4,0,0/Assets/Media/Images/43.3-HOT2-Handcart.jpg
a. Identify the parts of the handcart wheel that would represent congruent chords and congruent central angles. Explain why.
b. Find the radius of the wheel.
c. If the measure of the arc from A to B around the outside of the wheel were changed to 72�, what is the new angle the handle makes with the ground? Will the contents remain in the handcart at that angle? Will the handle rest on the ground?
d. If a pioneer pulling the handcart held the handle at a height of 48 inches off the ground, would the contents of the cart spill out the back? How high can the pioneer lift the handle off the ground before the contents started spilling out?
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

I don't see that it is possible to answer part a. There are no parts of the wheel that are chords. Each of the wheel spokes is a radius, but there are no chords.

As for the angles, it APPEARS that there are 12 spokes making congruent 30 degree central angles; but there is nothing in the statement of the problem that tells us so.

part b: If r is the radius, then

approximately

With the given measurements, we can also calculate the radius with

approximately

So the given measurements are consistent only to 1 or 2 decimal places. So let's call the radius 16.4 inches.

part c: If the arc AB is increased from 70 to 72 degrees, then the angle the handle makes with the ground is changed to 18 degrees, so the contents will remain on the cart. And since the handle doesn't change length, the end of the handle will no longer touch the ground.

In the original configuration, the angle the bed of the cart makes with the ground is 20 degrees and the radius of the wheel is 16.4 inches. When someone lifts the handle of the cart off the ground, the maximum height he can raise the handle, in order to keep the angle the bed makes with the ground at most 20 degrees, is twice the radius, or 32.8 inches.

So if the pioneer lifts the handle to 48 inches above the ground, the contents of the cart will spill out long before he gets the handle to that height.
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