Question

You work as 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 at an angle no more than 20 degrees with the ground so that nothing spills out of it. The distance from which 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 about 48 inches.

1. What is the radius of this wheel?
2. If the measure of the arc from A to B around the outside of the wheel was changed to 72 degrees, what is the new angle the handle makes with the ground? Would the contents spill or remain in the cart?
3. If a pioneer pulling the handcart held the handle at the height of 48 inches off the ground, would the contents spill out the back? How high can the pioneer lift the handle from the ground before contents start to spill?

Answer 1

Final Answers

1. The radius of the wheel is approximately 22.5 inches.

2. The new angle the handle makes with the ground is about 24 degrees. The contents would spill out of the cart.

3. If the pioneer lifts the handle to a height of 48 inches off the ground, the contents will spill out the back. The maximum height the pioneer can lift the handle before spillage is approximately 51.5 inches.

Explanation:

To find the radius of the wheel (question 1), we can use trigonometry. Given that the distance from the point where the handle rests on the ground to the wheel's center is 45 inches and the handle's total length is 48 inches, we utilize the sine function. Sin(θ) = opposite/hypotenuse. By rearranging the formula to find the radius (opposite side), the radius is calculated as 48 * sin(20 degrees) ≈ 22.5 inches.

For question 2, with the change in arc measure from 72 degrees (original) to 72 degrees, the new angle the handle makes with the ground can be calculated by applying arc length formula: arc length = radius * θ. Given the new arc length and radius, the handle angle is approximately 24 degrees, causing spillage since it exceeds the 20-degree limit.

Regarding question 3, if the pioneer lifts the handle to 48 inches, surpassing the 45-inch distance to the wheel center, the contents would spill. By calculating the maximum height, we consider the length of the handle from the wheel center. Adding the 45-inch distance to the allowed angle (20 degrees), the maximum lift height is around 51.5 inches, above which the contents will start spilling out.