Question

Level 4

3.
One of the largest sewing machines in the world has a flywheel (which turns as the machine sews) that is 5
feet in diameter. The highest point of the handle at the edge of the flywheel is 9 feet above the ground, and
the lowest point is 4 feet. The wheel makes a complete turn every 2 seconds. Write a model for the height h
(in feet) of the handle as a function of the time t (in seconds) given that the handle is at its lowest point when
t = 0.

Answer 1

To solve this problem, let's break it down step by step.

1. Find the difference in height between the highest and lowest point of the handle:

The difference in height is 9 feet - 4 feet = 5 feet.

2. Calculate the circumference of the flywheel:

The circumference of a circle can be found using the formula C = πd, where C is the circumference and d is the diameter.

The diameter is given as 5 feet, so the circumference is C = π(5) = 5π feet.

3. Determine the vertical distance covered by the handle in one revolution of the flywheel:

Since the handle is attached to the edge of the flywheel, it moves in a circular path as the flywheel rotates.

The circumference of the flywheel represents the distance covered by the handle in one revolution.

Therefore, the vertical distance covered is also 5π feet.

4. Calculate the speed of the handle:

The wheel makes a complete turn every 2 seconds, which means it completes one revolution in 2 seconds.

Therefore, the speed of the handle is the vertical distance covered (5π feet) divided by the time taken (2 seconds):

Speed = (5π feet) / (2 seconds) = (5/2)π feet per second.

So, the handle of the sewing machine moves at a speed of (5/2)π feet per second.