Problem 99 - Algebra and Trigonometry

Question

A water wheel has a radius of 12 feet. The wheel is rotating at 20 revolutions per minute. Find the linear speed, in feet per minute, of the water.

Step-by-Step Solution

Answer

The linear speed of the water is \(480\pi\) feet per minute.

Step 1: Understand the Problem and Gather the Data

We need to find the linear speed with which the water is moving. We know the radius of the wheel (\(12\) feet) and the rotational speed (\(20\) revolutions per minute). Note that the distance covered in one revolution is the circumference of the wheel.

Step 2: Calculate the Circumference of the Wheel

To find the linear speed, we first need to calculate the distance travelled in one revolution, which is the circumference of the wheel (circle). The formula for the circumference of a circle is \(C = 2\pi r\), where \(r\) is the radius. Substitute \(r = 12\) feet into the formula to find \(C = 2 \times \pi \times 12 = 24\pi\) feet.

Step 3: Determine the Linear Speed

Now that we have the circumference, we can calculate the distance the wheel 'moves' the water in one minute. Since the wheel goes through 20 revolutions per minute and one revolution moves the water 24\(\pi\) feet, in one minute, the wheel moves the water \(20\times24\pi = 480\pi\) feet. So, the linear speed of the water is \(480\pi\) feet per minute.