To find the truck's linear speed in feet per second, we follow these steps:

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Step 1: Linear Speed Formula

The linear speed $v$ of a point on the rim of a wheel is related to the angular speed $\omega$ and the radius $r$ by: $v = r \cdot \omega$

Here:

• $r$ is the radius of the wheel in feet.
• $\omega$ is the angular speed in radians per second.

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Step 2: Convert Radius to Feet

The radius is given as 15 inches. To convert to feet: $r = \frac{15}{12} = 1.25 \, \text{feet}.$

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Step 3: Convert Rotations per Minute to Radians per Second

The angular speed is given as $75 \, \text{rotations per minute (rpm)}$. Since one rotation corresponds to $2\pi \, \text{radians}$, we first convert to radians per minute: $\omega = 75 \, \text{rotations/minute} \cdot 2\pi \, \text{radians/rotation} = 150\pi \, \text{radians/minute}.$

Next, convert minutes to seconds: $\omega = \frac{150\pi}{60} = 2.5\pi \, \text{radians/second}.$

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Step 4: Calculate Linear Speed

Using the formula $v = r \cdot \omega$: $v = 1.25 \cdot 2.5\pi = 3.125\pi \, \text{feet/second}.$

Approximating $\pi \approx 3.1416$: $v \approx 3.125 \cdot 3.1416 \approx 9.82 \, \text{feet/second}.$

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Final Answer:

The truck's linear speed is approximately: $\boxed{9.82 \, \text{feet/second}}.$

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Would you like to go deeper into the calculations or learn about related concepts?

Here are 5 related questions for practice:

1. If the angular speed doubles, what happens to the linear speed?
2. How would the linear speed change if the wheel radius were 20 inches instead of 15 inches?
3. What is the angular speed in degrees per second instead of radians per second?
4. How far does the truck travel in one minute at this speed?
5. What is the formula for converting linear speed back to angular speed?

Tip: Always confirm units before starting any calculation!