Question

each of exercises 25–30 gives the speed of a revolving gear. find (a) the number of degrees per minute through which each gear turns and (b) the number of radians per minute. give answers to the nearest hundredth. 25. 35 rpm 26. 27 rpm 27. 2.5 rpm 28. 6.5 rpm 29. 14.6 rpm 30. 19.8 rpm

Explanation:

Let's solve these problems one by one. We'll take each gear speed (in rpm, revolutions per minute) and find the degrees per minute and radians per minute. Recall that one full revolution is \(360^\circ\) or \(2\pi\) radians.

Problem 25: 35 rpm

Part (a): Degrees per minute

A gear turning at 35 rpm means it makes 35 full revolutions per minute. Each revolution is \(360^\circ\), so we multiply the number of revolutions per minute by \(360^\circ\) per revolution.

Step 1: Calculate degrees per minute

Degrees per minute \(= 35 \times 360^\circ\)
\(= 12600^\circ\) per minute (since it's already a whole number, no need to round to the nearest hundredth here, but let's check the formula)

Part (b): Radians per minute

Each revolution is \(2\pi\) radians, so we multiply the number of revolutions per minute by \(2\pi\) radians per revolution.

Step 2: Calculate radians per minute

Radians per minute \(= 35 \times 2\pi\)
\(= 70\pi \approx 70 \times 3.1416 \approx 219.91\) radians per minute (rounded to the nearest hundredth)

Problem 26: 27 rpm

Part (a): Degrees per minute

Step 1: Calculate degrees per minute

Degrees per minute \(= 27 \times 360^\circ\)
\(= 9720^\circ\) per minute

Part (b): Radians per minute

Step 2: Calculate radians per minute

Radians per minute \(= 27 \times 2\pi\)
\(= 54\pi \approx 54 \times 3.1416 \approx 169.65\) radians per minute

Problem 27: 2.5 rpm

Part (a): Degrees per minute

Step 1: Calculate degrees per minute

Degrees per minute \(= 2.5 \times 360^\circ\)
\(= 900^\circ\) per minute

Part (b): Radians per minute

Step 2: Calculate radians per minute

Radians per minute \(= 2.5 \times 2\pi\)
\(= 5\pi \approx 5 \times 3.1416 \approx 15.71\) radians per minute

Problem 28: 6.5 rpm

Part (a): Degrees per minute

Step 1: Calculate degrees per minute

Degrees per minute \(= 6.5 \times 360^\circ\)
\(= 2340^\circ\) per minute

Part (b): Radians per minute

Step 2: Calculate radians per minute

Radians per minute \(= 6.5 \times 2\pi\)
\(= 13\pi \approx 13 \times 3.1416 \approx 40.84\) radians per minute

Problem 29: 14.6 rpm

Part (a): Degrees per minute

Step 1: Calculate degrees per minute

Degrees per minute \(= 14.6 \times 360^\circ\)
\(= 5256^\circ\) per minute

Part (b): Radians per minute

Step 2: Calculate radians per minute

Radians per minute \(= 14.6 \times 2\pi\)
\(= 29.2\pi \approx 29.2 \times 3.1416 \approx 91.73\) radians per minute

Problem 30: 19.8 rpm

Part (a): Degrees per minute

Step 1: Calculate degrees per minute

Degrees per minute \(= 19.8 \times 360^\circ\)
\(= 7128^\circ\) per minute

Part (b): Radians per minute

Step 2: Calculate radians per minute

Radians per minute \(= 19.8 \times 2\pi\)
\(= 39.6\pi \approx 39.6 \times 3.1416 \approx 124.41\) radians per minute

Summary of Answers:

• 25. (a) \(12600.00^\circ\) per minute, (b) \(\approx 219.91\) radians per minute
• 26. (a) \(9720.00^\circ\) per minute, (b) \(\approx 169.65\) radians per minute
• 27. (a) \(900.00^\circ\) per minute, (b) \(\approx 15.71\) radians per minute
• 28. (a) \(2340.00^\circ\) per minute, (b) \(\approx 40.84\) radians per minute
• 29. (a) \(5256.00^\circ\) per minute, (b) \(\approx 91.73\) radians per minute
• 30. (a) \(7128.00^\circ\) per minute, (b) \(\approx 124.41\) radians per minute

Answer:

Let's solve these problems one by one. We'll take each gear speed (in rpm, revolutions per minute) and find the degrees per minute and radians per minute. Recall that one full revolution is \(360^\circ\) or \(2\pi\) radians.

Problem 25: 35 rpm

Part (a): Degrees per minute

A gear turning at 35 rpm means it makes 35 full revolutions per minute. Each revolution is \(360^\circ\), so we multiply the number of revolutions per minute by \(360^\circ\) per revolution.

Step 1: Calculate degrees per minute

Degrees per minute \(= 35 \times 360^\circ\)
\(= 12600^\circ\) per minute (since it's already a whole number, no need to round to the nearest hundredth here, but let's check the formula)

Part (b): Radians per minute

Each revolution is \(2\pi\) radians, so we multiply the number of revolutions per minute by \(2\pi\) radians per revolution.

Step 2: Calculate radians per minute

Radians per minute \(= 35 \times 2\pi\)
\(= 70\pi \approx 70 \times 3.1416 \approx 219.91\) radians per minute (rounded to the nearest hundredth)

Problem 26: 27 rpm

Part (a): Degrees per minute

Step 1: Calculate degrees per minute

Degrees per minute \(= 27 \times 360^\circ\)
\(= 9720^\circ\) per minute

Part (b): Radians per minute

Step 2: Calculate radians per minute

Radians per minute \(= 27 \times 2\pi\)
\(= 54\pi \approx 54 \times 3.1416 \approx 169.65\) radians per minute

Problem 27: 2.5 rpm

Part (a): Degrees per minute

Step 1: Calculate degrees per minute

Degrees per minute \(= 2.5 \times 360^\circ\)
\(= 900^\circ\) per minute

Part (b): Radians per minute

Step 2: Calculate radians per minute

Radians per minute \(= 2.5 \times 2\pi\)
\(= 5\pi \approx 5 \times 3.1416 \approx 15.71\) radians per minute

Problem 28: 6.5 rpm

Part (a): Degrees per minute

Step 1: Calculate degrees per minute

Degrees per minute \(= 6.5 \times 360^\circ\)
\(= 2340^\circ\) per minute

Part (b): Radians per minute

Step 2: Calculate radians per minute

Radians per minute \(= 6.5 \times 2\pi\)
\(= 13\pi \approx 13 \times 3.1416 \approx 40.84\) radians per minute

Problem 29: 14.6 rpm

Part (a): Degrees per minute

Step 1: Calculate degrees per minute

Degrees per minute \(= 14.6 \times 360^\circ\)
\(= 5256^\circ\) per minute

Part (b): Radians per minute

Step 2: Calculate radians per minute

Radians per minute \(= 14.6 \times 2\pi\)
\(= 29.2\pi \approx 29.2 \times 3.1416 \approx 91.73\) radians per minute

Problem 30: 19.8 rpm

Part (a): Degrees per minute

Step 1: Calculate degrees per minute

Degrees per minute \(= 19.8 \times 360^\circ\)
\(= 7128^\circ\) per minute

Part (b): Radians per minute

Step 2: Calculate radians per minute

Radians per minute \(= 19.8 \times 2\pi\)
\(= 39.6\pi \approx 39.6 \times 3.1416 \approx 124.41\) radians per minute

Summary of Answers:

• 25. (a) \(12600.00^\circ\) per minute, (b) \(\approx 219.91\) radians per minute
• 26. (a) \(9720.00^\circ\) per minute, (b) \(\approx 169.65\) radians per minute
• 27. (a) \(900.00^\circ\) per minute, (b) \(\approx 15.71\) radians per minute
• 28. (a) \(2340.00^\circ\) per minute, (b) \(\approx 40.84\) radians per minute
• 29. (a) \(5256.00^\circ\) per minute, (b) \(\approx 91.73\) radians per minute
• 30. (a) \(7128.00^\circ\) per minute, (b) \(\approx 124.41\) radians per minute