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1. A philosopher paces around a circle with a diameter of 8 feet. After 21 minutes of pacing, she has gone [tex]2880^\circ[/tex] around the circle. How far, in feet, has she walked?

2. A sector with a central angle of [tex]252^\circ[/tex] is carved out of a circle with a radius of 14 cm. What is the area, in cm², of the sector?

3. A car travels at a constant speed around a circular track that has a circumference of 5 miles. The car completes 9 laps every 18 minutes. What is the angular speed in radians per minute? What is the linear speed in miles per minute?

4. A gear is spinning at 240 revolutions per minute. What is the angular speed of the gear in radians per minute?

5. A hamster is running on a wheel at a constant rate of 6 miles per hour. The wheel has a radius of 4 inches. What is the angular speed of the wheel in radians per second? Note: [tex]\frac{5280}{3600}=\frac{22}{15}[/tex].

Answer :

Final answer:

This detailed answer provides step-by-step solutions for questions involving pacing around a circle, carving out a sector, circular track, and spinning gear.

Explanation:

Question 1:

The circumference of a circle with a diameter of 8 feet is 8π feet. Since the philosopher paces 2880∘ around the circle, she covers a fraction of the circumference which is equivalent to 2880∘/360∘ = 8π/360 = 2π/45 of the total circumference. To find the distance walked, we multiply this fraction by the total circumference:

Distance walked = (2π/45) * 8π = 16π^2/45 feet.

Question 2:

The area of a sector can be found using the formula: Area = (central angle/360∘) * π * (radius^2). Substituting the given values, we have:

Area = (252∘/360∘) * π * (14 cm)^2 = (7/10) * π * 196 cm^2 = 1372π/10 cm^2.

Question 3:

The angular speed is determined by the number of laps the car completes in a given time period. In this case, the angular speed can be calculated as 9 laps/18 minutes = 0.5 laps/minute. The linear speed can be found by multiplying the angular speed by the circumference of the track: 0.5 laps/minute * 5 miles/lap = 2.5 miles/minute.

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