Question

A racecar is racing along a circular track. The car starts at the 3-o'clock position and travels CCW along the track. The car is constantly 8 feet from the center of the race track and travels at a constant speed and it takes the car 7.854 seconds to complete one full lap.

a. How many radians does the car sweep out per second? radians per second Preview

b. Write a function fthat determines the car's distance to the right of the center of the race track (in feet) in terms of the number of seconds t since the start of the race. f(t)

Answer 1

The car sweeps out \(\frac{16π}{7.854}\) radians per second and its distance to the right of the center of the race track can be determined using the function f(t) = 8(\frac{2π}{7.854})t.

Step-by-step explanation:

a. To determine the number of radians the car sweeps out per second, we need to find the circumference of the circular track. The circumference can be found using the formula C = 2πr, where r is the radius of the track. Since the car is constantly 8 feet from the center of the track, the radius of the track would be 8 feet. Therefore, the circumference of the track would be:

C = 2π(8) = 16π

Since the car completes one full lap in 7.854 seconds, we can find the number of radians the car sweeps out per second by dividing the circumference of the track by the time taken:

Radians per second = \(\frac{16π}{7.854}\)

b. To find the car's distance to the right of the center of the race track (in feet) in terms of the number of seconds t since the start of the race, we can use the concept of angular speed. The angular speed is defined as the change in angle divided by the change in time. In this case, the car is traveling at a constant speed, so the angular speed will remain the same throughout the race. The angular displacement can be found using the formula Δθ = ωΔt, where Δθ is the change in angle, ω is the angular speed, and Δt is the change in time.

since the car completes one full lap in a time of 7.854 seconds, the angular displacement for one full lap would be 2π radians. Therefore, the angular speed would be:

Angular speed = \(\frac{2π}{7.854}\)

To find the car's distance to the right of the center of the race track, we can use the concept of arc length. The arc length can be found using the formula s = rθ, where s is the arc length, r is the radius, and θ is the angle in radians. Since the car's distance from the center of the track is always 8 feet, the radius would be 8 feet. Therefore, the function f(t) that determines the car's distance to the right of the center of the race track in terms of the number of seconds t since the start of the race would be:

f(t) = 8θ = 8(\frac{2π}{7.854})t

Answer 2

a. The car sweeps out 0.799 radians per second.

b. f(t) = 8 cos (0.799t)

Step-by-step explanation:

Given that:

The radius r = 8 feet

a.

The time t it takes to complete one full lap = 7.854 seconds

Recall that:

Angular velocity ω = 2π / t

So if in 7.854 seconds, the car sweeps = 2π radian

∴

per second, the car will sweep ω = 2π/ 7.854

ω = 0.799 rad/sec

Thus, the car sweeps out 0.799 radians per second.

b.

Since, r = 8 feet.

Suppose the car travels an angle θ in time (t),

Then the general equation for the horizontal component is:

f(t) = rcos( ωt)

f(t) = 8 cos (0.799t)