suppose a ferris wheel is 250 feet in diameter. if a ferris wheel makes 1 revolution every 40 seconds, then…

suppose a ferris wheel is 250 feet in diameter. if a ferris wheel makes 1 revolution every 40 seconds, then the function ( h(t)=125sinleft(0.157t-\frac{pi}{2}\right)+125 ) represents the height ( h ), in feet, of a seat on the wheel as a function of time ( t ), where ( t ) is measured in seconds. for the particular seat under discussion, the ride begins when ( t = 0 ). complete parts (a) through (c) below.\n(a) during the first 40 seconds of the ride, at what time(s) ( t ) is an individual riding in the seat on the ferris wheel exactly 125 feet above the ground?\n( t=square ) (round to the nearest integer as needed. use a comma to separate answers as needed.)\n(b) during the first 80 seconds of the ride, at what time(s) ( t ) is an individual riding in the seat on the ferris wheel exactly 250 feet above the ground?\n( square ) (round to the nearest integer as needed. use a comma to separate answers as needed.)\n(c) during the first 40 seconds of the ride, over what interval of time ( t ) is an individual riding in the seat on the ferris wheel more than 125 feet above the ground?\nanswer in interval notation. use integers or decimals for any numbers in the expression. round to the nearest integer as needed.)

suppose a ferris wheel is 250 feet in diameter. if a ferris wheel makes 1 revolution every 40 seconds, then the function ( h(t)=125sinleft(0.157t-\frac{pi}{2}\right)+125 ) represents the height ( h ), in feet, of a seat on the wheel as a function of time ( t ), where ( t ) is measured in seconds. for the particular seat under discussion, the ride begins when ( t = 0 ). complete parts (a) through (c) below.\n(a) during the first 40 seconds of the ride, at what time(s) ( t ) is an individual riding in the seat on the ferris wheel exactly 125 feet above the ground?\n( t=square ) (round to the nearest integer as needed. use a comma to separate answers as needed.)\n(b) during the first 80 seconds of the ride, at what time(s) ( t ) is an individual riding in the seat on the ferris wheel exactly 250 feet above the ground?\n( square ) (round to the nearest integer as needed. use a comma to separate answers as needed.)\n(c) during the first 40 seconds of the ride, over what interval of time ( t ) is an individual riding in the seat on the ferris wheel more than 125 feet above the ground?\nanswer in interval notation. use integers or decimals for any numbers in the expression. round to the nearest integer as needed.)

Answer

Explanation:

Step1: Set up the equation

We want to find when (h(t)=125\sin(0.157t-\frac{\pi}{2}) + 125>125). Subtract 125 from both sides: (\sin(0.157t-\frac{\pi}{2})>0).

Step2: Solve the sine inequality

We know that (\sin(x)>0) when (2k\pi<x<(2k + 1)\pi,k\in\mathbb{Z}). Let (x = 0.157t-\frac{\pi}{2}). Then (2k\pi<0.157t-\frac{\pi}{2}<(2k + 1)\pi). First, add (\frac{\pi}{2}) to all parts: (2k\pi+\frac{\pi}{2}<0.157t<(2k + 1)\pi+\frac{\pi}{2}). Since (0.157\approx\frac{\pi}{20}), then (t=\frac{20}{\pi}(2k\pi+\frac{\pi}{2})=40k + 10) and (t=\frac{20}{\pi}((2k + 1)\pi+\frac{\pi}{2})=40k+30). For (k = 0) (within the first - 40 seconds): (10 < t<30).

Step3: For the second part (first 80 seconds)

For (k = 0): (10 < t<30); for (k = 1): (40 + 10<t<40+30) i.e. (50 < t<70).

Answer:

(a) ((10,30)) (b) ((10,30),(50,70))