The height, h, in metres, above the ground of a rider on a Ferris wheel can be modelled by the equation:
h= 10 sin ((pi/15 t) - 7.5) + 12
where t is the time, in seconds.
At t=0, the rider is at the lowest point. Determine the first two times that the rider is 20 m above the ground, to the nearest hundredth of a second.
Well, let's find the times when the rider is 20 meters above the ground by setting h equal to 20:
20 = 10 sin ((pi/15 t) - 7.5) + 12
Simplifying the equation, we get:
10 sin ((pi/15 t) - 7.5) = 8
To solve for t, we can isolate the sine function:
sin ((pi/15 t) - 7.5) = 8/10
sin ((pi/15 t) - 7.5) = 4/5
Now, let's take the inverse sine function of both sides to find the value inside the parentheses:
(pi/15 t) - 7.5 = arcsin(4/5)
Simplifying further:
(pi/15 t) = arcsin(4/5) + 7.5
Now, let's solve for t:
t = 15 * (arcsin(4/5) + 7.5) / pi
Using a calculator, we can find the value of t that gives us the first time the rider is 20 meters above the ground:
t ≈ 40.25 seconds
To find the second time, we can use the periodicity of the sine function. Since the sine function repeats every 2π, we can add 2π to the value inside the parentheses:
(pi/15 t) - 7.5 + 2π = arcsin(4/5) + 7.5 + 2π
Simplifying:
(pi/15 t) = arcsin(4/5) + 7.5 + 2π
t = 15 * (arcsin(4/5) + 7.5 + 2π) / pi
Using a calculator again:
t ≈ 56.75 seconds
So, the first two times that the rider is 20 meters above the ground, to the nearest hundredth of a second, are approximately 40.25 seconds and 56.75 seconds. Keep in mind, though, that this solution assumes the time starts at t = 0. If there is a specific interval mentioned, please let me know.
To find the first two times that the rider is 20 m above the ground, we need to solve the equation for h = 20.
The equation is:
h = 10sin((π/15 t) - 7.5) + 12
Setting h = 20, we have:
20 = 10sin((π/15 t) - 7.5) + 12
Rearranging the equation, we get:
10sin((π/15 t) - 7.5) = 8
Dividing both sides by 10, we have:
sin((π/15 t) - 7.5) = 0.8
To solve for t, we take the inverse sine (sin^(-1)) of both sides:
(π/15 t) - 7.5 = sin^(-1)(0.8)
Now, let's calculate sin^(-1)(0.8):
sin^(-1)(0.8) ≈ 53.13°
Converting 53.13° to radians:
53.13° = (π/180) * 53.13 ≈ 0.9273 radians
Substituting this value back into the equation:
(π/15 t) - 7.5 = 0.9273
Adding 7.5 to both sides:
(π/15 t) = 8.4273
Multiplying both sides by 15/π:
t = (8.4273) * (15/π)
t ≈ 35.96 seconds
So, the first time the rider is 20 m above the ground is approximately 35.96 seconds.
To find the second time, we can use the property of the sine function that sin(x) = sin(π - x), which means that if one solution is x, then another solution is (π - x).
Therefore, the second time will be:
t = π - t1
t = π - 35.96
t ≈ 2.24 seconds
So, the second time the rider is 20 m above the ground is approximately 2.24 seconds.
To summarize, the first two times that the rider is 20 m above the ground are approximately 35.96 seconds and 2.24 seconds, respectively.
To determine the first two times that the rider is 20m above the ground, we need to solve the equation for h = 20.
The given equation is:
h = 10 sin((π/15t) - 7.5) + 12
Let's substitute h with 20 in the equation and solve for t:
20 = 10 sin((π/15t) - 7.5) + 12
Simplify the equation:
8 = 10 sin((π/15t) - 7.5)
Now, we need to isolate sin((π/15t) - 7.5). Divide both sides of the equation by 10:
8/10 = sin((π/15t) - 7.5)
0.8 = sin((π/15t) - 7.5)
To solve for (π/15t) - 7.5, we need to take the inverse sine (sin^(-1)) of both sides:
sin^(-1)(0.8) = (π/15t) - 7.5
Now, solve for (π/15t):
(π/15t) = sin^(-1)(0.8) + 7.5
Multiply both sides by 15/π:
t = (15/π) * (sin^(-1)(0.8) + 7.5)
Evaluate the right side of the equation to find t. Using a calculator with the sine inverse (sin^(-1)) function:
t ≈ (15/π) * (0.927 + 7.5)
t ≈ (15/π) * 8.427
Calculating the approximate value:
t ≈ 34.174 seconds
The first time that the rider is approximately 20m above the ground is t = 34.174 seconds.
To find the second time, we need to find the next solution.
Using the periodic nature of the sine function, we know that sin(x + 2π) = sin(x). Thus, we can add 2π to the value of x to find the next solution.
t ≈ 34.174 + (2π) seconds
Using the approximation π ≈ 3.14:
t ≈ 34.174 + (2 * 3.14) seconds
t ≈ 34.174 + 6.28 seconds
Calculating the approximate value:
t ≈ 40.454 seconds
Therefore, the second time that the rider is approximately 20m above the ground is t ≈ 40.454 seconds.
To summarize:
The first time the rider is approximately 20m above the ground is at t = 34.17 seconds (to the nearest hundredth of a second).
The second time the rider is approximately 20m above the ground is at t = 40.45 seconds (to the nearest hundredth of a second).
we want h to be 20
20 = 10 sin ((pi/15 t) - 7.5) + 12
8 = 10 sin ((pi/15 t) - 7.5)
.8 = sin ((pi/15 t) - 7.5)
(pi/15 t) - 7.5) = .927295 or (pi/15 t) - 7.5) = pi - .927295 = 2.214297
Case 1: (pi/15 t) - 7.5) = .927295
pi/15 t = 8.427295
t = 40.237
case 2: (pi/15 t) - 7.5) = 2.214297
t = 46.28235
But the period of your wheel is 2pi/(pi/15) = 30 seconds, so my answers are for the second rotation.
Let’s subtract 30 seconds, to get
times of 10.24 sec and 16.28 seconds
check: if t = 10.24
h = 10sin(15/pi*10.24 - 7.5) + 12
= 20.016 (pretty close)
My other answer also works.