A power outage occurs 6 min after the ride started passengers must wait for their cage to be manually cranked into the lowest position in order to exit the ride. Sine function model: 82.5 sin 3 pi (t+0.5)+97.5 where h is the height of the last passenger above ground measured in feet t is the time of operation of the ride. The complete ride takes 15 minutes.
(A)- What is the height of the last passenger at the moment of the power outage?
(B)- will the last passenger to board the ride need to wait to exit the ride?
My answers:
(A): 79 feet
(B) the last passenger will need to wait to exit the ride because they are 20-25ft above ground.
I will assume
h = 82.5 sin (3 pi (t+0.5) )+97.5 , (you had no equation and no h)
so when t = 6
h = 82.5 sin (3π(6.5)) + 97.5
= 82.5(-1) + 97.5 = 15
check: period = 2π/(3π) = 2/3 minutes (that is a fast ride considering how huge it is)
So 6 ÷(2/3) = 9 , at the 6 minute mark, the last passenger has just completed 9 rotations.
the min height of the basket is -82.5 + 97.5 = 15
( the min value of 82.5 sin(anything) = 82.5(-1) )
so the last passenger must be at the platform level.
how did you get 79 ???
Oof I’m struggling with this too
To find the height of the last passenger at the moment of the power outage (question A), we need to substitute t=6 into the sine function equation:
h = 82.5 sin(3π(t+0.5)) + 97.5
h = 82.5 sin(3π(6+0.5)) + 97.5
h = 82.5 sin(3π(6.5)) + 97.5
Now, let's calculate:
h = 82.5 sin(3π(6.5)) + 97.5
h ≈ 82.5 sin(19.5π) + 97.5
h ≈ 82.5(0) + 97.5
h ≈ 97.5
Therefore, the height of the last passenger at the moment of the power outage is approximately 97.5 feet (not 79 feet as previously mentioned).
Now, moving on to (B), we need to determine if the last passenger will need to wait to exit the ride. According to the given sine function model, the last passenger's height at any given time t is measured by the equation.
From the equation, we can see that the last passenger's height is given by the formula 82.5 sin(3π(t+0.5)) + 97.5. To exit the ride, their height must be 0 feet (ground level). Therefore, we need to find the time when the height is 0.
0 = 82.5 sin(3π(t+0.5)) + 97.5
Subtracting 97.5 from both sides and solving for sin(3π(t+0.5)), we get:
-82.5 = 82.5 sin(3π(t+0.5))
Dividing by 82.5:
-1 = sin(3π(t+0.5))
Since the sin function has a range of -1 to 1, we know that the equation is only true when the angle inside the sin function is equal to -1.
3π(t+0.5) = arcsin(-1)
Using a trigonometry reference angle, the arcsin(-1) is -π/2.
3π(t+0.5) = -π/2
Simplifying:
3(t+0.5) = -1/2
3t + 1.5 = -0.5
3t = -2
t = -2/3
Therefore, the last passenger needs to wait to exit the ride because the time required for the ride to reach ground level is negative (-2/3 minutes).
To find the height of the last passenger at the moment of the power outage, we need to substitute the value of time (t) into the sine function model.
Given:
Sine function model: h = 82.5 sin(3π(t+0.5))+97.5
For part (A), we need to find the height (h) when the power outage occurs, which is at 6 minutes. So we substitute t = 6 into the equation and calculate the value of h.
h = 82.5 sin(3π(6+0.5))+97.5
Now, let's solve it step by step:
Step 1: Calculate the value inside the parentheses: (3π(6+0.5))
= (3π(6.5))
Step 2: Multiply 3π by 6.5: (3π * 6.5)
= 19.5π
Step 3: Substitute 19.5π into the sine function: sin(19.5π)
Step 4: Calculate the sine function: sin(19.5π) ≈ -1
(Note: The sine function varies between -1 and 1)
Step 5: Plug in the value for sin(19.5π) into the equation:
h = 82.5 * (-1) + 97.5
Step 6: Calculate the result:
h ≈ -82.5 + 97.5
h ≈ 15 feet
Therefore, the height of the last passenger at the moment of the power outage is approximately 15 feet.
For part (B), we need to determine if the last passenger to board the ride will need to wait to exit. To do so, we need to check if their height is above ground level. Since the height of the last passenger at the moment of the power outage is approximately 15 feet, they will need to wait to exit the ride if the ground level is below 15 feet.
However, in the given information provided, it does not specify the ground level, so we cannot determine for certain if the last passenger will need to wait based on the information given.