the height in meters, of a passenger on a Ferris wheel is given by the function h(t)=11-10cos(0.1pit), where t is the time in seconds. How long will it take a passenger to reach a height of 5 meters the first time? Express your answer to 3 decimal places.
so
5 = 11-10cos(.1pi t)
cos (.1 pi t) = 6/10 = .6
set your calculator to radians
.1pi t = .927295
t = 2.95 seconds
To find the time it takes for a passenger to reach a height of 5 meters the first time, we need to solve the equation h(t) = 5.
Given that h(t) = 11 - 10cos(0.1πt), we set up the equation:
11 - 10cos(0.1πt) = 5
To solve for t, we can rearrange the equation:
10cos(0.1πt) = 11 - 5
10cos(0.1πt) = 6
Next, we isolate cos(0.1πt) by dividing both sides by 10:
cos(0.1πt) = 6/10
cos(0.1πt) = 0.6
To find the value of t, we can take the inverse cosine (cos^(-1)) of both sides:
0.1πt = cos^(-1)(0.6)
Simplifying the right side, we have:
0.1πt ≈ 0.927
Finally, to solve for t, we divide both sides by 0.1π:
t = 0.927 / (0.1π)
Evaluating this expression, we find:
t ≈ 2.952
Therefore, it will take approximately 2.952 seconds for the passenger to reach a height of 5 meters the first time.
To find out how long it will take a passenger to reach a height of 5 meters for the first time, we need to solve the equation h(t) = 5.
The given function is h(t) = 11 - 10cos(0.1πt). Setting h(t) equal to 5, we have:
5 = 11 - 10cos(0.1πt)
Rearranging the equation, we get:
10cos(0.1πt) = 11 - 5
10cos(0.1πt) = 6
Dividing both sides of the equation by 10:
cos(0.1πt) = 0.6
Now, we need to find the inverse cosine (also known as the arccosine or cos⁻¹) of 0.6. Using a calculator or any software that provides trigonometric functions, we can evaluate cos⁻¹(0.6) to find the value of the angle:
cos⁻¹(0.6) ≈ 0.9273 radians
So, now we have:
0.1πt = 0.9273
Simplifying further:
0.1t = 0.9273/π
Dividing by 0.1:
t ≈ (0.9273/π) / 0.1
Calculating this expression:
t ≈ 0.2954 seconds
Therefore, it will take approximately 0.295 seconds (rounded to 3 decimal places) for the passenger to reach a height of 5 meters for the first time.