Karen is riding a ferris wheel. the function h(t)=7.5cos(8t-240)+10 represents the height of Karen in metres above the ground where t is time in seconds.
a) how high is karen after 2.5 minutes?
b) how long is 4 revolutions of the ferris wheel?
a) To find Karen's height after 2.5 minutes, we need to convert 2.5 minutes to seconds. There are 60 seconds in a minute, so 2.5 minutes is equal to 2.5 * 60 = <<2.5*60=150>>150 seconds.
Substituting t = 150 into the function h(t), we have h(150) = 7.5cos(8(150)-240)+10.
Simplifying further, we have h(150) = 7.5cos(1200-240)+10.
Next, calculate inside the parentheses: h(150) = 7.5cos(960)+10.
Using a calculator, find the cosine of 960 degrees, which is approximately -0.866. So h(150) = 7.5(-0.866) + 10.
Multiply 7.5 by -0.866: h(150) = -6.495 + 10.
Finally, add -6.495 to 10: h(150) = 3.505.
Therefore, Karen is 3.505 meters above the ground after 2.5 minutes.
b) To find the time for 4 revolutions of the Ferris wheel, we need to determine the time it takes for the height function h(t) to complete one full period.
The period of a cosine function is given by T = 2π/b, where b is the coefficient of t in the function.
In this case, the coefficient of t is 8, so the period is T = 2π/8 = π/4.
Therefore, it takes π/4 seconds for the height function h(t) to complete one full period or one revolution.
To find the time for 4 revolutions, we multiply the period by 4: 4 * (π/4) = π.
So, 4 revolutions of the Ferris wheel take π seconds or approximately 3.14 seconds.