Mean Value Theorem for Integrals

[finalproject]

So I'm not sure if I've attached this correctly, but this is my final problem for a calculus project I've been working on. I'm suspicious that I haven't actually solved the problem, and was hoping someone here could direct me as to what I need to do / what I've done wrong.

Edit: Clarifying the text, the question reads -
The Mean Value Theorem for Integrals states that for a continuous and positive function f on a closed interval [a,b] there must exist a number c in [a,b] such that f(c): 1/(b-a) (definite integral from a to b) f(x)dx.
Find all values of c in the interval [0,90], which satisfies the MVT for Integrals for the function R(t) given in Question 9. (( R(t) = 0.4t-20cos((pi*t)/45)+40 )). Show the equation needed to be solved to find the value of c. (Note: You may find c using any method you choose. That is, you can solve the equation graphically and/or with a CAS.)

 

[DrPhil]

View attachment 3079
So I'm not sure if I've attached this correctly, but this is my final problem for a calculus project I've been working on. I'm suspicious that I haven't actually solved the problem, and was hoping someone here could direct me as to what I need to do / what I've done wrong.

Edit: Clarifying the text, the question reads -
The Mean Value Theorem for Integrals states that for a continuous and positive function f on a closed interval [a,b] there must exist a number c in [a,b] such that f(c): 1/(b-a) (definite integral from a to b) f(x)dx.
Find all values of c in the interval [0,90], which satisfies the MVT for Integrals for the function R(t) given in Question 9. (( R(t) = 0.4t-20cos((pi*t)/45)+40 )). Show the equation needed to be solved to find the value of c. (Note: You may find c using any method you choose. That is, you can solve the equation graphically and/or with a CAS.)

Having determined that the mean value \(\displaystyle \hat{f} = 58\), you still need to solve for \(\displaystyle c\) by any method you wish.

\(\displaystyle \hat{f} = f(c) = 0.4c - 20\ \cos \dfrac{\pi\ c}{45} + 40 = 58\)

By the mean-value theorem, there must be at least one solution for \(\displaystyle c\) in the interval \(\displaystyle [0,90]\).