Please help me with math?

Question

If C(r)=2(pi)r and r=f(t)=1/(t^2)+1 what is (C x f)(t) when t=3. Please show work

Answer 1

C(r) = 2 pi r
r = 1/(t^2) + 1

Step 1: Substitute the value for r to find C(t)
C(t) = 2 pi (t^-2 + 1) = 2 pi /t^2 + 2 pi
f(t) = t^-2 + 1

(C x f)(t) = C(t) * f(t)
(C x f)(t) = (2 pi /t^2 + 2 pi) * (t^-2 + 1)

Step 2: Distribute it out:
(C x f)(t) = 2pi/t^4 + 2pi/t^2 + 2pi/t^2 + 2pi
(C x f)(t) = 2pi (t^-4 + 4t^-2 + 1)

Step 3: Substitute 3 for t:
(C x f)(3) = 2pi (3^-4 + 4(3)^-2 + 1)
(C x f)(3) = 2pi (1/81 + 4/9 + 1)
(C x f)(3) = 2pi (1/81 + 36/81+ 81/81)
(C x f)(3) = 2pi (118/81)
(C x f)(3) = 236/81 * pi
(C x f)(3) = 2.914 * pi = 9.153 (solution!)

Answer 2

(2(pi)r * 1/(t^2+1) = 2(pi)(1/(t^2)+1) / ((t^2) + 1))
= 2(pi)/((t^2)+1)^2
(C x f) (3) = 2pi/100
= 0.0628