The total worldwide box-office receipts for a long-running movie are approximated by the following function where T(x) is measured in millions of dollars and x is the number of years since the movie's release.
T(x) = (120x^2)/(x^2 + 4)
How fast are the total receipts changing 1 yr, 4 yr, and 8 yr after its release? (Round your answers to two decimal places.)
after 1 yr $ million/year
after 4 yr $ million/year
after 8 yr $ million/year
To find how fast the total receipts are changing after a certain number of years, we need to calculate the derivative of the function T(x) with respect to x.
Let's differentiate T(x) = (120x^2)/(x^2 + 4) using the quotient rule.
The quotient rule states that for two functions u(x) and v(x), the derivative of their quotient is given by:
(u(x)/v(x))' = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2
In this case, u(x) = 120x^2 and v(x) = (x^2 + 4).
Let's compute the derivatives of u(x) and v(x) separately:
u'(x) = d/dx (120x^2) = 240x
v'(x) = d/dx (x^2 + 4) = 2x
Now, applying the quotient rule:
T'(x) = [(u'(x)v(x)) - (u(x)v'(x))] / (v(x))^2
= [(240x)(x^2 + 4) - (120x^2)(2x)] / (x^2 + 4)^2
= (240x^3 + 960x - 240x^3) / (x^2 + 4)^2
= 960x / (x^2 + 4)^2
Now, we can evaluate T'(x) at specific values of x.
After 1 year (x = 1):
T'(1) = 960(1) / (1^2 + 4)^2
= 960 / (1 + 4)^2
= 960 / 25
≈ 38.4 million/year
After 4 years (x = 4):
T'(4) = 960(4) / (4^2 + 4)^2
= 3840 / (16 + 4)^2
= 3840 / 400
= 9.6 million/year
After 8 years (x = 8):
T'(8) = 960(8) / (8^2 + 4)^2
= 7680 / (64 + 4)^2
= 7680 / 4356
≈ 1.77 million/year
Therefore, the rates of change of the total receipts are approximately:
- After 1 year: 38.4 million/year
- After 4 years: 9.6 million/year
- After 8 years: 1.77 million/year
To calculate the rate at which the total receipts are changing after a certain number of years, we need to find the derivative of the function T(x) with respect to x. Then, we can evaluate the derivatives at specific points to find the rates of change.
Step 1: Find the derivative of T(x)
To find the derivative of T(x), we can use the quotient rule:
T'(x) = [(d/dx)(120x^2)(x^2 + 4) - (120x^2)(d/dx)(x^2 + 4)] / (x^2 + 4)^2
T'(x) = [(240x(x^2 + 4) - (120x^2)(2x)) / (x^2 + 4)^2
T'(x) = [(240x^3 + 960x - 240x^3) / (x^2 + 4)^2
T'(x) = (960x) / (x^2 + 4)^2
Step 2: Evaluate the derivatives at specific points
a) After 1 year (x = 1):
T'(1) = (960(1)) / (1^2 + 4)^2
= 960 / (1 + 4)^2
= 960 / 25
= 38.40 million/year
After 1 year, the total receipts are changing at a rate of approximately $38.40 million per year.
b) After 4 years (x = 4):
T'(4) = (960(4)) / (4^2 + 4)^2
= 3840 / (16 + 4)^2
= 3840 / (20)^2
= 3840 / 400
= 9.60 million/year
After 4 years, the total receipts are changing at a rate of approximately $9.60 million per year.
c) After 8 years (x = 8):
T'(8) = (960(8)) / (8^2 + 4)^2
= 7680 / (64 + 4)^2
= 7680 / (68)^2
= 7680 / 4624
≈ 1.66 million/year
After 8 years, the total receipts are changing at a rate of approximately $1.66 million per year.