Tangent Lines and Rates of Change — Question 9

Question 9 - Solution

Given: $$R(t) = 200t - 3t^2$$

(a) Average rate of change on $[2, 6]$: $$R(2) = 200(2) - 3(2)^2 = 400 - 12 = 388$$ $$R(6) = 200(6) - 3(6)^2 = 1200 - 108 = 1092$$ $$\text{Average rate} = \frac{1092 - 388}{6 - 2} = \frac{704}{4} = 176 \text{ dollars per hundred items}$$

(b) Instantaneous rate of change at $t = 2$: $$R'(t) = 200 - 6t \quad \Rightarrow \quad R'(2) = 200 - 12 = 188$$

(c) Interpretation: At $t = 2$, revenue is increasing faster than the average over the interval $[2,6]$. The tangent line shows this steeper slope.