Video game sales that level off

Name: ______________________ Date: ____________ Video game sales that level off (6 points)

Table

A video game studio tracks the total number of copies of a new game sold since its release. The table gives the total sales

S

, in thousands of copies,

t

weeks after release. The total sales are modeled by the function

S(t) = a + b\ln t

, whose graph passes through both data points.

Total sales S, in thousands of copies, t weeks after release
$t$ (weeks)	1	8
$S$ (thousands of copies)	12	37

(A(i)) Use the given data to write two equations that can be used to find the values of the constants $a$ and $b$ in the expression for $S(t)$ . [1 point]
(A(ii)) Find the values of $a$ and $b$ as decimal approximations. [1 point]
(B(i)) Use the given data to find the average rate of change of $S$ , in thousands of copies per week, from $t = 1$ to $t = 8$ . Express your answer as a decimal approximation. Show the computations that lead to your answer. [1 point]
(B(ii)) Use the average rate of change from part B(i) to estimate the total sales at $t = 4$ . Is this estimate an overestimate or an underestimate of the value of the model $S(4)$ ? Explain your reasoning using the concavity of the graph of $S$ . [1 point]
(C(i)) According to the model, when will the total sales reach $60$ thousand copies? Give your answer correct to three decimal places. [1 point]
(C(ii)) The studio expects sales of the game to level off after a few months, as most interested players have bought it. Explain why the model's error is likely to grow for values of $t$ greater than $8$ , and what that means for the answer to part C(i). [1 point]

The question