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The marketing division of a large firm has found that it can model the sales generated by an advertising campaign as:

\[ s(u) = 0.75u + 1.5 \]

million dollars, when the firm invests \( u \) thousand dollars in advertising. The firm plans to invest:

\[ v(x) = -2.4x^2 + 53x + 150 \]

thousand dollars each month, where \( x \) is the number of months after the beginning of the advertising campaign.

(a) Write the function for the model that gives the sales in million dollars generated when \( x \) is the number of months since the beginning of the ad campaign.

\[ s(u(x)) = \] million dollars

(b) Write the formula for the rate of change of predicted sales \( x \) months into the campaign.

\[ \frac{dS}{dx} = \] million dollars per month

(c) What will be the rate of change of sales when \( x = 17 \)? (Round your answer to three decimal places.)

\[ \] million dollars per month

Answer :

(a) The function for the model that gives the sales in million dollars generated when x is the number of months since the beginning of the ad campaign is s(u(x)) = 0.75v(x) + 1.5 million dollars.

(b) The formula for the rate of change of predicted sales x months into the campaign is dx/dS = 1 million dollars per month.

(c) The rate of change of sales when x = 17 is 1 million dollars per month.

a) The function that gives the sales in million dollars generated when x is the number of months since the beginning of the ad campaign can be obtained by substituting v(x) into s(u).

So, s(u(x)) = s(v(x)) = 0.75v(x) + 1.5 million dollars.

(b) The rate of change of predicted sales x months into the campaign can be found by differentiating s(u(x)) with respect to x. Let's find dx/dS. Using the chain rule, dx/dS = (dx/du) * (du/dS).

The derivative of v(x) with respect to x is dv(x)/dx = -4.8x + 53. The derivative of u(x) with respect to S is du(x)/dS = 1/(dv(x)/dx) = 1/(-4.8x + 53).

So, dx/dS = (dx/du) * (du/dS) = (-4.8x + 53) * (1/(-4.8x + 53)) = 1.

(c) To find the rate of change of sales when x = 17, we substitute x = 17 into dx/dS.

So, dx/dS = 1.

Therefore, the rate of change of sales when x = 17 is 1 million dollars per month.

In summary, (a) The function for the model that gives the sales in million dollars generated when x is the number of months since the beginning of the ad campaign is s(u(x)) = 0.75v(x) + 1.5 million dollars.

(b) The formula for the rate of change of predicted sales x months into the campaign is dx/dS = 1 million dollars per month.

(c) The rate of change of sales when x = 17 is 1 million dollars per month.

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