QuestionJuly 24, 2025

14. (11 pts) The profit P (in thousands of dollars) for a company spending an amount s (in thousands of dollars) on advertising is given by the equation: P(s)=-(1)/(10)s^3+9s^2+400 a. Find the amount of money the company should spend on advertising in order to make a maximum profit. Also, what is the maximum profit the company makes? If the company spends underline ( )on advertising the maximum profit is underline ( ). b. The point of diminishing returns is the point at which the rate of growth of the profit begins to decline. Find the point of diminishing returns. square The point of diminishing returns is underline ( )

14. (11 pts) The profit P (in thousands of dollars) for a company spending an amount s (in thousands of dollars) on advertising is given by the equation: P(s)=-(1)/(10)s^3+9s^2+400 a. Find the amount of money the company should spend on advertising in order to make a maximum profit. Also, what is the maximum profit the company makes? If the company spends underline ( )on advertising the maximum profit is underline ( ). b. The point of diminishing returns is the point at which the rate of growth of the profit begins to decline. Find the point of diminishing returns. square The point of diminishing returns is underline ( )
14. (11 pts) The profit P (in thousands of dollars) for a company spending an amount s (in thousands of dollars) on advertising is given by the equation: P(s)=-(1)/(10)s^3+9s^2+400 a. Find the amount of money the company should spend on advertising in order to make a maximum profit. Also, what is the maximum profit the company makes? If the company spends underline ( )on advertising the maximum profit is underline ( ). b. The point of diminishing returns is the point at which the rate of growth of the profit begins to decline. Find the point of diminishing returns. square The point of diminishing returns is underline ( )

Solution
3.1(244 votes)

Answer

If the company spends \60,000 on advertising, the maximum profit is \26,800,000. ### The point of diminishing returns is \30,000. Explanation 1. Find the derivative of P(s) Calculate P'(s) = -\frac{3}{10}s^2 + 18s. 2. Set the derivative to zero for maximum profit Solve -\frac{3}{10}s^2 + 18s = 0. Factor to get s(-\frac{3}{10}s + 18) = 0, giving s = 0 or s = 60. 3. Determine the maximum using the second derivative test Calculate P''(s) = -\frac{6}{10}s + 18. Evaluate at s = 60: P''(60) = -6 + 18 = 12 > 0, confirming a local maximum. 4. Calculate maximum profit Substitute s = 60 into P(s): P(60) = -\frac{1}{10}(60)^3 + 9(60)^2 + 400 = 26800 (in thousands). 5. Find the point of diminishing returns Set P''(s) = 0: -\frac{6}{10}s + 18 = 0. Solve for s: s = 30.

Explanation

1. Find the derivative of P(s)<br /> Calculate $P'(s) = -\frac{3}{10}s^2 + 18s$.<br /><br />2. Set the derivative to zero for maximum profit<br /> Solve $-\frac{3}{10}s^2 + 18s = 0$. Factor to get $s(-\frac{3}{10}s + 18) = 0$, giving $s = 0$ or $s = 60$.<br /><br />3. Determine the maximum using the second derivative test<br /> Calculate $P''(s) = -\frac{6}{10}s + 18$. Evaluate at $s = 60$: $P''(60) = -6 + 18 = 12 > 0$, confirming a local maximum.<br /><br />4. Calculate maximum profit<br /> Substitute $s = 60$ into $P(s)$: $P(60) = -\frac{1}{10}(60)^3 + 9(60)^2 + 400 = 26800$ (in thousands).<br /><br />5. Find the point of diminishing returns<br /> Set $P''(s) = 0$: $-\frac{6}{10}s + 18 = 0$. Solve for $s$: $s = 30$.
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