QuestionThe monthly revenue $R$ achieved by selling $x$ wristwatches is figured to be $R(x)=75 x-0.2 x^{2}$ . The monthly cost $C$ of selling $x$ wristwatches is $C(x)=30 x+1700$ . (a) How many wristwatches must the firm sell to maximize revenue? What is the maximum revenue? (b) Profit is given as $\mathrm{P}(\mathrm{x})=\mathrm{R}(\mathrm{x})-\mathrm{C}(\mathrm{x})$ . What is the profit function? (c) How many wristwatches must the firm sell to maximize profit? What is the maximum profit? (d) Provide a reasonable explanation as to why the answers found in parts (a) and (c) differ. Explain why a quadratic function is a reasonable model for revenue. (b) The profit function is $\mathrm{P}(\mathrm{x})=45 \mathrm{x}-0.2 \mathrm{x}^{2}-1700$ . (Type an expression using $x$ as the variable.) (c) The firm must sell 113 wristwatches to maximize profit. (Round to the nearest integer as needed.) The maximum profit is $\$ 831.20$ . (Round to two decimal places as needed.) (d) Why do the answer found in part (a) and part (c) differ? Choose the correct answer below. A. The parts differ because part (a) uses the revenue function which is equal to the sum of the profit function and the cost function. B. The parts differ because part (c) uses the profit function which is equal to the product of the revenue function and the cost function. C. The parts differ because part (c) uses the revenue function which is equal to the sum of the profit function and the cost function. D. The parts differ because part (a) uses the profit function which is equal to the difference of the revenue function and the cost function.
Explain why a quadratic function is a reasonable model for revenue. Choose the correct answer below. A. Revenue is the profit of the item times the number $x$ of units actually sold. So the revenue, $R$ , is a quadratic function of the price $p$ . B. Revenue is the profit of the item plus the number $x$ of units actually sold. So the revenue, $R$ , is a quadratic function of the price $p$ . C. Revenue is the selling price of the item times the number $x$ of units actually sold. So the revenue, $R$ , is a quadratic function of the price $p$ . D. Revenue is the selling price of the item plus the number $x$ of units actually sold. So the revenue, $R$ , is a quadratic function of the price $p$ .

Studdy Solution

STEP 1

1. The revenue function $R(x) = 75x - 0.2x^2$ is a quadratic function, which typically has a maximum or minimum point.
2. The cost function $C(x) = 30x + 1700$ is a linear function.
3. The profit function $P(x) = R(x) - C(x)$ is derived by subtracting the cost function from the revenue function.
4. Maximizing revenue or profit involves finding the vertex of the quadratic function.

STEP 2

1. Find the number of wristwatches that maximizes revenue and calculate the maximum revenue.
2. Derive the profit function.
3. Find the number of wristwatches that maximizes profit and calculate the maximum profit.
4. Explain the difference between the answers in parts (a) and (c) and why a quadratic function is a reasonable model for revenue.

STEP 3

To find the number of wristwatches that maximizes revenue, we need to find the vertex of the quadratic function $R(x) = 75x - 0.2x^2$ .
The formula for the vertex $x$ of a quadratic function $ax^2 + bx + c$ is given by:
$x = -\frac{b}{2a}$
For $R(x) = -0.2x^2 + 75x$ , $a = -0.2$ and $b = 75$ .
$x = -\frac{75}{2(-0.2)} = \frac{75}{0.4} = 187.5$
Since the number of wristwatches must be a whole number, we round $187.5$ to the nearest integer, which is $188$ .

STEP 4

Calculate the maximum revenue by substituting $x = 188$ back into the revenue function:
$R(188) = 75(188) - 0.2(188)^2$
$R(188) = 14100 - 0.2(35344)$
$R(188) = 14100 - 7068.8$
$R(188) = 7031.2$
The maximum revenue is $\$7031.20$ .

STEP 5

Derive the profit function $P(x)$ by subtracting the cost function from the revenue function:
$P(x) = R(x) - C(x)$
$P(x) = (75x - 0.2x^2) - (30x + 1700)$
$P(x) = 75x - 0.2x^2 - 30x - 1700$
$P(x) = 45x - 0.2x^2 - 1700$

STEP 6

To find the number of wristwatches that maximizes profit, find the vertex of the quadratic profit function $P(x) = -0.2x^2 + 45x - 1700$ .
Using the vertex formula:
$x = -\frac{b}{2a}$
For $P(x) = -0.2x^2 + 45x - 1700$ , $a = -0.2$ and $b = 45$ .
$x = -\frac{45}{2(-0.2)} = \frac{45}{0.4} = 112.5$
Since the number of wristwatches must be a whole number, we round $112.5$ to the nearest integer, which is $113$ .

STEP 7

Calculate the maximum profit by substituting $x = 113$ back into the profit function:
$P(113) = 45(113) - 0.2(113)^2 - 1700$
$P(113) = 5085 - 0.2(12769) - 1700$
$P(113) = 5085 - 2553.8 - 1700$
$P(113) = 831.2$
The maximum profit is $\$831.20$ .

STEP 8

Explain why the answers in parts (a) and (c) differ:
The answers differ because part (a) uses the revenue function, which is focused on maximizing the total income without considering costs, while part (c) uses the profit function, which considers both revenue and costs to maximize the net gain. Therefore, the correct explanation is:
A. The parts differ because part (a) uses the revenue function which is equal to the sum of the profit function and the cost function.

STEP 9

Explain why a quadratic function is a reasonable model for revenue:
A quadratic function is a reasonable model for revenue because it reflects the reality that revenue is the selling price of the item times the number $x$ of units actually sold, and as more units are sold, the price may decrease due to factors like discounts or market saturation, leading to a quadratic relationship. Therefore, the correct explanation is:
C. Revenue is the selling price of the item times the number $x$ of units actually sold. So the revenue, $R$ , is a quadratic function of the price $p$ .

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