Math / Algebra

Questionskipped part.
Tutorial Exercise each profit (in dollars). (a) What is the cost function? (b) What is the revenue function? (c) What is the profit function? (d) Compute the profit (loss) corresponding to production levels of 9,000 and 12,000 units. Click here to begin! Submit Answer

Studdy Solution

STEP 1

1. The fixed cost per month is $31,500.<br />2. The variable cost per unit produced is$ 9.
3. The selling price per unit is $12.
4. The production levels of interest are 9,000 and 12,000 units.

STEP 2

1. Determine the cost function.
2. Determine the revenue function.
3. Determine the profit function.
4. Compute the profit (or loss) at production levels of 9,000 and 12,000 units.

STEP 3

The cost function $C(x)$ is composed of the fixed cost and the variable cost per unit. Let $x$ be the number of units produced.
$C(x) = \text{fixed cost} + (\text{variable cost per unit} \times \text{number of units})$

STEP 4

Substitute the given values into the cost function formula.
$C(x) = 31500 + 9x$

STEP 5

The revenue function $R(x)$ is determined by the selling price per unit multiplied by the number of units sold. Let $x$ be the number of units produced and sold.
$R(x) = \text{selling price per unit} \times \text{number of units}$

STEP 6

Substitute the given values into the revenue function formula.
$R(x) = 12x$

STEP 7

The profit function $P(x)$ is the difference between the revenue function and the cost function.
$P(x) = R(x) - C(x)$

STEP 8

Substitute the expressions for $R(x)$ and $C(x)$ into the profit function formula.
$P(x) = 12x - (31500 + 9x)$

STEP 9

Simplify the profit function.
$P(x) = 12x - 31500 - 9x$ $P(x) = 3x - 31500$

STEP 10

Compute the profit for a production level of 9,000 units by substituting $x = 9000$ into the profit function.
$P(9000) = 3(9000) - 31500$

STEP 11

Calculate the value.
$P(9000) = 27000 - 31500$ $P(9000) = -4500$ The profit (or loss) for 9,000 units is a loss of $4500.

STEP 12

Compute the profit for a production level of 12,000 units by substituting $x = 12000$ into the profit function.
$P(12000) = 3(12000) - 31500$

STEP 13

Calculate the value.
$P(12000) = 36000 - 31500$ $P(12000) = 4500$ The profit (or loss) for 12,000 units is a profit of $4500.
Hence, the cost function, revenue function, and profit function are: - Cost function: $C(x) = 31500 + 9x$ - Revenue function: $R(x) = 12x$ - Profit function: $P(x) = 3x - 31500$
The profit (loss) for 9,000 units is $-4500$ , and for 12,000 units is $4500$ .

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Questionskipped part.
Tutorial Exercise each profit (in dollars). (a) What is the cost function? (b) What is the revenue function? (c) What is the profit function? (d) Compute the profit (loss) corresponding to production levels of 9,000 and 12,000 units. Click here to begin! Submit Answer

Studdy Solution

STEP 1

1. The fixed cost per month is $31,500.<br />2. The variable cost per unit produced is$ 9.
3. The selling price per unit is $12.
4. The production levels of interest are 9,000 and 12,000 units.

STEP 2

1. Determine the cost function.
2. Determine the revenue function.
3. Determine the profit function.
4. Compute the profit (or loss) at production levels of 9,000 and 12,000 units.

STEP 3

The cost function $C(x)$ is composed of the fixed cost and the variable cost per unit. Let $x$ be the number of units produced.
$C(x) = \text{fixed cost} + (\text{variable cost per unit} \times \text{number of units})$

STEP 4

Substitute the given values into the cost function formula.
$C(x) = 31500 + 9x$

STEP 5

The revenue function $R(x)$ is determined by the selling price per unit multiplied by the number of units sold. Let $x$ be the number of units produced and sold.
$R(x) = \text{selling price per unit} \times \text{number of units}$

STEP 6

Substitute the given values into the revenue function formula.
$R(x) = 12x$

STEP 7

The profit function $P(x)$ is the difference between the revenue function and the cost function.
$P(x) = R(x) - C(x)$

STEP 8

Substitute the expressions for $R(x)$ and $C(x)$ into the profit function formula.
$P(x) = 12x - (31500 + 9x)$

STEP 9

Simplify the profit function.
$P(x) = 12x - 31500 - 9x$ $P(x) = 3x - 31500$

STEP 10

Compute the profit for a production level of 9,000 units by substituting $x = 9000$ into the profit function.
$P(9000) = 3(9000) - 31500$

STEP 11

Calculate the value.
$P(9000) = 27000 - 31500$ $P(9000) = -4500$ The profit (or loss) for 9,000 units is a loss of $4500.

STEP 12

Compute the profit for a production level of 12,000 units by substituting $x = 12000$ into the profit function.
$P(12000) = 3(12000) - 31500$

STEP 13

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Questionskipped part.Tutorial Exercise each profit (in dollars). (a) What is the cost function? (b) What is the revenue function? (c) What is the profit function? (d) Compute the profit (loss) corresponding to production levels of 9,000 and 12,000 units. Click here to begin! Submit Answer

Studdy Solution

STEP 1

1. The fixed cost per month is 31,500.<br/>2.Thevariablecostperunitproducedis31,500.<br />2. The variable cost per unit produced is 31,500.<br/>2.Thevariablecostperunitproducedis9.3. The selling price per unit is $12.4. The production levels of interest are 9,000 and 12,000 units.

STEP 2

1. Determine the cost function.2. Determine the revenue function.3. Determine the profit function.4. Compute the profit (or loss) at production levels of 9,000 and 12,000 units.

STEP 3

STEP 4

Substitute the given values into the cost function formula.C(x)=31500+9x C(x) = 31500 + 9x C(x)=31500+9x

STEP 5

STEP 6

Substitute the given values into the revenue function formula.R(x)=12x R(x) = 12x R(x)=12x

STEP 7

The profit function P(x) P(x) P(x) is the difference between the revenue function and the cost function.P(x)=R(x)−C(x) P(x) = R(x) - C(x) P(x)=R(x)−C(x)

STEP 8

Substitute the expressions for R(x) R(x) R(x) and C(x) C(x) C(x) into the profit function formula.P(x)=12x−(31500+9x) P(x) = 12x - (31500 + 9x) P(x)=12x−(31500+9x)

STEP 9

Simplify the profit function.P(x)=12x−31500−9x P(x) = 12x - 31500 - 9x P(x)=12x−31500−9x P(x)=3x−31500 P(x) = 3x - 31500 P(x)=3x−31500

STEP 10

Compute the profit for a production level of 9,000 units by substituting x=9000 x = 9000 x=9000 into the profit function.P(9000)=3(9000)−31500 P(9000) = 3(9000) - 31500 P(9000)=3(9000)−31500

STEP 11

Calculate the value.P(9000)=27000−31500 P(9000) = 27000 - 31500 P(9000)=27000−31500 P(9000)=−4500 P(9000) = -4500 P(9000)=−4500 The profit (or loss) for 9,000 units is a loss of $4500.

STEP 12

Compute the profit for a production level of 12,000 units by substituting x=12000 x = 12000 x=12000 into the profit function.P(12000)=3(12000)−31500 P(12000) = 3(12000) - 31500 P(12000)=3(12000)−31500

STEP 13

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Questionskipped part.
Tutorial Exercise each profit (in dollars). (a) What is the cost function? (b) What is the revenue function? (c) What is the profit function? (d) Compute the profit (loss) corresponding to production levels of 9,000 and 12,000 units. Click here to begin! Submit Answer

1. The fixed cost per month is $31,500.<br />2. The variable cost per unit produced is$ 9.
3. The selling price per unit is $12.
4. The production levels of interest are 9,000 and 12,000 units.

1. Determine the cost function.
2. Determine the revenue function.
3. Determine the profit function.
4. Compute the profit (or loss) at production levels of 9,000 and 12,000 units.

Substitute the given values into the cost function formula.
$C(x) = 31500 + 9x$

Substitute the given values into the revenue function formula.
$R(x) = 12x$

The profit function $P(x)$ is the difference between the revenue function and the cost function.
$P(x) = R(x) - C(x)$

Substitute the expressions for $R(x)$ and $C(x)$ into the profit function formula.
$P(x) = 12x - (31500 + 9x)$

Simplify the profit function.
$P(x) = 12x - 31500 - 9x$ $P(x) = 3x - 31500$

Compute the profit for a production level of 9,000 units by substituting $x = 9000$ into the profit function.
$P(9000) = 3(9000) - 31500$

Calculate the value.
$P(9000) = 27000 - 31500$ $P(9000) = -4500$ The profit (or loss) for 9,000 units is a loss of $4500.

Compute the profit for a production level of 12,000 units by substituting $x = 12000$ into the profit function.
$P(12000) = 3(12000) - 31500$