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The Pear company sells pPhones. The cost to manufacture apPhone C(x) = -23x² +51000x + 18997 dollars (this includes overhead costs and production costs for each pPhone). If the company sells x pPhones for the maximum price they can fetch, the revenue function will be R(x) = -30x² + 177000 dollars. How many pPhones should the Pear company produce and sell to maximimze profit? (Remember that profit=revenue-cost.) X=

The Pear company sells pPhones. The cost to manufacture apPhone C(x) = -23x² +51000x + 18997 dollars (this includes overhead costs and production costs for each pPhone). If the company sells x pPhones for the maximum price they can fetch, the revenue function will be R(x) = -30x² + 177000 dollars. How many pPhones should the Pear company produce and sell to maximimze profit? (Remember that profit=revenue-cost.) X=

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
ChapterP: Prerequisites
SectionP.7: A Library Of Parent Functions
Problem 47E
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**Understanding Profit Maximization for the Pear Company** The Pear company sells pPhones. The cost to manufacture \( x \) pPhones is \[ C(x) = -23x^2 + 51000x + 18997 \text{ dollars} \] (This includes overhead costs and production costs for each pPhone). If the company sells \( x \) pPhones for the maximum price they can fetch, the revenue function will be \[ R(x) = -30x^2 + 177000x \text{ dollars}. \] **Question** How many pPhones should the Pear company produce and sell to maximize profit? (Remember that profit = revenue - cost.) **Solution Approach** To maximize profit, we need to calculate the profit function \( P(x) \) which is the difference between the revenue function \( R(x) \) and the cost function \( C(x) \): \[ P(x) = R(x) - C(x) \] Substituting the given functions: \[ P(x) = (-30x^2 + 177000x) - (-23x^2 + 51000x + 18997) \] Simplify \[ P(x) = -30x^2 + 177000x + 23x^2 - 51000x - 18997 \] \[ P(x) = (-30x^2 + 23x^2) + (177000x - 51000x) - 18997 \] \[ P(x) = -7x^2 + 126000x - 18997 \] To find the number of pPhones that maximizes profit, we need to find the vertex of this quadratic function. The vertex form of a quadratic function \( ax^2 + bx + c \) occurs at \[ x = -\frac{b}{2a} \] Here, \( a = -7 \) and \( b = 126000 \), thus \[ x = -\frac{126000}{2(-7)} \] \[ x = \frac{126000}{14} \] \[ x = 9000 \] Hence, the Pear company should produce and sell **9000 pPhones** to maximize profit. *[Insert interactive graph or diagram illustrating the profit function P(x) and its maximum point.]* Please submit your solution or further questions below the input box.
Transcribed Image Text:**Understanding Profit Maximization for the Pear Company** The Pear company sells pPhones. The cost to manufacture \( x \) pPhones is \[ C(x) = -23x^2 + 51000x + 18997 \text{ dollars} \] (This includes overhead costs and production costs for each pPhone). If the company sells \( x \) pPhones for the maximum price they can fetch, the revenue function will be \[ R(x) = -30x^2 + 177000x \text{ dollars}. \] **Question** How many pPhones should the Pear company produce and sell to maximize profit? (Remember that profit = revenue - cost.) **Solution Approach** To maximize profit, we need to calculate the profit function \( P(x) \) which is the difference between the revenue function \( R(x) \) and the cost function \( C(x) \): \[ P(x) = R(x) - C(x) \] Substituting the given functions: \[ P(x) = (-30x^2 + 177000x) - (-23x^2 + 51000x + 18997) \] Simplify \[ P(x) = -30x^2 + 177000x + 23x^2 - 51000x - 18997 \] \[ P(x) = (-30x^2 + 23x^2) + (177000x - 51000x) - 18997 \] \[ P(x) = -7x^2 + 126000x - 18997 \] To find the number of pPhones that maximizes profit, we need to find the vertex of this quadratic function. The vertex form of a quadratic function \( ax^2 + bx + c \) occurs at \[ x = -\frac{b}{2a} \] Here, \( a = -7 \) and \( b = 126000 \), thus \[ x = -\frac{126000}{2(-7)} \] \[ x = \frac{126000}{14} \] \[ x = 9000 \] Hence, the Pear company should produce and sell **9000 pPhones** to maximize profit. *[Insert interactive graph or diagram illustrating the profit function P(x) and its maximum point.]* Please submit your solution or further questions below the input box.
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