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A manufacturer has total cost function C(x) = 45000 + 100x + x³ (in dollars) and revenue function R(x) = 7903x (in dollars), where x is the production level (or # of items.) A. Find and simplify the profit function, P(x). P(x) = -x³ +7803x 45000 B. Find the derivative of the profit function (i.e. the Marginal Profit function.) P'(x) = -3x² +7803 C. What level of production will maximize profit? X = 51 (Round to the nearest whole number if needed.) D. What is the maximum profit? Maximum Profit: $ (Round to the nearest dollar if needed.) ▼ ▼ Part 2 of 4 Part 3 of 4 Part 4 of 4

A manufacturer has total cost function C(x) = 45000 + 100x + x³ (in dollars) and revenue function R(x) = 7903x (in dollars), where x is the production level (or # of items.) A. Find and simplify the profit function, P(x). P(x) = -x³ +7803x 45000 B. Find the derivative of the profit function (i.e. the Marginal Profit function.) P'(x) = -3x² +7803 C. What level of production will maximize profit? X = 51 (Round to the nearest whole number if needed.) D. What is the maximum profit? Maximum Profit: $ (Round to the nearest dollar if needed.) ▼ ▼ Part 2 of 4 Part 3 of 4 Part 4 of 4

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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A manufacturer has a total cost function \( C(x) = 45000 + 100x + x^3 \) (in dollars) and a revenue function \( R(x) = 7903x \) (in dollars), where \( x \) is the production level (or number of items). **A.** Find and simplify the profit function, \( P(x) \). \[ P(x) = -x^3 + 7803x - 45000 \] **B.** Find the derivative of the profit function (i.e., the Marginal Profit function). \[ P'(x) = -3x^2 + 7803 \] **C.** What level of production will maximize profit? \[ x = 51 \] (Round to the nearest whole number if needed.) **D.** What is the maximum profit? Maximum Profit: \[ \$ \] (Round to the nearest dollar if needed.)
Transcribed Image Text:A manufacturer has a total cost function \( C(x) = 45000 + 100x + x^3 \) (in dollars) and a revenue function \( R(x) = 7903x \) (in dollars), where \( x \) is the production level (or number of items). **A.** Find and simplify the profit function, \( P(x) \). \[ P(x) = -x^3 + 7803x - 45000 \] **B.** Find the derivative of the profit function (i.e., the Marginal Profit function). \[ P'(x) = -3x^2 + 7803 \] **C.** What level of production will maximize profit? \[ x = 51 \] (Round to the nearest whole number if needed.) **D.** What is the maximum profit? Maximum Profit: \[ \$ \] (Round to the nearest dollar if needed.)
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