Math Problem Statement

Homework:Section 4.2 Enhanced Homework

Question 19, 4.2.32-BE

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Part 1

A note card company has found that the marginal cost per card of producing x note cards is given by the function​ below, where

Upper C prime left parenthesis x right parenthesisC′(x)

is the marginal​ cost, in​ cents, per card. Find the total cost of producing

700700

​cards, disregarding any fixed costs.

Upper C prime left parenthesis x right parenthesisC′(x)equals=negative 0.08 x plus 86−0.08x+86​,

for x less than or equals 1000x≤1000

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Part 1

The total cost is

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cents

The total cost iscents

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Solution

To find the total cost of producing 700 cards when the marginal cost is given by C(x)=0.08x+86C'(x) = -0.08x + 86, we need to integrate the marginal cost function over the interval [0,700][0, 700]. This integral gives the total cost:

C(x)=C(x)dxC(x) = \int C'(x) \, dx

Step 1: Set up the integral

C(x)=0700(0.08x+86)dxC(x) = \int_{0}^{700} (-0.08x + 86) \, dx

Step 2: Integrate the function

The integral of 0.08x-0.08x is: 0.08xdx=0.04x2\int -0.08x \, dx = -0.04x^2

The integral of 8686 is: 86dx=86x\int 86 \, dx = 86x

So the total cost function is: C(x)=0.04x2+86x+C0C(x) = -0.04x^2 + 86x + C_0

Since we're finding the cost disregarding fixed costs, we omit C0C_0.

Step 3: Evaluate the definite integral

We calculate C(x)C(x) from x=0x = 0 to x=700x = 700: C(700)=0.04(700)2+86(700)C(700) = -0.04(700)^2 + 86(700)

First, compute 7002700^2: 7002=490000700^2 = 490000

Now calculate: 0.04(490000)=19600-0.04(490000) = -19600 86(700)=6020086(700) = 60200

Add these together: C(700)=19600+60200=40600centsC(700) = -19600 + 60200 = 40600 \, \text{cents}

Final Answer:

The total cost of producing 700 cards is: 40600cents\boxed{40600 \, \text{cents}}

Do you have any questions, or would you like more details?


Here are 5 related questions to explore further:

  1. How does the total cost change if the number of cards increases to 1000?
  2. What is the average cost per card for 700 cards?
  3. How does the marginal cost affect the total cost function?
  4. Can we include fixed costs into the calculation? How would that change the result?
  5. What happens to the total cost if the marginal cost function changes to C(x)=0.05x+90C'(x) = -0.05x + 90?

Tip: Always check the limits of integration and ensure units (cents, dollars, etc.) are consistent!

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Math Problem Analysis

Mathematical Concepts

Calculus
Integration
Marginal Cost

Formulas

Integral of a linear function: ∫(ax + b) dx = (a/2)x^2 + bx + C
Definite integral: ∫[a, b] f(x) dx = F(b) - F(a)

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12, Undergraduate (Introductory Calculus)