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A company produces very unusual CD's for which the variable cost is $ 14 per CD and the fixed costs are $ 35000. They will sell the CD's for $ 93 each. Let x be the number of CD's produced. Write the total cost C as a function of the number of CD's produced. C =$ Write the total revenue R as a function of the number of CD's produced. R=$ Write the total profit P as a function of the number of CD's produced. P=$ Find the number of CD's which must be produced to break even. The number of CD's which must be produced to break even is

A company produces very unusual CD's for which the variable cost is $ 14 per CD and the fixed costs are $ 35000. They will sell the CD's for $ 93 each. Let x be the number of CD's produced. Write the total cost C as a function of the number of CD's produced. C =$ Write the total revenue R as a function of the number of CD's produced. R=$ Write the total profit P as a function of the number of CD's produced. P=$ Find the number of CD's which must be produced to break even. The number of CD's which must be produced to break even is

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter3: The Derivative
Section3.4: Definition Of The Derivative
Problem 49E
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### Problem Statement: Cost and Revenue Functions for CD Production A company produces very unusual CDs for which the variable cost is $14 per CD and the fixed costs are $35,000. They will sell the CDs for $93 each. Let \( x \) be the number of CDs produced. **Questions:** **1. Write the total cost \( C \) as a function of the number of CDs produced.** \[ C = \] **2. Write the total revenue \( R \) as a function of the number of CDs produced.** \[ R = \] **3. Write the total profit \( P \) as a function of the number of CDs produced.** \[ P = \] **4. Find the number of CDs which must be produced to break even.** \[ \text{The number of CDs which must be produced to break even is } \] ### Solutions: 1. **Total Cost \( C \)**: The total cost includes both fixed and variable costs. The fixed cost is $35,000 and the variable cost per CD is $14. Therefore, the total cost \( C \) can be expressed as: \[ C = 35,000 + 14x \] where \( x \) is the number of CDs produced. 2. **Total Revenue \( R \)**: Revenue is earned from selling the CDs. If each CD is sold for $93, the total revenue \( R \) can be written as: \[ R = 93x \] 3. **Total Profit \( P \)**: Profit is calculated as the difference between revenue and total cost. Therefore, the total profit \( P \) can be given by: \[ P = R - C = 93x - (35,000 + 14x) = 93x - 35,000 - 14x = 79x - 35,000 \] 4. **Break Even Point**: To find the break-even point, set the total profit \( P \) to 0 and solve for \( x \): \[ 0 = 79x - 35,000 \] \[ 79x = 35,000 \] \[ x = \frac{35,000}{79} \] \[ x \approx 443.04 \] Since you can't produce a fraction of a CD
Transcribed Image Text:### Problem Statement: Cost and Revenue Functions for CD Production A company produces very unusual CDs for which the variable cost is $14 per CD and the fixed costs are $35,000. They will sell the CDs for $93 each. Let \( x \) be the number of CDs produced. **Questions:** **1. Write the total cost \( C \) as a function of the number of CDs produced.** \[ C = \] **2. Write the total revenue \( R \) as a function of the number of CDs produced.** \[ R = \] **3. Write the total profit \( P \) as a function of the number of CDs produced.** \[ P = \] **4. Find the number of CDs which must be produced to break even.** \[ \text{The number of CDs which must be produced to break even is } \] ### Solutions: 1. **Total Cost \( C \)**: The total cost includes both fixed and variable costs. The fixed cost is $35,000 and the variable cost per CD is $14. Therefore, the total cost \( C \) can be expressed as: \[ C = 35,000 + 14x \] where \( x \) is the number of CDs produced. 2. **Total Revenue \( R \)**: Revenue is earned from selling the CDs. If each CD is sold for $93, the total revenue \( R \) can be written as: \[ R = 93x \] 3. **Total Profit \( P \)**: Profit is calculated as the difference between revenue and total cost. Therefore, the total profit \( P \) can be given by: \[ P = R - C = 93x - (35,000 + 14x) = 93x - 35,000 - 14x = 79x - 35,000 \] 4. **Break Even Point**: To find the break-even point, set the total profit \( P \) to 0 and solve for \( x \): \[ 0 = 79x - 35,000 \] \[ 79x = 35,000 \] \[ x = \frac{35,000}{79} \] \[ x \approx 443.04 \] Since you can't produce a fraction of a CD
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