Question
Let the demand function for a product be given by the function D(q)= -1.55q+210, where q is the quantity of items in demand and D(q) is the price per item, in dollars, that can be charged when q units are sold. Suppose fixed costs of production for this item are $2,000 and variable costs are $9 per item produced. If 54 items are produced and sold, find the following:
A) The total revenue from selling 54
items (to the nearest penny).
Answer: $
B) The total costs to produce 54
items (to the nearest penny).
Answer: $
C) The total profits to produce 54
items (to the nearest penny. Profits may or may not be negative.).
Answer: $
A) The total revenue from selling 54
items (to the nearest penny).
Answer: $
B) The total costs to produce 54
items (to the nearest penny).
Answer: $
C) The total profits to produce 54
items (to the nearest penny. Profits may or may not be negative.).
Answer: $
Answers
GPT-4o mini
Let's solve the problem step by step.
### A) Total Revenue from Selling 54 Items
The total revenue \( R \) from selling \( q \) items is calculated using the demand function \( D(q) \):
1. First, we need to find the price \( D(54) \) when 54 items are sold. Using the demand function:
\[
D(54) = -1.55(54) + 210
\]
\[
D(54) = -83.7 + 210 = 126.3
\]
So, the price per item when selling 54 items is $126.30.
2. The total revenue \( R \) is:
\[
R(54) = D(54) \times q = 126.3 \times 54
\]
\[
R(54) = 126.3 \times 54 = 6819.2
\]
**Total Revenue: $6819.20**
### B) Total Costs to Produce 54 Items
The total cost \( C \) comprises fixed costs and variable costs.
1. Fixed costs are given as $2,000.
2. The variable cost per item is $9, hence for 54 items:
\[
\text{Variable Costs} = 9 \times 54 = 486
\]
3. Therefore, the total costs are:
\[
C = \text{Fixed Costs} + \text{Variable Costs} = 2000 + 486 = 2486
\]
**Total Costs: $2486.00**
### C) Total Profits from Producing 54 Items
Total profit \( P \) is calculated as total revenue minus total costs:
1. We already found:
- Total Revenue \( R(54) = 6819.2 \)
- Total Costs \( C = 2486.0 \)
2. Thus, the total profit \( P \) is:
\[
P = R - C = 6819.2 - 2486 = 4333.2
\]
**Total Profits: $4333.20**
### Summary of Answers:
A) The total revenue from selling 54 items: **$6819.20**
B) The total costs to produce 54 items: **$2486.00**
C) The total profits to produce 54 items: **$4333.20**
### A) Total Revenue from Selling 54 Items
The total revenue \( R \) from selling \( q \) items is calculated using the demand function \( D(q) \):
1. First, we need to find the price \( D(54) \) when 54 items are sold. Using the demand function:
\[
D(54) = -1.55(54) + 210
\]
\[
D(54) = -83.7 + 210 = 126.3
\]
So, the price per item when selling 54 items is $126.30.
2. The total revenue \( R \) is:
\[
R(54) = D(54) \times q = 126.3 \times 54
\]
\[
R(54) = 126.3 \times 54 = 6819.2
\]
**Total Revenue: $6819.20**
### B) Total Costs to Produce 54 Items
The total cost \( C \) comprises fixed costs and variable costs.
1. Fixed costs are given as $2,000.
2. The variable cost per item is $9, hence for 54 items:
\[
\text{Variable Costs} = 9 \times 54 = 486
\]
3. Therefore, the total costs are:
\[
C = \text{Fixed Costs} + \text{Variable Costs} = 2000 + 486 = 2486
\]
**Total Costs: $2486.00**
### C) Total Profits from Producing 54 Items
Total profit \( P \) is calculated as total revenue minus total costs:
1. We already found:
- Total Revenue \( R(54) = 6819.2 \)
- Total Costs \( C = 2486.0 \)
2. Thus, the total profit \( P \) is:
\[
P = R - C = 6819.2 - 2486 = 4333.2
\]
**Total Profits: $4333.20**
### Summary of Answers:
A) The total revenue from selling 54 items: **$6819.20**
B) The total costs to produce 54 items: **$2486.00**
C) The total profits to produce 54 items: **$4333.20**