Question

Let the demand function for a product be given by the function D(q)= -1.95q+230, 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 109 items are produced and sold, find the following:

A) The total revenue from selling 109
items (to the nearest penny).
Answer: $


B) The total costs to produce 109
items (to the nearest penny).
Answer: $


C) The total profits to produce 109
items (to the nearest penny. Profits may or may not be negative.).
Answer: $

Answer 1

To solve the problem, let's break it down step by step.

### A) Total Revenue from selling 109 items

The demand function is given by:
\[
D(q) = -1.95q + 230
\]
First, we will find the price when \( q = 109 \):
\[
D(109) = -1.95(109) + 230
\]
Calculating this:
\[
D(109) = -1.95 \times 109 + 230 = -212.55 + 230 = 17.45
\]
Now, to find the total revenue (TR) from selling 109 items, we use the formula:
\[
TR = \text{Price per item} \times \text{Quantity}
\]
\[
TR = 17.45 \times 109
\]
Calculating this:
\[
TR = 1904.05
\]
Thus, the total revenue from selling 109 items is:
\[
\text{Total Revenue} = \boxed{1904.05}
\]

### B) Total Costs to produce 109 items

Total costs (TC) consist of fixed costs plus variable costs. The fixed costs are given as $2,000, and the variable cost per item is $9.

First, let's calculate the total variable cost (TVC) for producing 109 items:
\[
TVC = \text{Variable Cost per item} \times \text{Quantity} = 9 \times 109
\]
Calculating this:
\[
TVC = 981
\]
Now, we can add the fixed costs:
\[
TC = \text{Fixed Costs} + \text{Total Variable Costs} = 2000 + 981 = 2981
\]
Thus, the total costs to produce 109 items is:
\[
\text{Total Costs} = \boxed{2981}
\]

### C) Total Profits to produce 109 items

Total profit (P) is calculated as:
\[
P = \text{Total Revenue} - \text{Total Costs}
\]
Using the values we calculated:
\[
P = 1904.05 - 2981
\]
Calculating this:
\[
P = -1076.95
\]
Thus, the total profits (which could be negative) to produce 109 items is:
\[
\text{Total Profits} = \boxed{-1076.95}
\]

In summary:
A) Total Revenue: \( \boxed{1904.05} \)
B) Total Costs: \( \boxed{2981} \)
C) Total Profits: \( \boxed{-1076.95} \)