Question

Cost, Revenue & Profit
For these problems, x will represent the number of items and y will represent the money.

The fixed costs for a certain item are $140 per week. The cost to produce each item is $8 per item.

Using this information, what is the cost equation? Give your answer in slope-intercept form: y=

The retailer intends to sell each item for $21/item.

Using this information, what is the revenue equation? Give your answer in slope-intercept form: y=



If in this week 8 items are made, and all items are sold in the week, what are the total costs to the retailer?
Cost = $


What is the revenue from selling 8 items?
Revenue = $


Finally, what is the profit for this retailer?
Profit = $

Answer 1

To solve the problems, we will create equations based on the provided information.

### Cost Equation

The fixed costs are $140 per week, and the variable costs are $8 per item produced.

Let \( C \) represent the total cost and \( x \) represent the number of items. Hence, we can write the cost equation as:

\[
C = \text{Fixed Costs} + (\text{Variable Cost per Item} \times x)
\]

Substituting in the values:

\[
C = 140 + 8x
\]

This is in the form \( y = mx + b \), where \( m = 8 \) (the slope) and \( b = 140 \) (the y-intercept).

**Cost equation:**

\[
y = 8x + 140
\]

---

### Revenue Equation

The selling price for each item is $21. Thus, the revenue \( R \) can be calculated as:

\[
R = \text{Price per Item} \times x
\]

Substituting in the value:

\[
R = 21x
\]

This is also in slope-intercept form, with \( m = 21 \) and \( b = 0 \).

**Revenue equation:**

\[
y = 21x
\]

---

### Total Costs for 8 Items

To find the total cost when 8 items are made:

\[
C = 140 + 8(8) = 140 + 64 = 204
\]

**Total costs:**

\[
\text{Cost} = 204 \text{ dollars}
\]

---

### Revenue from Selling 8 Items

To find the revenue from selling 8 items:

\[
R = 21(8) = 168
\]

**Revenue:**

\[
\text{Revenue} = 168 \text{ dollars}
\]

---

### Profit Calculation

Profit can be calculated as:

\[
\text{Profit} = \text{Revenue} - \text{Cost}
\]

Substituting the values we calculated:

\[
\text{Profit} = 168 - 204 = -36
\]

**Profit:**

\[
\text{Profit} = -36 \text{ dollars}
\]

---

### Summary

- Cost Equation: \( y = 8x + 140 \)
- Revenue Equation: \( y = 21x \)
- Total Cost for 8 Items: $204
- Revenue from Selling 8 Items: $168
- Profit: $-36 (a loss of $36)