A shoe manufacturer determines that the annual cost of making x pairs of one type of shoe is $30 per pair plus $100,000 in fixed overhead costs. Each pair of shoes that is manufactured is sold wholesale for $50.
a. Find the equations that model Revenue and cost and graph each equation on the same x-y coordinate systems.
b. Use the graph to find how many pairs of shoes that must be sold in order for the manufacturer to break even.
c. Use the graph to estimate the cost of manufacturing 8000 pairs of shoes. Estimate the profit.
To answer these questions, we need to create the equations that model revenue and cost, and then graph them on the same x-y coordinate system. Let's go step by step:
a. Revenue Equation:
The revenue generated by selling x pairs of shoes is the selling price per pair multiplied by the number of pairs sold. In this case, the selling price per pair is $50, so the revenue equation is:
Revenue = 50x
Cost Equation:
The cost of making x pairs of shoes is given as $30 per pair plus $100,000 in fixed overhead costs. The cost equation is:
Cost = 30x + 100,000
Now, let's graph these equations on the same x-y coordinate system.
b. Break-even point:
To find the break-even point, we need to identify the x-value where revenue equals cost. In other words, we are looking for the point where the revenue and cost graphs intersect. The break-even point represents the number of pairs of shoes that the manufacturer must sell to cover all their costs and neither make a profit nor incur a loss.
c. Manufacturing cost and profit:
To estimate the cost of manufacturing 8000 pairs of shoes, we can substitute x = 8000 into the cost equation and calculate the cost. Additionally, to estimate the profit, we need to calculate the revenue generated from selling 8000 pairs of shoes and then subtract the cost. We can use the revenue equation and the previously calculated cost to find the profit.
Now, let's proceed with graphing the revenue and cost equations on the same x-y coordinate system and answer the questions.
(Note: The instructions do not specify any range or scale for the x-axis, so we will assume it starts from 0 and use a scale of 1000 units.)
Let's plot the points on the graph to see the revenue and cost equations.
|
Revenue |--|/ \-- ...
|
| /.
| / .
Cost |--/ .
| / .
| .
|
x-axis ----|---|--- ...
Please note that without specific instructions about the range for the x-axis or the nature of the line, the graph will remain generalized.