A Specialty shop owner used a revenue function and a cost-volume function to analyze his monthly sales. one month he found that with a sales volume of 1465 items he had revenues of $32,962.50 and a total cost of $26,405.50. Another month he had total costs of $17,638 on a sales volume of 940 items.
(a) Find the linear revenue function.
(b) Find the linear cost function.
(c) Find the break-even point.
To find the linear revenue function, we need to find the equation that represents the relationship between the sales volume (number of items sold) and the revenue generated.
In the first month, we know that with a sales volume of 1465 items, the revenue was $32,962.50. We can use this information to find the slope of the linear revenue function.
Let's use the formula for the slope of a line, which is given by:
slope = (y2 - y1) / (x2 - x1)
Using the coordinates (1465, $32,962.50) and (0, 0), we can find the slope:
slope = (32,962.50 - 0) / (1465 - 0)
slope = 32,962.50 / 1465
slope ≈ 22.56
The slope represents the amount of revenue generated for each item sold. Therefore, the linear revenue function can be written as:
R(x) = 22.56x
where R(x) represents the revenue generated and x represents the sales volume (number of items sold).
Next, let's find the linear cost function using the given information.
In the first month, we know that with a sales volume of 1465 items, the total cost was $26,405.50. Using the formula for the slope of a line, we can find the slope.
slope = (26,405.50 - 0) / (1465 - 0)
slope = 26,405.50 / 1465
slope ≈ 18.03
The slope represents the cost per item sold. Therefore, the linear cost function can be written as:
C(x) = 18.03x
where C(x) represents the total cost and x represents the sales volume (number of items sold).
To find the break-even point, we need to determine the sales volume at which the revenue equals the cost. In other words, we need to find the x-value where R(x) = C(x).
Setting R(x) and C(x) equal to each other:
22.56x = 18.03x
Dividing both sides of the equation by x:
22.56 = 18.03
This shows that the break-even point occurs when the sales volume (x) is equal to 18.03.
Therefore, the break-even point is 18.03 items sold.