53. a specialty shop owner used a revenue function and a cost - volume function to analyze his monthly…

53. 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.

53. 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.

Answer

Explanation:

Step1: Find the revenue - per - item for the linear revenue function

The linear revenue function is of the form $R(x)=ax$, where $a$ is the price per item and $x$ is the number of items sold. Given that when $x = 1465$ items, $R(1465)=32962.50$. Then $a=\frac{R(x)}{x}=\frac{32962.50}{1465}=22.5$. So the linear revenue function is $R(x)=22.5x$.

Step2: Find the linear cost function

The linear cost function is of the form $C(x)=bx + c$, where $b$ is the variable cost per item and $c$ is the fixed cost. We have two data - points: $(x_1,C(x_1))=(1465,26405.50)$ and $(x_2,C(x_2))=(940,17638)$. First, find the variable cost per item $b$ using the formula $b=\frac{C(x_1)-C(x_2)}{x_1 - x_2}=\frac{26405.50 - 17638}{1465 - 940}=\frac{8767.5}{525}=16.7$. Then substitute $b = 16.7$, $x = 940$ and $C(940)=17638$ into $C(x)=bx + c$ to find $c$. So $17638=16.7\times940 + c$, $17638 = 15698 + c$, and $c=17638 - 15698 = 1940$. The linear cost function is $C(x)=16.7x + 1940$.

Step3: Find the break - even point

The break - even point occurs when $R(x)=C(x)$. So $22.5x=16.7x + 1940$. Subtract $16.7x$ from both sides: $22.5x-16.7x=1940$, $5.8x = 1940$. Solve for $x$: $x=\frac{1940}{5.8}=334.48\approx334$ (rounded to the nearest whole number). To find the break - even revenue, substitute $x$ into $R(x)$: $R(334)=22.5\times334 = 7515$.

Answer:

(a) $R(x)=22.5x$ (b) $C(x)=16.7x + 1940$ (c) The break - even point is approximately $(334,7515)$