The cost for a guitar maker to handcraft x guitars is given by the function $C ( x )$ . The revenue taken in with the sales of x guitars is given by the function $R ( x )$ . Find the number of guitars that must be sold so that the cost equals the revenue. $C ( x ) = \frac { 1 } { 8 } x ^ { 2 } + x + 16$ , $R ( x ) = \frac { 1 } { 4 } x ^ { 2 }$
A) 16 guitars
B) 15 guitars
C) 13 guitars
D) 17 guitars
E) 14 guitars

Question

Quizplus · Accepted Answer

To find the number of guitars that must be sold so that the cost equals the revenue, we need to solve the equation $C ( x ) = R ( x )$.Substituting the given functions, we get:$$\frac { 1 } { 8 } x ^ { 2 } + x + 16 = \frac { 1 } { 4 } x ^ { 2 }$$Simplifying:$$\frac { 1 } { 8 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 2 } + x + 16 = 0$$$$\frac { - 1 } { 8 } x ^ { 2 } + x + 16 = 0$$Multiplying both sides by -8 to clear the fraction:$$- x ^ { 2 } + 8 x + 128 = 0$$Factoring:$$-( x ^ { 2 } - 8 x - 128 ) = 0$$$$-( x - 16 )( x + 8 ) = 0$$Setting each factor to zero:$$x - 16 = 0$$$$x = 16$$$$x + 8 = 0$$$$x = -8$$Since the number of guitars cannot be negative, the only valid solution is $x = 16$.Therefore, the guitar maker must sell 16 guitars so that the cost equals the revenue.

The Cost for a Guitar Maker to Handcraft X Guitars C(x)C ( x )C(x)

The Cost for a Guitar Maker to Handcraft X Guitars $C ( x )$