Babette came up with the equation Ix by calculating Px middo

Babette came up with the equation I(x) by calculating P(x) middot Q(x), where P(x) is the profit she makes off of each chair if the price is set at x dollars and Q(x) is the number of chairs that she will sell if the price is set at x dollars. Using the fact that P(x) = x - 30, find an equation for Q(x). b) Find and interpret the zeros of I(x) in the context of the problem. Write your interpretation in a complete sentence with units.

Solution

(a) We have I(x)=P(x).Q(x)= -2(x-30)(x-75) and P(x)=(x-30). Then Q(x)=I(x)/P(x)= -2(x-30)(x-75)/(x-30) =-2(x-75) ( if x 30). Thus, if x 30, we have Q(x) = -2(x-75). ( when x = 30, x -30 = 0 and division by 0 is not defined).

(b) We have I(x) =-2(x-30)(x-75). Thus I(x) = 0 when x = 30 or x = 75. This means that 30 and 75 are the zeros of I(x). Further, since I(x) is the profit made by Babette on sale of x number of chairs, hence Babette will not make any profit is she sells either 30 or 75 chairs.