Question

An economist for a sporting goods company estimates the revenue and cost functions for the production of a new snowboard. These functions are R (x) = -x^2+10x and C (x) = 4x+5, respectively, where x is the number of snowboards produced, in thousands. The average profit is defined by the function AP(x) =P(x) / x, where P(x) is the profit function. Determine the production levels that make AP(x) > 0

Answer 1

Answer:

Between 1000 and 5000 snowboards will make the function AP(x) >0.

Step-by-step explanation:

Since x can only take possitive values, we have that AP(x) = P(x)/x > 0 if and only if P(x) > 0.

In order to find when P(x) > 0, we find the values from where it is 0 and then we use the Bolzano Theorem.

P(x) = R(x) - C(x) = -x²+10x - (4x+5) = -x²+6x - 5. the roots of P can be found using the quadratic formula:

$r_1,r_2 = \frac{-6 ^+_- \sqrt{6^2-4*(-1)*(-5)} }{2*(-1)} = \frac{-6^+_-\sqrt{16}}{-2} = \{1, 5\}$

Therefore, P(1) = P(5) = 0. Lets find intermediate values to apply Bolzano Theorem:

• P(0) = -5 < 0 ( P is negative in (-∞ , 1) )
• P(2) = -4+6*2-5 = 3 > 0 (P is positive in (1,5) )
• P(6) = -36+36-5 = -5 < 0 (P is negative in (5, +∞) )

The production levels that make AP(x) >0 are between 1000 and 5000 snowboards (because we take x by thousands)