Question

The revenue (R(x)) of producing and selling x Awesome Hearing Aids is modeled by the function (R(x) = -4x^2 + 56x). How many hearing aids need to be produced and sold to maximize revenue? What is the revenue at its maximum?

a) To produce and sell 7 hearing aids; $196 revenue
b) To produce and sell 14 hearing aids; $112 revenue
c) To produce and sell 7 hearing aids; $112 revenue
d) To produce and sell 14 hearing aids; $196 revenue

Answer 1

To maximize revenue, 7 Awesome Hearing Aids need to be produced and sold, which will yield a maximum revenue of $196. The revenue function's vertex gives us these values, calculated using the vertex formula for a quadratic equation.

Step-by-step explanation:

To find the number of Awesome Hearing Aids that need to be produced and sold to maximize revenue, and what that maximum revenue is, we can analyze the revenue function R(x) = -4x^2 + 56x. This function is a quadratic with a negative leading coefficient, meaning it will graph into a downward-facing parabola. The vertex of this parabola gives us the maximum point — the maximum revenue and the corresponding quantity of hearing aids.

To find the vertex, we can use the vertex formula for a quadratic equation, which is x = -b / (2a). In our case, 'a' is -4 and 'b' is 56. Plugging these values into the formula gives us x = -56 / (2 * -4) = 7. Therefore, to maximize revenue, 7 hearing aids need to be produced and sold.

Now, to find the maximum revenue, we substitute x back into the revenue equation: R(7) = -4(7)^2 + 56(7) = -4(49) + 392 = -196 + 392 = 196. Hence, the maximum revenue is $196.

The correct answer is a) To produce and sell 7 hearing aids; $196 revenue.