A company finds that if they price their product at $55, they can sell 690 items of it. For every dollar increase in the price, the number of items sold will decrease by 10 . What is the maximum revenue possible in this situation? (Do not use commas when entering the answer) $ ◻ What price will guarantee the maximum revenue? $ ◻

Question

A company finds that if they price their product at  $55, they can sell 690 items of it. For every dollar increase in the price, the number of items sold will decrease by 10 . What is the maximum revenue possible in this situation? (Do not use commas when entering the answer)  $  ◻ What price will guarantee the maximum revenue? $  ◻

Bytelearn · Accepted Answer

Define revenue function:  Define the revenue function based on the given information. Let p be the price of the product. The initial price is $55, and the initial quantity sold is 690 items. For every $1 increase in price, the quantity sold decreases by 10 items. The revenue R can be expressed as R = p * q, where q is the quantity sold at price p. Express quantity as function:  Express quantity q as a function of price p. Since the quantity decreases by 10 items for each $1 increase in price from $55, we have q = 690 - 10(p - 55). Simplifying, q = 690 - 10p + 550, which further simplifies to q = 1240 - 10p. Substitute quantity in revenue:  Substitute q in the revenue function. So, R = p * (1240 - 10p) = 1240p - 10p^2. This is a quadratic equation in terms of p, where the coefficient of p^2 is negative, indicating a downward opening parabola. Find vertex for maximum revenue:  Find the vertex of the parabola to determine the maximum revenue. The vertex formula for a parabola ax^2 + bx + c is given by x = -b/(2a). Here, a = -10 and b = 1240, so p = -1240 / (2 * -10) = 62. Calculate maximum revenue:  Calculate the maximum revenue by substituting p = 62 into the revenue equation. R = 62 * (1240 - 10 * 62) = 62 * 620 = 38440.

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