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 $10$ . $\newline$ What is the maximum revenue possible in this situation? (Do not use commas when entering the answer) $\$$ $\newline$ What price will guarantee the maximum revenue? $\$$ ____

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Q. 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

\newline

What is the maximum revenue possible in this situation? (Do not use commas when entering the answer)

\$

________

\newline

What price will guarantee the maximum revenue?

\$

____________

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 $R = p \times q$ , 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, $R = p \times (1240 - 10p) = 1240p - 10p^2$ . 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 $p = -1240 / (2 \times -10) = 62$ .
Calculate maximum revenue: Calculate the maximum revenue by substituting $p = 62$ into the revenue equation. $R = 62 \times (1240 - 10 \times 62) = 62 \times 620 = 38440$ .