Use the given price-demand function to answer parts \( (A) \), (B), and (C). \( p(x)=75-2 x \quad 1 \leq x \leq 25 \)

The price-demand function given is \( p(x) = 75 - 2x \), where \( p(x) \) represents the price at which \( x \) units are demanded, and \( x \) is the quantity demanded. The function is valid for \( 1 \leq x \leq 25 \). Let's address parts (A), (B), and (C) based on this function. ### Part (A): Find the price when \( x = 10 \). To find the price when \( x = 10 \), we substitute \( x \) into the price-demand function: \[ p(10) = 75 - 2(10) = 75 - 20 = 55 \] So, the price when \( x = 10 \) is **$55**. ### Part (B): Find the quantity demanded when the price is $35. To find the quantity demanded when the price is $35, we set \( p(x) = 35 \) and solve for \( x \): \[ 35 = 75 - 2x \] Rearranging the equation: \[ 2x = 75 - 35 \] \[ 2x = 40 \] \[ x = 20 \] So, the quantity demanded when the price is $35 is **20 units**. ### Part (C): Determine the maximum price and the quantity at which it occurs. The maximum price occurs when \( x \) is at its minimum value, which is \( x = 1 \): \[ p(1) = 75 - 2(1) = 75 - 2 = 73 \] Thus, the maximum price is **$73**, and it occurs when the quantity demanded is **1 unit**. ### Summary of Results: - (A) Price when \( x = 10 \): **$55** - (B) Quantity demanded when price is $35: **20 units** - (C) Maximum price: **$73** at **1 unit**.