Questions: For groups of 25 or more people, a museum charges 21 per person for the first 25 people and 14 for each additional person. Which function f gives the total charge, in dollars, for a group with n people, where n ≥ 25? (A) f(n)=14 n+175 (B) f(n)=14 n+525 (C) f(n)=35 n-350 (D) f(n)=14 n+21

Solution Steps

To determine the correct function \( f(n) \) that gives the total charge for a group with \( n \) people, where \( n \geq 25 \), we need to consider the pricing structure: $21 per person for the first 25 people and $14 for each additional person. The total cost for the first 25 people is fixed at \( 25 \times 21 = 525 \). For any additional person beyond the first 25, the cost is \( 14 \) per person. Therefore, the function can be expressed as the sum of the fixed cost for the first 25 people and the variable cost for the additional people.

For a group of \( n \) people where \( n \geq 25 \), the cost for the first 25 people is calculated as: \[ \text{Fixed Cost} = 25 \times 21 = 525 \]

For any additional people beyond the first 25, the cost is \( 14 \) per person. If \( n \) is the total number of people, the number of additional people is \( n - 25 \). Therefore, the additional cost can be expressed as: \[ \text{Additional Cost} = (n - 25) \times 14 \]

The total charge \( f(n) \) for \( n \) people is the sum of the fixed cost and the additional cost: \[ f(n) = \text{Fixed Cost} + \text{Additional Cost} = 525 + (n - 25) \times 14 \]

Substituting \( n = 30 \) into the total charge function: \[ f(30) = 525 + (30 - 25) \times 14 = 525 + 5 \times 14 = 525 + 70 = 595 \]

Final Answer