Question

1. connection to real life your school purchases a math software application. the program has an initial cost of $500 plus $20 for each student who subscribes.

part a. how many students must subscribe for the average cost per student to fall to $30?

part b. what happens to the average cost as more students subscribe?

Explanation:

Part A

Step1: Define variables and cost function

Let \( n \) be the number of students. The total cost \( C \) is the initial cost plus the cost per student times the number of students, so \( C = 500 + 20n \). The average cost per student \( \bar{C} \) is \( \frac{C}{n}=\frac{500 + 20n}{n} \).

Step2: Set up the equation for average cost

We want the average cost to be \( \$30 \), so we set up the equation \( \frac{500 + 20n}{n}=30 \).

Step3: Solve the equation

Multiply both sides by \( n \): \( 500 + 20n = 30n \). Subtract \( 20n \) from both sides: \( 500 = 10n \). Then divide both sides by 10: \( n = 50 \).

The average cost function is \( \bar{C}=\frac{500 + 20n}{n}=\frac{500}{n}+20 \). As \( n \) (number of students) increases, \( \frac{500}{n} \) approaches 0 (since the numerator is constant and the denominator gets larger). So the average cost \( \bar{C} \) approaches \( 20 \) (because \( \frac{500}{n}\to0 \) as \( n\to\infty \), so \( \bar{C}\to0 + 20 = 20 \)).

Answer:

50 students

Part B