Final answer:
The drama club must sell at least 47 but no more than 48 student tickets to meet the show's expenses and not exceed the auditorium capacity, considering they have already sold 103 adult tickets.
Step-by-step explanation:
To calculate the number of student tickets the drama club must sell, we can establish an inequality based on their ticket revenue goal. Let's denote the number of student tickets sold as x. Since each student ticket is sold at $8 and each adult ticket at $10, we can express the total revenue R from ticket sales as R = 8x + 10(103). The drama club aims to make no less than $1400, so we have the inequality 8x + 10(103) ≥ 1400.
To find all possible values of x, let's solve the inequality:
- 8x + 10(103) ≥ 1400
- 8x ≥ 1400 - 10(103)
- 8x ≥ 1400 - 1030
- 8x ≥ 370
- x ≥ 370 / 8
- x ≥ 46.25
Since you cannot sell a fraction of a ticket, the smallest number of student tickets the drama club must sell is 47. However, since the auditorium can hold a maximum of 151 people, we must also consider this constraint. With 103 adult tickets sold, this leaves 151 - 103 = 48 seats for student tickets. Therefore, the drama club can sell between 47 and 48 student tickets.