At a certain university, a student's senior thesis is acceptable if at least \( \frac{3}{4} \) of the student's committee votes in its favor. What is the smallest number of favorable votes needed to accept a thesis if the committee has 3 members? 4 members? 5 members? 6 members? 7 members? 8 members? What is the smallest number of favorable votes needed to accept a thesis if the committee has 3 members?

To determine the smallest number of favorable votes needed for a thesis to be accepted based on the committee size, we can calculate \( \frac{3}{4} \) of the total votes and round up to the nearest whole number, as fractional votes are not possible. For a committee of 3 members, calculating \( \frac{3}{4} \times 3 = 2.25 \). Rounding up gives us 3 votes needed. For a committee of 4 members, \( \frac{3}{4} \times 4 = 3 \). Since 3 is already a whole number, 3 votes are required. For 5 members, \( \frac{3}{4} \times 5 = 3.75 \), rounded up gives us 4 votes. For 6 members, \( \frac{3}{4} \times 6 = 4.5 \), rounded up gives us 5 votes. For 7 members, \( \frac{3}{4} \times 7 = 5.25 \), rounded up gives us 6 votes. For 8 members, \( \frac{3}{4} \times 8 = 6 \). Thus, 6 votes are needed. To summarize: - 3 members: 3 votes - 4 members: 3 votes - 5 members: 4 votes - 6 members: 5 votes - 7 members: 6 votes - 8 members: 6 votes