Question

The keytones choir is practicing for a show. to feel prepared, they will practice for more than 43 hours. they also don't want to overuse their voices, so they will practice for less than 67 hours. so far, they have practiced a total of 11 hours. let x be the number of hours per week they will practice for the remaining weeks. (suppose they will decide to practice the same amount of time each week.)

Answer 1

The keytones choir will practice for 12 hours per week for the remaining weeks to be adequately prepared for the show.

Step-by-step explanation:

To find the number of hours the keytones choir will practice per week (represented by x), we set up an inequality based on the given conditions. The total practice time needs to be more than 43 hours but less than 67 hours. Given that they have already practiced for 11 hours, the inequality becomes:

`\[11 + 12x > 43 \quad \text{and} \quad 11 + 12x < 67\]`

Solving for x, we find that the choir will practice for 12 hours per week:

`\[11 + 12x > 43 \implies 12x > 32 \implies x > (8)/(3)\]`

`\[11 + 12x < 67 \implies 12x < 56 \implies x < (14)/(3)\]`

Since the choir can't practice a fractional number of hours, we round up to the nearest whole number, and thus, they will practice for 12 hours per week. This ensures they meet the criteria of practicing more than 43 hours and less than 67 hours overall, while maintaining consistency in their weekly practice sessions. This approach strikes a balance between adequate preparation and avoiding overuse of their voices, contributing to a successful performance.

Answer 2

The possible numbers of hours per week they will practice is any value between 4 and 8.375, inclusive.

To find the possible numbers of hours per week the Keytones Choir will practice for the remaining 8 weeks, we need to consider the given constraints.

The total number of hours they want to practice is at least 43 and at most 67.

They have already practiced for 11 hours, so the remaining number of hours they need to practice is:

Total number of hours - Hours already practiced = 43 - 11 = 32

Now, let's solve for x, which represents the number of hours per week they will practice.

To practice for 8 weeks, the total number of hours they will practice is:

Number of hours per week * Number of weeks = 8x

Since they want to practice at least 32 hours and at most 67 hours, we can set up a compound inequality:

32 ≤ 8x ≤ 67

Now, we can solve this compound inequality for x by dividing each term by 8:

32/8 ≤ x ≤ 67/8

Simplifying further, we get:

4 ≤ x ≤ 8.375

Keep in mind that since the number of hours per week needs to be a whole number, the Keytones Choir can practice for 4, 5, 6, 7, or 8 hours per week.

Complete Question:

The Keytones Choir is practicing for a show. To feel prepared, they will practice for at least 43 hours. They also don't want to overuse their voices, so they will practice for at most 67 hours. So far, they have practiced a total of 11 hours. Let x be the number of hours per week they will practice for the remaining 8 weeks. (Suppose they will decide to practice the same amount of time each week.)

(a) Find the possible numbers of hours per week they will practice. Write your answer as a compound inequality solved for x.