Question

tyler filled up his bathtub, took a bath, and then drained the tub. the function b gives the depth of the water, in inches, t minutes after tyler began to fill the bathtub.
explain the meaning of each statement in this situation.

b(0) = 0

b(1) < b(7)

b(9) = 11

b(10) = b(22)

b(20) > b(40)

the bathtub was empty when tyler began to fill the tub.
the depth of the water in the bathtub 1 minute after tyler began to fill the tub was less than the depth after 7 minutes.
the depth of the water in the bathtub 9 minutes after tyler began to fill the tub was 11 inches.
the depth of the water in the bathtub 20 minutes after tyler began to fill the tub was greater than the depth after 40 minutes.
the depth of the water in the bathtub 10 minutes after tyler began to fill the tub was the same as the depth after 22 minutes.

possible points: 2

Explanation:

• For \( B(0) = 0 \): The function \( B(t) \) represents water depth at \( t \) minutes after filling starts. At \( t = 0 \) (start), depth is 0, so the tub was empty initially.
• For \( B(1) < B(7) \): \( B(1) \) is depth at 1 minute, \( B(7) \) at 7 minutes. The inequality means depth at 1 minute is less than at 7 minutes.
• For \( B(9) = 11 \): \( t = 9 \) minutes, so depth is 11 inches at 9 minutes after starting to fill.
• For \( B(10) = B(22) \): \( B(10) \) (10 minutes) and \( B(22) \) (22 minutes) have equal depth, so depth at 10 and 22 minutes is the same.
• For \( B(20) > B(40) \): \( B(20) \) (20 minutes) and \( B(40) \) (40 minutes) depth; the inequality means depth at 20 minutes is greater than at 40 minutes.

Now, match each statement:

• \( B(0) = 0 \) matches "The bathtub was empty when Tyler began to fill the tub."
• \( B(1) < B(7) \) matches "The depth of the water in the bathtub 1 minute after Tyler began to fill the tub was less than the depth after 7 minutes."
• \( B(9) = 11 \) matches "The depth of the water in the bathtub 9 minutes after Tyler began to fill the tub was 11 inches."
• \( B(10) = B(22) \) matches "The depth of the water in the bathtub 10 minutes after Tyler began to fill the tub was the same as the depth after 22 minutes."
• \( B(20) > B(40) \) matches "The depth of the water in the bathtub 20 minutes after Tyler began to fill the tub was greater than the depth after 40 minutes."

Answer:

• \( B(0) = 0 \): The bathtub was empty when Tyler began to fill the tub.
• \( B(1) < B(7) \): The depth of the water in the bathtub 1 minute after Tyler began to fill the tub was less than the depth after 7 minutes.
• \( B(9) = 11 \): The depth of the water in the bathtub 9 minutes after Tyler began to fill the tub was 11 inches.
• \( B(10) = B(22) \): The depth of the water in the bathtub 10 minutes after Tyler began to fill the tub was the same as the depth after 22 minutes.
• \( B(20) > B(40) \): The depth of the water in the bathtub 20 minutes after Tyler began to fill the tub was greater than the depth after 40 minutes.