The problem describes Tyler filling up a bathtub, taking a bath, and then draining the tub. The function $B(t)$ gives the depth of the water in inches, $t$ minutes after Tyler began to fill the bathtub. The problem asks to explain the meaning of the following statements: a. $B(0) = 0$ b. $B(1) < B(7)$ c. $B(9) = 11$ d. $B(10) = B(22)$ e. $B(20) > B(40)$

1. Problem Description

The problem describes Tyler filling up a bathtub, taking a bath, and then draining the tub. The function $B(t)$ gives the depth of the water in inches, $t$ minutes after Tyler began to fill the bathtub. The problem asks to explain the meaning of the following statements:

a. $B(0) = 0$

b. $B(1) < B(7)$

c. $B(9) = 11$

d. $B(10) = B(22)$

e. $B(20) > B(40)$

2. Solution Steps

a. $B(0) = 0$

This means that at the beginning ($t=0$ minutes), the depth of the water in the bathtub is 0 inches. In other words, the bathtub is initially empty.

b. $B(1) < B(7)$

This means that the depth of the water in the bathtub at 1 minute is less than the depth of the water in the bathtub at 7 minutes. The water level at 1 minute is lower than at 7 minutes, which implies that the tub is still filling between minute 1 and minute

7.

c. $B(9) = 11$

This means that the depth of the water in the bathtub at 9 minutes is 11 inches.

d. $B(10) = B(22)$

This means that the depth of the water in the bathtub at 10 minutes is the same as the depth of the water at 22 minutes. Since the tub is filling initially, and then draining, this likely means that the tub is draining at $t=22$ and the depth has decreased to equal the depth at $t=10$. This statement may also represent a scenario where the tub is full at $t=10$ and $t=22$, which would mean Tyler kept the water at the same depth between 10 and 22 minutes.

e. $B(20) > B(40)$

This means that the depth of the water in the bathtub at 20 minutes is greater than the depth of the water in the bathtub at 40 minutes. This suggests that the tub is draining and the water level is decreasing over time.

3. Final Answer

a. The bathtub is empty at the start.

b. The depth of the water at 1 minute is less than the depth at 7 minutes.

c. The depth of the water at 9 minutes is 11 inches.

d. The depth of the water at 10 minutes is the same as the depth at 22 minutes.

e. The depth of the water at 20 minutes is greater than the depth at 40 minutes.