A pool is being filled. The following function gives the pool's water level (in centimeters) after $t$ seconds: $\newline$ $W(t)=4 \sqrt{t}+\ln (t+1)$ $\newline$ What is the instantaneous rate of change of the pool's water level after $100$ seconds? $\newline$ Choose $1$ answer: $\newline$ (A) $0$ . $21$ centimeters $\newline$ (B) $0$ . $21$ centimeters per second $\newline$ (C) $44$ . $6$ centimeters $\newline$ (D) $44$ . $6$ centimeters per second

Full solution

Q. A pool is being filled. The following function gives the pool's water level (in centimeters) after

t

seconds:

\newline

W(t)=4 \sqrt{t}+\ln (t+1)

\newline

What is the instantaneous rate of change of the pool's water level after

100

seconds?

\newline

Choose

1

answer:

\newline

(A)

0

21

centimeters

\newline

(B)

0

21

centimeters per second

\newline

(C)

44

6

centimeters

\newline

(D)

44

6

centimeters per second

Differentiate Function: To find the instantaneous rate of change, we need to differentiate the function $W(t)$ with respect to $t$ .
Evaluate Derivative at $t=100$ : Differentiate $W(t) = 4\sqrt{t} + \ln(t+1)$ . The derivative of $4\sqrt{t}$ is $\frac{2}{\sqrt{t}}$ and the derivative of $\ln(t+1)$ is $\frac{1}{(t+1)}$ . So, $W'(t) = \frac{2}{\sqrt{t}} + \frac{1}{(t+1)}$ .
Calculate Instantaneous Rate: Now we need to evaluate $W'(t)$ at $t = 100$ seconds to find the instantaneous rate of change at that moment.
Plug $t=100$ into $W'(t)$ : Plug $t = 100$ into $W'(t)$ to get $W'(100) = \frac{2}{\sqrt{100}} + \frac{1}{100+1}$ .
Add Results: Calculate $W'(100) = \frac{2}{10} + \frac{1}{101}$ . $W'(100) = 0.2 + 0.00990099$ .
Round to Two Decimal Places: Add $0.2$ and $0.00990099$ to get the instantaneous rate of change. $\newline$ $W'(100) = 0.20990099$ .
Determine Units: Round the answer to two decimal places since the choices are given in two decimal places. $\newline$ $W'(100) \approx 0.21$ .
Correct Answer: The units for the rate of change should be in centimeters per second because it's the change in water level over time.
Correct Answer: The units for the rate of change should be in centimeters per second because it's the change in water level over time.The correct answer is $0.21$ centimeters per second, which corresponds to choice $(B)$ .