AI Math Solver
Resources
Questions
Pricing
Login
Register
Home
>
Questions
>
Interpret $W'(2) > 3$ for a water tank filling rate | AP Calculus AB
Mathematics (AP Calculus AB)
Grade 12 (Senior High School)
Question Content
At time $t = 0$, a storage tank is empty and begins filling with water. For $t > 0$ hours, the depth of the water in the tank is increasing at a rate of $W(t)$ feet per hour. Which of the following is the best interpretation of the statement $W'(2) > 3$?
Correct Answer
(D) Over the first two hours after the tank begins filling with water, the rate at which the depth of the water is rising is always increasing at a rate greater than 3 feet per hour per hour.
Detailed Solution Steps
1
Step 1: Identify the meaning of $W(t)$: $W(t)$ is the rate of change of the water depth (in feet per hour) at time t hours.
2
Step 2: Identify the meaning of $W'(t)$: $W'(t)$ is the derivative of $W(t)$, which represents the rate of change of the rate of water depth increase, measured in feet per hour per hour.
3
Step 3: Interpret $W'(2) > 3$: This means that at $t=2$ hours, the rate at which the depth is rising ($W(t)$) is increasing at a rate greater than 3 feet per hour per hour. This corresponds to option D, which states that over the first two hours, the rate of increase of the depth is always increasing at a rate greater than 3 ft/h².
Knowledge Points Involved
1
Interpretation of Derivatives of Rates
If a function $f(t)$ represents a rate of change (e.g., speed, filling rate), its derivative $f'(t)$ represents the rate of change of that rate (e.g., acceleration, rate of change of filling rate). The units of $f'(t)$ are the units of $f(t)$ divided by time.
2
Contextual Derivative Interpretation
In real-world contexts, derivatives describe rates of change. For a function representing a physical quantity, its derivative describes how that quantity changes over time, and the second derivative describes how that rate of change itself changes.
3
Function Notation for Rates
When a function is defined as a rate (e.g., $W(t)$ is the rate of water depth increase), the notation $W'(t)$ specifically refers to the derivative of that rate function, not the derivative of the depth function itself.
Loading solution...
