A rectangular bathtub with a base of 18 ft^2 is filling with water at a rate of ..7ft^3 per minute. How fast is the water level rising after 3 minutes?
To find how fast the water level is rising after 3 minutes, we need to calculate the rate of change of the height of the water level over time. Let's break down the problem and solve it step by step:
1. Start by finding the volume of the bathtub:
Since the base area of the bathtub is 18 ft^2, and the height of the water level is the variable we are looking for, we can define the volume of the bathtub as V = 18h, where h represents the height of the water level.
2. Determine the rate of change of the volume with respect to time:
Differentiate both sides of the volume equation with respect to time (t), using the product rule of differentiation:
dV/dt = 18(dh/dt)
Here, dV/dt represents the rate at which the volume changes with respect to time, and dh/dt represents the rate at which the height of the water level changes with respect to time.
3. Now, substitute the given values into the equation:
We are given that water is filling the bathtub at a rate of 0.7 ft^3 per minute, so we can substitute dh/dt = 0.7 into the differentiation equation:
dV/dt = 18(0.7)
4. Calculate the rate of change of the water level after 3 minutes:
Now, substitute t = 3 into the equation, as we want to find the rate of change after 3 minutes:
dV/dt = 18(0.7) = 12.6
Therefore, the rate at which the water level is rising after 3 minutes is 12.6 ft^2 per minute.
To find the rate at which the water level is rising after 3 minutes, we can use the formula:
Rate of change of volume = (Rate of change of height) * (Length) * (Width)
Given:
Base area of the bathtub = 18 ft^2
Rate of change of volume = 0.7 ft^3/min
To find the rate at which the water level is rising after 3 minutes, we need to find the rate of change of height.
Step 1: Find the length and width of the rectangular bathtub.
Since the base area of the bathtub is given as 18 ft^2, we can find the length and width by factoring 18 into pairs:
1 x 18 = 18
2 x 9 = 18
3 x 6 = 18
From these factors, we can see that the possible dimensions of the bathtub are:
Length = 6 ft
Width = 3 ft
Step 2: Calculate the rate of change of height.
Rate of change of height = Rate of change of volume / (Length * Width)
Rate of change of height = 0.7 ft^3/min / (6 ft * 3 ft)
Rate of change of height = 0.7 ft^3/min / 18 ft^2
Rate of change of height = 0.0389 ft/min (approximately)
Step 3: Calculate the rate at which the water level is rising after 3 minutes.
The rate at which the water level is rising after 3 minutes is equal to the rate of change of height.
Rate at which the water level is rising after 3 minutes = 0.0389 ft/min
Therefore, the water level is rising at a rate of approximately 0.0389 ft/min after 3 minutes.