The base of the fish tank below has a length of 40 inches, a width of 20 inches, and height of 1.5 feet. The tank starts off with a water depth of 7 inches. You then add water to the tank at a constant rate such that the water level increases 1 inch every 2 minutes. Wirte a funcion representing the fish tank's water volume in terms of t minutes.
the water surface has a constant area of 800 in^2.
So, every inch of water adds a volume of 800 in^3.
So, after t minutes, the tank holds
v(t) = 7*800 + 800t in^3
You should probably restrict the domain of t so that the tank does not overflow.
To find the function representing the fish tank's water volume in terms of time, we need to first calculate the initial volume when the water depth is 7 inches. Then, we can use the given rate of 1 inch every 2 minutes to determine how much the volume increases over time.
Here are the steps to get the answer:
1. Convert the dimensions to a consistent unit:
- The length is 40 inches, so we'll keep it in inches.
- The width is 20 inches, so we'll keep it in inches.
- The height is 1.5 feet, which is equal to 18 inches since 1 foot equals 12 inches.
2. Calculate the initial volume of the tank when the water depth is 7 inches:
- The formula for calculating the volume of a rectangular prism (or fish tank) is V = length × width × height.
- Since the water depth is 7 inches, the height of the tank with respect to the water is 7 inches.
- Thus, the initial volume V0 when the water depth is 7 inches can be calculated as V0 = 40 inches × 20 inches × 7 inches.
3. Determine the rate of increase in volume per minute:
- The rate of increase in volume per minute can be calculated by dividing the change in volume by the change in time.
- The change in volume is equal to the cross-sectional area of the tank multiplied by the rate of increase in water depth.
- As the water depth increases by 1 inch every 2 minutes, the rate of increase in water depth is 1 inch / 2 minutes.
- The cross-sectional area of the tank is equal to the length multiplied by the width, which is 40 inches × 20 inches.
- Hence, the rate of increase in volume per minute, dV/dt, is given by (40 inches × 20 inches) × (1 inch / 2 minutes).
4. Write the function representing the fish tank's water volume in terms of time:
- Let t represent the time in minutes.
- The water volume at any given time t, V(t), can be found by adding the rate of increase in volume per minute multiplied by the time t to the initial volume: V(t) = V0 + (dV/dt) × t.
Therefore, the function representing the fish tank's water volume in terms of t minutes is V(t) = (40 inches × 20 inches × 7 inches) + [(40 inches × 20 inches) × (1 inch / 2 minutes)] × t.