Consider a spherical tank of radius 4 m that is filling with water. Let V be the volume of water in the tank (in cubic meters) at a given time, and h the depth of the water (in meters) at the same time. It can be shown using calculus that V is a function of h according to the rule
V = f (h) = π/3*h^2(12 − h).
a. What values of h make sense to consider in the context of this function? What
values of V make sense in the same context?
b. What is the domain of the function f in the context of the spherical tank? Why?
What is the corresponding codomain? Why?
c. Determine and interpret (with appropriate units) the values f (2), f (4), and f (8).
What is important about the value of f (8)?
(a) well, the radius is 4, so 0<=h<=8
(b) the domain is all real numbers; in this context, clearly see (a)
see google for codomain
(c) these are all just volumes of water at various heights. Naturally, with a radius of 4, f(8) is the volume when full. You can check to see that it is indeed 4/3 πr^3
a. In the context of this function, the values of h that make sense to consider are those within the range of the tank's depth. Since the radius of the tank is 4 meters, the depth of the water can range from 0 to 8 meters (assuming the tank is completely filled). Values of V that make sense in the same context are those that represent the volume of water in the tank, which can range from 0 to the maximum volume the tank can hold.
b. The domain of the function f in the context of the spherical tank is the range of the variable h, which is from 0 to 8 meters. This is because the depth of the water cannot be negative or exceed the height of the tank. The corresponding codomain is the range of the variable V, which can also be from 0 to the maximum volume the tank can hold.
c. To determine the values f(2), f(4), and f(8), we substitute these values into the function:
- f(2) = π/3 * (2^2) * (12 - 2) = (4π/3) * 10 = 40π/3 cubic meters
- f(4) = π/3 * (4^2) * (12 - 4) = (16π/3) * 8 = 128π/3 cubic meters
- f(8) = π/3 * (8^2) * (12 - 8) = (64π/3) * 4 = 256π/3 cubic meters
The interpretation of these values is that f(2) represents the volume of water at a depth of 2 meters, f(4) represents the volume of water at a depth of 4 meters, and f(8) represents the maximum volume of water the tank can hold when it is completely filled. The value of f(8) is important because it indicates the maximum capacity of the tank and helps determine if it is being filled to its limit.
a. In the context of this function, the values of h that make sense are between 0 and 12, inclusive. This is because the depth of the water in the tank cannot be negative or exceed the height of the tank, which is 12 meters. Similarly, the values of V that make sense in this context are also between 0 and the maximum volume of the tank, which we can calculate.
b. The domain of the function f in the context of the spherical tank is the set of all possible values for h, which is [0, 12]. This is because the depth of the water cannot be negative or exceed the height of the tank. The corresponding codomain is the set of all possible values for V, which is also [0, V_max]. V_max can be calculated by plugging in h=12 into the function f.
c. To determine the values f(2), f(4), and f(8), we can substitute these values of h into the function f.
- f(2) = π/3 * 2^2 * (12 - 2) = 48π/3 = 16π
- This means that when the depth of the water is 2 meters, the volume of water in the tank is 16π cubic meters.
- f(4) = π/3 * 4^2 * (12 - 4) = 144π/3 = 48π
- This means that when the depth of the water is 4 meters, the volume of water in the tank is 48π cubic meters.
- f(8) = π/3 * 8^2 * (12 - 8) = 384π/3 = 128π
- This means that when the depth of the water is 8 meters, the volume of water in the tank is 128π cubic meters.
The value of f(8) = 128π is important because it represents the maximum volume that the tank can hold. It occurs when the depth of the water is equal to the height of the tank (12 meters). Any further increase in depth would result in the water overflowing.
a. In the context of this function, the values of h that make sense to consider are the depths of water in the tank. Since we are considering a spherical tank of radius 4 m, the depths of water can range from 0 to 8 m. Beyond this range, the water would either overflow or the tank would be empty.
For the volume of water, V, the values that make sense are non-negative values. It is impossible to have a negative volume of water in the tank.
b. The domain of the function f in the context of the spherical tank is the range of valid values for the input variable h, which in this case is the depth of water. As mentioned earlier, the valid values for h range from 0 to 8 m since the tank has a radius of 4 m.
The corresponding codomain of the function is the range of possible output values for the volume of water, which can be any non-negative real number. However, since we are dealing with a physical situation of a real tank, it would not make sense to have an infinite volume of water. Therefore, the codomain could be limited to positive real numbers.
c. To determine the values f(2), f(4), and f(8), we can substitute these values into the function:
f(2) = (π/3)*(2^2)*(12 - 2) = π/3 * 4 * 10 = 40π/3
The value of f(2) is 40π/3 cubic meters. This represents the volume of water when the depth is 2 meters.
f(4) = (π/3)*(4^2)*(12 - 4) = π/3 * 16 * 8 = 128π
The value of f(4) is 128π cubic meters. This represents the volume of water when the depth is 4 meters.
f(8) = (π/3)*(8^2)*(12 - 8) = π/3 * 64 * 4 = 256π
The value of f(8) is 256π cubic meters. This represents the volume of water when the depth is 8 meters.
The value of f(8) is important because it represents the maximum volume of water that the tank can hold. At a depth of 8 meters, the tank is completely filled.