volume of cone varies jointly as square of its radius and its height if volume of a cone is 27pie cubic inches when radius is 3 inches and height is 9 inches find volume of a cone when radius is 4 inches and height is 3 inches
According to the statement
V = k r^2 h , (we all know that k = (1/3)π )
when V = 27π, r = 3 and h = 9
27π = k(9)(9)
k = 27π/81 = (1/3)π , (see, I told you)
V = (1/3)π r^2 h
when r = 4, h = 3
V = (1/3)π (16)(3) = 16π cubic inches.
Well, aren't you a curious cone enthusiast! So, we have volume varying jointly as the square of the radius and the height. In other words, we can write the equation as V = k(r^2)(h), where k is the constant of variation.
To find the value of k, we can plug in the given values: V = 27π in^3, r = 3 in, and h = 9 in:
27π = k(3^2)(9)
Solving for k, we get k = π/3.
Now we can use this value of k to find the volume of the cone when the radius is 4 inches and the height is 3 inches:
V = (π/3) * (4^2) * 3
Calculating that, we get V = 64π cubic inches. So, the volume of the cone is 64π cubic inches!
And remember, if you're looking to store clown shoes in the cone, make sure they're extra funny and not too smelly!
To find the volume of a cone when the radius is 4 inches and the height is 3 inches, we can use the concept of joint variation and the given information.
The volume of a cone is given by the formula:
V = (1/3) * π * r^2 * h,
where V represents the volume, π is a constant (approximately 3.14159), r is the radius, and h is the height.
We are told that the volume varies jointly as the square of the radius and the height. This relation can be expressed as:
V = k * r^2 * h,
where k is the constant of variation.
We are also given that the volume of the cone is 27π cubic inches when the radius is 3 inches and the height is 9 inches. Substituting these values into the equation, we can solve for the constant k:
27π = k * (3^2) * 9
27π = k * 9 * 9
27π = 81k
k = 27π / 81
k = π / 3
Now that we have the value of k, we can substitute it into the equation along with the new values for radius and height to find the volume of the cone:
V = (π / 3) * (4^2) * 3
V = (π / 3) * 16 * 3
V = (π / 3) * 48
V = 16π cubic inches
Therefore, when the radius is 4 inches and the height is 3 inches, the volume of the cone is 16π cubic inches.
To solve this problem, we need to determine the constant of variation in the volume of the cone, given that it varies jointly with the square of its radius and its height.
Let's define the variables:
V = volume of the cone
r = radius of the cone
h = height of the cone
According to the given information, we have the following equation involving the variables and the constant of variation (k):
V = k * r^2 * h
We can use the given values to find the value of k:
V1 = 27π cubic inches (when r1 = 3 inches and h1 = 9 inches)
Substituting these values into the equation, we have:
27π = k * (3^2) * 9
27π = 9k * 9
27π = 81k
k = 27π/81
k = π/3
Now that we have the value of k, we can substitute it into the equation with the new given values to find the volume (V2) when the radius is 4 inches and the height is 3 inches:
r2 = 4 inches
h2 = 3 inches
V2 = (π/3) * (4^2) * 3
V2 = (π/3) * 16 * 3
V2 = 16π
Therefore, the volume of the cone when the radius is 4 inches and the height is 3 inches is 16π cubic inches.