A can with the ratio of the height of the radius is 4 to 1.
a. express the volume of the can as a function of the radius, r.
b. express the volume of the can as a function of the height, h.
c. if you want the can to have a volume of 100cm cubed, what does the height have to equal (to the nearest thousandth)?
To answer these questions, we need to understand the formula for the volume of a cylinder. The volume of a cylinder is given by the formula V = πr^2h, where V represents the volume, r represents the radius, and h represents the height.
a. Express the volume of the can as a function of the radius, r:
Since we know the ratio of the height to the radius is 4:1, we can express the height as h = 4r.
Substituting the value of h into the formula, we get:
V = πr^2 * 4r
V = 4πr^3
Therefore, the volume of the can as a function of the radius, r, is V = 4πr^3.
b. Express the volume of the can as a function of the height, h:
Since the height is already given, the formula remains the same, which is V = πr^2h.
c. To find the height that corresponds to a volume of 100 cm^3, we can rearrange the formula for volume to solve for h:
V = πr^2h
100 = πr^2h
To solve for h, divide both sides of the equation by (πr^2):
h = 100 / (πr^2)
Now let's calculate the height to the nearest thousandth. Let's assume the value of π is approximately 3.14159:
h ≈ 100 / (3.14159 * r^2)
Let's say the radius, r, is given as a specific value, say 2 cm:
h ≈ 100 / (3.14159 * 2^2)
h ≈ 100 / (3.14159 * 4)
h ≈ 100 / 12.56636
h ≈ 7.957747 cm
Therefore, if the volume of the can is 100 cm^3, the height must be approximately 7.958 cm to the nearest thousandth.