the diameter of a cylinder is one half the height. Express the total surface area as a function of the height h.
Can someone please help me with this question
D = h/2
Total surface area:
A = pi D h + 2 pi D^2
= pi D(h + 2 D)
Now substitute h/2 for D in that equation.
A = pi h/2 (2 h)
= pi h^2
To find the total surface area of a cylinder in terms of its height, we need to understand the components of the cylinder's surface area.
A cylinder has two circular bases and a curved surface. The curved surface is a rectangle that wraps around the cylinder, and its height is equal to the height of the cylinder (h) and its width is equal to the circumference of one of the circular bases.
Let's assume that the diameter of the cylinder is "d" and the radius is "r". According to the given information, the diameter is half the height, so we can write:
d = h/2
From the circumference formula of a circle, we know that:
Circumference = 2πr
Since the diameter is twice the radius, we can rewrite this as:
Circumference = πd
Substituting the value of the diameter we found earlier:
πd = π(h/2) = (πh)/2
Now, the curved surface area is equal to the product of the circumference and the height (since it is a rectangle) which can be calculated as:
Curved Surface Area = Circumference * Height
= (πh/2) * h
= πh^2/2
The total surface area of the cylinder includes both circular bases and the curved surface. The area of each circular base is given by:
Base Area = πr^2
Substituting the diameter relation again, we have:
πr^2 = π(d/2)^2 = (π/4)d^2 = (π/4)(h^2/4) = (π/16)h^2
Since there are two bases, the total area of the bases is:
Total Base Area = 2 * Base Area
= 2 * (π/16)h^2
= (π/8)h^2
Therefore, the total surface area is the sum of the curved surface area and the base area:
Total Surface Area = Curved Surface Area + 2 * Base Area
= πh^2/2 + (π/8)h^2
= (5π/8)h^2
So, the total surface area of the cylinder is given by the function (5π/8)h^2 in terms of the height h.
Sure, I can help you with that! To express the total surface area of a cylinder as a function of the height, we need to understand the components of the cylinder's surface.
A cylinder has two circular bases and a curved surface connecting them. The total surface area is the sum of these two parts.
Let's start by finding the dimensions of the cylinder. You mentioned that the diameter is one half the height. Let's define the height as "h" units. Therefore, the diameter would be (1/2)h.
The formula for the surface area of a cylinder is:
Surface Area = 2πr^2 + 2πrh
In this formula, "r" represents the radius of the circular base.
Since the diameter is equal to (1/2)h, the radius can be written as (1/2) of the diameter:
radius = (1/2) * (1/2)h = (1/4)h
Replacing the value of the radius in the surface area formula, we get:
Surface Area = 2π((1/4)h)^2 + 2π((1/4)h)h
Simplifying this expression further:
Surface Area = 2π(h^2/16) + 2π(h^2/8)
Combining the terms:
Surface Area = πh^2/8 + πh^2/4
Combining the fractions with the same denominator:
Surface Area = (3πh^2)/8
Therefore, the total surface area of the cylinder can be expressed as a function of the height:
Surface Area = (3πh^2)/8