Which has the greater effect on the surface area of a right circular cylinder - doubling the radius or doubling the height? how can I explain this
To determine which has a greater effect on the surface area of a right circular cylinder - doubling the radius or doubling the height - we can use the formula for the surface area of a cylinder.
The surface area (A) of a right circular cylinder is given by the formula:
A = 2πr² + 2πrh
Let's consider each case individually:
1. Doubling the radius:
If we double the radius (r), the new radius will be 2r. Plugging this new value into the surface area formula, we get:
A = 2π(2r)² + 2π(2r)h
A = 4πr² + 4πrh
2. Doubling the height:
If we double the height (h), the new height will be 2h. Plugging this new value into the surface area formula, we have:
A = 2πr² + 2πr(2h)
A = 2πr² + 4πrh
Now, let's examine the two cases side by side:
Doubling the radius: A = 4πr² + 4πrh
Doubling the height: A = 2πr² + 4πrh
Comparing these two equations, we can observe that the term "4πrh" appears in both cases, but "4πr²" is only present when we double the radius. This shows that doubling the radius has a greater effect on the surface area of the right circular cylinder compared to doubling the height, since it introduces an additional term with a larger impact.
Therefore, we can explain that doubling the radius of the cylinder has a greater effect on the surface area than doubling the height.
When it comes to the surface area of a right circular cylinder, doubling the radius has a greater effect compared to doubling the height.
The formula for finding the surface area of a right circular cylinder is:
Surface Area = 2πr² + 2πrh
In this formula, "r" refers to the radius, and "h" refers to the height of the cylinder.
To understand why doubling the radius has a greater effect on the surface area than doubling the height, we can examine the impact each change has on the different terms in the formula.
1. Doubling the radius (2r): When we double the radius, the term 2πr² is affected. Since the radius is squared in this term, doubling it will result in a fourfold increase in the area of the circular base (2² = 4). This means that the area of the base increases by a factor of 4.
2. Doubling the height (2h): Doubling the height affects the term 2πrh. However, the height is not squared in this term. Doubling the height will simply result in a twofold increase in the height. This means that the area of the curved surface increases by a factor of 2.
Comparing the two effects, we can see that the increase in the area of the base (fourfold) due to doubling the radius is greater than the increase in the curved surface area (twofold) due to doubling the height. This is why doubling the radius has a greater effect on the overall surface area of a right circular cylinder.