Question

Explaining the effect of doubling a measure on a cylinder's volume:

a. Doubling the height has the same effect as doubling the radius because both impact the base area.

b. Doubling the radius has a greater effect on volume than doubling the height due to the squared relationship.

c. Doubling the height and doubling the radius have equal effects on the volume of a cylinder.

d. Doubling the radius affects the volume more than doubling the height, as it influences the cylinder's overall size.

Answer 1

Explaining the effect of doubling a measure on a cylinder's volume:

b. Doubling the radius has a greater effect on volume than doubling the height due to the squared relationship.

Step-by-step explanation:

The volume V of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height. Doubling the radius involves changing it to 2r, while doubling the height involves changing it to 2h. Substituting these values into the formula, we get Vₙₑₓ₋ᵣₐᵢdᵤₛ = π(2r)²h and Vₙₑₓ₋hₑᵢgₕₜ = πr²(2h). Simplifying these expressions, we get Vₙₑₓ₋ᵣₐᵢdᵤₛ = 4πr²h and Vₙₑₓ₋hₑᵢgₕₜ = 2πr²h.

Comparing these two results, we see that doubling the radius (resulting in 4r²) has a squared impact on the volume, whereas doubling the height (resulting in 2h) only has a linear impact. Therefore, doubling the radius has a greater effect on the volume than doubling the height due to the squared relationship in the formula.

In summary, the explanation involves substituting the doubled values into the volume formula for a cylinder, highlighting the impact of the squared term when doubling the radius. This analysis demonstrates why doubling the radius has a greater effect on the volume compared to doubling the height.

Answer 2

The correct answer is B.

Doubling the radius and leaving the height the same quadruples the volume of the cylinder. Doubling the height and leaving the radius the same doubles the volume of the cylinder.