A spherical balloon has a circumference of 10 inches. If it expands to twice the volume, what happens to its surface area?
twice the volume means the radius grows by ∛2
So, the area grows by (∛2)2 or ∛4
To answer this question, we need to understand the relationship between the circumference, volume, and surface area of a sphere.
The circumference of a sphere is given by the formula C = 2πr, where C is the circumference and r is the radius. Since the balloon has a circumference of 10 inches, we can rearrange the formula to solve for the radius: r = C / (2π) = 10 / (2π) ≈ 1.5915 inches.
The volume of a sphere is given by the formula V = (4/3)πr³. Since the balloon expands to twice the volume, the new volume would be 2V = 2 * (4/3)πr³ = (8/3)πr³.
Now, let's find the new radius of the balloon after it expands. We can use the formula for volume and rearrange it to solve for the radius:
r = (3V / (4π)) ^ (1/3)
= (3 * (8/3)πr³) / (4π) ^ (1/3)
= (2r³)^(1/3)
Let's simplify this equation:
r = 2^(1/3) * r
Since we're interested in the relationship between the surface area and volume, we need to calculate the surface area of the original and expanded balloon.
The surface area of a sphere is given by the formula A = 4πr². Using this formula, we can calculate the surface area of the original balloon:
A_original = 4πr²
= 4π * (1.5915 inches)²
≈ 31.7056 square inches
Now, let's calculate the surface area of the expanded balloon:
A_expanded = 4π * (2^(1/3) * r)²
= 4π * 2^(2/3) * r²
= 4π * 2^(2/3) * (1.5915 inches)²
≈ 50.3254 square inches
Comparing the surface areas, we can see that the expanded balloon has a surface area of approximately 50.3254 square inches. This is larger than the original balloon's surface area of approximately 31.7056 square inches. Therefore, when the balloon expands to twice the volume, its surface area increases as well.