A sphere has a surface area of 32in².If the diameter is halved, find the surface area.
Surface area is proportional to Diameter^2, so the area will be (1/2)^2 = 1/4 of the original area, which is 8 in^2.
To find the surface area of a sphere, we can use the formula:
Surface Area = 4πr²
Where π is a mathematical constant approximately equal to 3.14159 and r is the radius of the sphere.
Given that the surface area of the original sphere is 32in², we can set up the equation:
32 = 4πr²
To find the surface area of the sphere when the diameter is halved, we need to find the new radius. Since the diameter is halved, the new radius will be half of the original radius.
Let's solve the original equation for the radius:
32 = 4πr²
Divide both sides by 4π:
8 = πr²
Divide both sides by π:
r² = 8/π
Take the square root of both sides to find the radius:
r ≈ √(8/π)
Now that we have the radius of the original sphere, we can find the radius of the new sphere when the diameter is halved:
r_new = r/2
To find the surface area of the new sphere, we can use the same formula:
Surface Area = 4πr_new²
Let's substitute the radius values into the formula:
Surface Area = 4π(r/2)²
Surface Area = π(r²/4)
Now, substitute the value of r ≈ √(8/π) into the formula:
Surface Area ≈ π((√(8/π))²/4)
The original question did not specify if an approximation is acceptable, so let's calculate the exact value.
Simplify the expression under the square root:
Surface Area ≈ π((8/π)/4)
Surface Area ≈ π(8/4π)
Surface Area ≈ 2
Therefore, when the diameter is halved, the new sphere will have a surface area of approximately 2 square inches.