A “gazing ball” is a spherical garden ornament with a mirrored surface that reflects its surroundings. The surface area of the ball is approximately 314 square inches. What is its volume to the nearest cubic inch?
A=4πr2
314 = 4 * 3.14 * r^2
314 = 12.56 * r^2
314/12.56 = r^2
25 = r^2
5 = r
V=4/3πr^3
V = (4/3) * 3.14 * 5^3
V = ________ cubic inches
To find the volume of the gazing ball to the nearest cubic inch, we need to know the formula for the volume of a sphere. The formula is given by:
V = (4/3) * π * r³
Where:
V is the volume of the sphere,
π is a mathematical constant approximately equal to 3.14159,
and r is the radius of the sphere.
However, we are given the surface area of the ball instead of the radius. To solve for the radius, we can use the formula for the surface area of a sphere:
A = 4 * π * r²
Where:
A is the surface area of the sphere.
Given that the surface area of the gazing ball is approximately 314 square inches, we can substitute this value into the surface area formula:
314 = 4 * π * r²
Now, we can solve for the radius (r):
314 / (4 * π) = r²
r = √(314 / (4 * π))
Using a calculator, we find that r is approximately 5.00 inches (rounded to two decimal places).
Now that we have the radius, we can substitute it into the formula for the volume of a sphere to find the volume:
V = (4/3) * π * (5.00)³
V ≈ 523.33 cubic inches
Therefore, the volume of the gazing ball to the nearest cubic inch is approximately 523 cubic inches.