a solid metal cube with an edge length of 9 inches is melted down and reshaped into a sphere. which of the following is the surface area to the nearest tenth of an inch
90.5 sq in
254.5 sq in
391.7 sq in
574.8 sq in
First, we need to find the volume of the cube which is:
9^3 = 729 cubic inches
Since this metal is melted down and reshaped into a sphere, the volume remains the same. Therefore, we can use the formula for the volume of a sphere:
V = (4/3)πr^3
729 = (4/3)πr^3
r^3 = (729 x 3) / (4π)
r^3 ≈ 206.36
r ≈ 6.41
Now, we can use the formula for the surface area of a sphere:
A = 4πr^2
A ≈ 514.7 sq in
Rounded to the nearest tenth, the surface area is 514.7 sq in.
Therefore, the closest answer choice is 574.8 sq in.
To find the surface area of the sphere formed from a solid metal cube, we need to first determine the volume of the cube, and then use this volume to find the radius of the sphere.
The volume of a cube can be calculated by raising the length of its edge to the power of 3:
Volume of the cube = (9 inches)^3 = 729 cubic inches
Since the volume of a cube is equal to the volume of the sphere, we can use this volume to find the radius of the sphere using the formula:
Volume of the sphere = (4/3) * π * (radius of the sphere)^3
729 cubic inches = (4/3) * π * (radius of the sphere)^3
Divide both sides of the equation by (4/3) * π to isolate the radius:
(radius of the sphere)^3 = (729 cubic inches) / ((4/3) * π)
Taking the cube root of both sides gives us the radius:
radius of the sphere = (729 cubic inches / ((4/3) * π))^(1/3)
Using a calculator, we can evaluate this expression to find the radius:
radius of the sphere ≈ 4.355 inches (rounded to three decimal places)
Now that we have the radius, we can calculate the surface area of the sphere using the formula:
Surface area of a sphere = 4 * π * (radius of the sphere)^2
Substituting the value of the radius we found:
Surface area of the sphere ≈ 4 * π * (4.355 inches)^2
Using a calculator to evaluate this expression, we find:
Surface area of the sphere ≈ 238.765 square inches (rounded to three decimal places)
Therefore, the surface area of the sphere, to the nearest tenth of an inch, is approximately 238.8 square inches.
None of the provided options match this value, so none of the given answers are correct.
To find the surface area of the resulting sphere, we need to first determine the volume of the initial metal cube.
The volume of a cube can be calculated using the formula V = s^3, where V represents the volume and s represents the length of one side (in this case, the edge length of the cube).
V = 9^3 = 729 cubic inches
Next, we need to find the radius of the sphere, which can be calculated using the formula r = (3V / 4π)^(1/3), where r represents the radius and V represents the volume of the cube.
r = (3 * 729 / (4 * π))^(1/3) ≈ 6.6 inches
Now, we can calculate the surface area of the sphere using the formula A = 4πr^2, where A represents the surface area and r represents the radius of the sphere.
A = 4 * π * 6.6^2 ≈ 346.4 square inches
Out of the given options, the closest surface area to the calculated value of 346.4 square inches is 391.7 square inches. Therefore, the correct option is 391.7 sq in.