Two castings of the same metal have the same surface area. One castings is in the form of a sphere and the other is a cube. What is the ratio of the solidification time for the sphere to that of the cube?a)3/4b)6/πc)1πd)3/πCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Surface Area of a Sphere:
The surface area of a sphere is given by the formula 4πr^2, where r is the radius of the sphere.

Surface Area of a Cube:
The surface area of a cube is given by the formula 6s^2, where s is the length of one side of the cube.

Given:
The two castings have the same surface area.

Let's assume:
Let's assume that the surface area of both the sphere and the cube is equal to A.

Equating the Surface Areas:
4πr^2 = 6s^2

Simplifying the Equation:
Dividing both sides by 2, we get:
2πr^2 = 3s^2

Ratio of Solidification Time:
The solidification time for a casting is directly proportional to its volume. The volume of a sphere is given by the formula (4/3)πr^3, and the volume of a cube is given by the formula s^3.

Let's assume:
Let's assume that the solidification time for the sphere is T1 and the solidification time for the cube is T2.

Equating the Volumes:
(4/3)πr^3 = s^3

Simplifying the Equation:
Dividing both sides by π:
(4/3)r^3 = (s^3)/π

Ratio of Solidification Time:
Since the solidification time is directly proportional to the volume, we can write:
T1/T2 = (4/3)r^3 / (s^3)/π

Substituting the Value of s^2:
Using the equation 2πr^2 = 3s^2, we can substitute the value of s^2 in the ratio equation:
T1/T2 = (4/3)r^3 / (2πr^2)^(3/2)

Simplifying the Equation:
T1/T2 = (4/3) / (2π)^(3/2) * (r^3 / r^3)
T1/T2 = (4/3) / (2π)^(3/2)

Approximating the Value:
Using the approximation π ≈ 3.14, we can simplify the equation further:
T1/T2 ≈ (4/3) / (2 * 3.14)^(3/2)

Calculating the Value:
T1/T2 ≈ (4/3) / (2 * 3.14)^(3/2)
T1/T2 ≈ (4/3) / (2 * 5.61)
T1/T2 ≈ 0.238

Final Answer:
The ratio of the solidification time for the sphere to that of the cube is approximately 0.238, which is equivalent to 6/π. Therefore, the correct answer is option B.

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