Question

Объем первого шара в 27 раз больше объема второго. во сколько раз площадь поверхности первого шара больше площади поверхности второго?

Answer 1

Объем пропорционален кубу радиуса, т.о. радиус первого шара в ∛27=3, больше радиуса второго.
площадь поверхности пропорциональна квадрату радиуса, т.е. будет больше в 3^2
=9 раз

Answer 2

Calculating the Surface Area Ratio of Two Spheres

To calculate the ratio of the surface areas of two spheres when the volume of the first sphere is 27 times larger than the volume of the second sphere, we can use the following approach:

1. Volume and Surface Area Relationship: - The volume of a sphere is given by the formula: \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the sphere. - The surface area of a sphere is given by the formula: \( A = 4 \pi r^2 \).

2. Using the Volume Ratio to Find the Radius Ratio: - If the volume of the first sphere is 27 times larger than the volume of the second sphere, then the radius of the first sphere is \( 3 \) times larger than the radius of the second sphere.

3. Calculating the Surface Area Ratio: - Since the surface area is directly proportional to the square of the radius, the ratio of the surface areas of the two spheres will be the square of the ratio of their radii.

Applying the Formulas to Find the Surface Area Ratio

Let's apply the formulas to find the ratio of the surface areas of the two spheres.

The volume ratio is \( 27:1 \), which means the radius ratio is \( 3:1 \).

Using the formula for the surface area of a sphere, we can calculate the surface area ratio:

For the first sphere: - Radius: \( R_1 \) - Surface Area: \( A_1 = 4 \pi R_1^2 \)

For the second sphere: - Radius: \( R_2 \) - Surface Area: \( A_2 = 4 \pi R_2^2 \)

The ratio of the surface areas \( A_1:A_2 \) will be \( (R_1^2):(R_2^2) \).

Conclusion

The ratio of the surface areas of the two spheres, when the volume of the first sphere is 27 times larger than the volume of the second sphere, is \( 9:1 \). This means that the surface area of the first sphere is 9 times larger than the surface area of the second sphere.