Question

Spheres A and B are similar.

The volume of Sphere A x 27 = the volume of Sphere B

How many times greater is the radius of sphere A?

Answer 1

[AI-generated answer — Bot]

The radius of Sphere A is three times greater than the radius of Sphere B.

Answer 2

Yeahhh! let's hear it for the robot, it got one right!!!

Answer 3

[AI-generated answer — Step-by-Step Bot]

To find out how many times greater the radius of Sphere A is compared to Sphere B, we need to use the relationship between the volumes of similar spheres.

The volumes of similar spheres are proportional to the cube of their radii. So, if the volume of Sphere A is 27 times the volume of Sphere B, then the ratio of their volumes can be expressed as:

(Volume of Sphere A) / (Volume of Sphere B) = 27

Knowing that the volume of a sphere is given by the formula:

Volume = (4/3) * π * (radius)^3

Let's denote the radius of Sphere B as r. So, the volume of Sphere B can be written as:

Volume of Sphere B = (4/3) * π * r^3

We can plug this into the equation and solve for the radius of Sphere A:

(4/3) * π * (radius of Sphere A)^3 = 27 * (4/3) * π * r^3

Cancel out the common terms:

(radius of Sphere A)^3 = 27 * r^3

Take the cube root of both sides:

radius of Sphere A = ∛(27 * r^3)

Simplifying:

radius of Sphere A = 3 * r

Therefore, the radius of Sphere A is three times greater than the radius of Sphere B.

Answer 4

[AI-generated answer — Explain Bot]

To determine how many times greater the radius of Sphere A is compared to Sphere B, we can use the relationship between the volumes of the spheres.

The volume of a sphere is calculated using the formula V = (4/3)πr³, where V is the volume and r is the radius.

Let's denote the radius of Sphere A as rA and the radius of Sphere B as rB.

Since Sphere A and Sphere B are similar, we know that the ratio of their volumes is equal to the cube of the ratio of their radii:

(Volume of Sphere A) / (Volume of Sphere B) = (rA³) / (rB³)

Given that the volume of Sphere A times 27 is equal to the volume of Sphere B, we can write the equation:

27 * (4/3)πrA³ = (4/3)πrB³

By canceling out the common terms, we get:

27 * rA³ = rB³

To find the ratio of the radii, we can take the cube root of both sides of the equation:

∛27 * ∛(rA³) = ∛rB³

Simplifying further:

3 * rA = rB

Now we can determine how many times greater the radius of Sphere A is compared to Sphere B by dividing the radius of Sphere A by the radius of Sphere B:

(rA / rB) = (3 * rA) / rB

(rA / rB) = 3

Hence, the radius of Sphere A is three times greater than the radius of Sphere B.