the diagonal of a rectangle with side A,B(B>A)makes an angle 60 degree with one of its sides.the ratio of the volumes of the cylinders generated by revolving the rectangle about each of the adjacent sides is

Question

the diagonal of a rectangle with side A,B(B>A)makes an angle 60 degree with one of its sides.the ratio of the volumes of the cylinders generated by revolving the rectangle about each of the adjacent sides is

Answer 1

b/a = tan 60° = √3

The ratio of the volumes is thus

πb^2a/πa^2b = b/a = √3

Answer 2

To find the ratio of the volumes of the cylinders generated by revolving the rectangle about each of its adjacent sides, we first need to calculate the volumes of these cylinders.

Let's consider a rectangle with sides A and B, where B is greater than A. The diagonal of this rectangle forms a right triangle with one of its sides. Given that the angle between the diagonal and one side of the rectangle is 60 degrees, we can solve this problem by using trigonometric ratios.

In the right triangle formed, the side opposite the 60 degree angle is A, and the hypotenuse is the diagonal, which we can denote as D. The side adjacent to the 60 degree angle can be found using the cosine function:

cos(60) = A/D

To find the length of the side adjacent to the 60 degree angle, we rearrange the equation:

D = A / cos(60)

Now, let's revolve the rectangle about the side with length A. This generates a cylinder with height B and a radius equal to A. The volume of this cylinder can be found using the formula:

Volume1 = π * (A^2) * B

Next, let's revolve the rectangle about the side with length D. This generates a cylinder with height A and a radius equal to D (which is A / cos(60)). The volume of this cylinder can be calculated as:

Volume2 = π * ((A / cos(60))^2) * A

Now we can calculate the ratio of the volumes:

Ratio = Volume1 / Volume2
= (π * (A^2) * B) / (π * ((A / cos(60))^2) * A)

Simplifying the equation:

Ratio = (A^2 * B) / (((A / cos(60))^2) * A)
= (A * A * B) / (((A^2) / (cos(60))^2) * A)
= A / ((A^2) / (cos(60))^2)
= A * (cos(60))^2 / A^2
= (cos(60))^2 / A

Simplifying further:

Ratio = (3/4) / A

Therefore, the ratio of the volumes of the cylinders generated by revolving the rectangle about each of the adjacent sides is (3/4) divided by the length of side A.