The total surface area of a solid hemisphere is equal to the curved surface area of a cylinder. The radius of the hemisphere is r cm. The radius of the cylinder is twice the radius... The total surface area of a solid hemisphere is equal to the curved surface area of a cylinder. The radius of the hemisphere is r cm. The radius of the cylinder is twice the radius of the hemisphere. Given that the volume of hemisphere : volume of cylinder = 1 : m, find the value of m. Read more

Steps to Solve

1. Understand the surface area condition

The total surface area of the hemisphere is given by:

$$ \text{Total Surface Area of Hemisphere} = 2\pi r^2 + \pi r^2 = 3\pi r^2 $$

The curved surface area of the cylinder, with radius (2r) and height (h), is:

$$ \text{Curved Surface Area of Cylinder} = 2\pi (2r) h = 4\pi rh $$

Setting these equal gives:

$$ 3\pi r^2 = 4\pi rh $$

2. Simplify the equation

Cancel out (\pi) from both sides:

$$ 3r^2 = 4rh $$

Now, we can solve for (h):

$$ h = \frac{3r}{4} $$

3. Calculate the volumes

The volume of the hemisphere is given by:

$$ V_{\text{hemisphere}} = \frac{2}{3}\pi r^3 $$

The volume of the cylinder is given by:

$$ V_{\text{cylinder}} = \pi (2r)^2 h = \pi (4r^2) \left(\frac{3r}{4}\right) = 3\pi r^3 $$

4. Set up the ratio of the volumes

Now, we can find the ratio of the volumes:

$$ \text{Volume Ratio} = \frac{V_{\text{hemisphere}}}{V_{\text{cylinder}}} = \frac{\frac{2}{3}\pi r^3}{3\pi r^3} $$

5. Simplify the ratio

Cancel out (\pi r^3):

$$ \text{Volume Ratio} = \frac{\frac{2}{3}}{3} = \frac{2}{9} $$

From the problem, this is given as (1 : m).

6. Find the value of (m)

From the ratio:

$$ 1 : m = \frac{2}{9} $$

So, we set this equal to (1 : m):

$$ m = \frac{9}{2} $$