The problem provided in the image is about a solid consisting of a cone and a hemisphere that share a common base. The radius of the base is given as $35$ cm. The volume of the cone is stated to be $\frac{1}{5}$ of the volume of the hemisphere. The question asks to find:

(i) The height of the cone.

Let's solve this step-by-step:

Step 1: Find the Volume of the Hemisphere

The volume $V$ of a hemisphere with radius $r$ is given by:

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

Given $r = 35$ cm:

$V_{\text{hemisphere}} = \frac{2}{3} \pi (35)^3$

Step 2: Set Up the Equation for the Cone's Volume

The volume of the cone $V_{\text{cone}}$ is given to be $\frac{1}{5}$ of the volume of the hemisphere:

$V_{\text{cone}} = \frac{1}{5} V_{\text{hemisphere}}$

Substitute the value of $V_{\text{hemisphere}}$:

$V_{\text{cone}} = \frac{1}{5} \left( \frac{2}{3} \pi (35)^3 \right)$

Simplify this:

$V_{\text{cone}} = \frac{2}{15} \pi (35)^3$

Step 3: Find the Formula for the Volume of the Cone

The volume of a cone is given by:

$V_{\text{cone}} = \frac{1}{3} \pi r^2 h$

where $r = 35$ cm and $h$ is the height of the cone.

Step 4: Equate the Two Expressions for the Volume of the Cone

We have two expressions for the volume of the cone. Setting them equal to each other gives:

$\frac{1}{3} \pi (35)^2 h = \frac{2}{15} \pi (35)^3$

Step 5: Solve for the Height $h$

Simplify the equation:

$\frac{1}{3} \pi (35)^2 h = \frac{2}{15} \pi (35)^3$

Cancel out $\pi (35)^2$ from both sides:

$\frac{1}{3} h = \frac{2}{15} \times 35$

Multiply both sides by 3:

$h = \frac{2 \times 35}{5} = 14$

So, the height $h$ of the cone is 14 cm.

Would you like more details, or do you have any other questions?

Related Questions:

1. How do you find the volume of a sphere and how does it relate to a hemisphere?
2. What would the total volume of the solid be (cone + hemisphere)?
3. If the radius were doubled, how would the volume of the cone and hemisphere change?
4. How does changing the height of the cone affect the volume ratio?
5. Can you calculate the surface area of the solid formed by the cone and hemisphere?

Tip:

When working with volumes, always ensure you understand the geometric formulas and how they relate to the dimensions provided in the problem.