Question 3

The toy is made of two parts joined together — a cone on top and a hemisphere (half of a sphere) at the bottom.

Total surface area = curved surface area (CSA) of the cone + curved surface area (CSA) of the hemisphere. (The flat circular faces are hidden inside where the two parts join, so we do not count them.)

Step 1 — Find the height of the cone

Let's first note down what is given:

• Radius of the cone, $r = 3.5\text{ cm}$
• Radius of the hemisphere $= 3.5\text{ cm}$ (same as the cone)
• Total height of the toy $= 15.5\text{ cm}$

For a hemisphere, its height is equal to its radius. So the hemisphere is $3.5\text{ cm}$ tall.

Looking at the figure, the total height is the cone's height plus the hemisphere's height. So to find the cone's height $h$, we subtract:

$h = \text{Total height} - \text{Height of hemisphere}$

$h = 15.5 - 3.5$

$\boxed{h = 12 \text{ cm}}$

Step 2 — Find the slant height of the cone

To find the CSA of a cone, we first need its slant height $l$ (the distance along the sloping side).

The radius, the height, and the slant height form a right-angled triangle, where the slant height is the hypotenuse. So by the Pythagoras theorem:

$(\text{hypotenuse})^2 = (\text{height})^2 + (\text{base})^2$

$l^2 = h^2 + r^2$

Taking the square root on both sides:

$l = \sqrt{h^2 + r^2}$

$l = \sqrt{(12)^2 + (3.5)^2}$

$l = \sqrt{144 + 12.25}$

$l = \sqrt{156.25}$

$\boxed{l = 12.5 \text{ cm}}$

Step 3 — Find the curved surface areas

Now we find the CSA of each part. We will use $\pi = \dfrac{22}{7}$.

Cone — the formula is CSA $= \pi r l$:

$\text{CSA of cone} = \frac{22}{7} \times 3.5 \times 12.5$

Here $3.5 \div 7 = 0.5$, which makes the multiplication simpler:

$\text{CSA of cone} = 22 \times 0.5 \times 12.5$

$\boxed{\text{CSA of cone} = 137.5 \text{ cm}^2}$

Hemisphere — the formula is CSA $= 2\pi r^2$:

$\text{CSA of hemisphere} = 2 \times \frac{22}{7} \times (3.5)^2$

$\text{CSA of hemisphere} = 2 \times \frac{22}{7} \times 12.25$

Again $12.25 \div 7 = 1.75$:

$\text{CSA of hemisphere} = 2 \times 22 \times 1.75$

$\boxed{\text{CSA of hemisphere} = 77 \text{ cm}^2}$

Step 4 — Find the total surface area

Finally, we add the two curved surface areas to get the total surface area of the toy:

$\text{Total Surface Area} = \text{CSA of cone} + \text{CSA of hemisphere}$

$\text{Total Surface Area} = 137.5 + 77$

$\boxed{\text{Total Surface Area} = 214.5 \text{ cm}^2}$

Answer

The total surface area of the toy is 214.5 cm².