Q3:

A toy is in the form of a cone of radius $3.5$ cm mounted on a hemisphere of same radius. The total height of the toy is $15.5$ cm. Find the total surface area of the toy.

Given:

1. The toy consists of a cone mounted on a hemisphere.
2. The radius of the cone ($r$) = $3.5$ cm.
3. The radius of the hemisphere ($r$) = $3.5$ cm.
4. The total height of the toy ($H$) = $15.5$ cm.

To Find:

The total surface area (TSA) of the toy.

Step 1: Determine the height of the conical part ($h$)

The total height of the toy is the sum of the height of the cone ($h$) and the radius of the hemisphere ($r$).
$H = h + r$
$15.5 = h + 3.5$
$h = 15.5 - 3.5$
$h = 12$ cm

Step 2: Calculate the slant height of the cone ($l$)

The formula for the slant height of a cone is $l = \sqrt{r^2 + h^2}$.
$l = \sqrt{(3.5)^2 + (12)^2}$
$l = \sqrt{12.25 + 144}$
$l = \sqrt{156.25}$
$l = 12.5$ cm

Step 3: Formulate the Total Surface Area (TSA) of the toy

The total surface area of the toy is the sum of the curved surface area (CSA) of the cone and the curved surface area (CSA) of the hemisphere.
$TSA = \text{CSA of cone} + \text{CSA of hemisphere}$
$TSA = \pi rl + 2\pi r^2$
$TSA = \pi r (l + 2r)$

Step 4: Perform the calculation

Substitute the values $r = 3.5$ cm, $l = 12.5$ cm, and $\pi = \frac{22}{7}$:
$TSA = \frac{22}{7} \times 3.5 \times (12.5 + 2(3.5))$
$TSA = \frac{22}{7} \times 3.5 \times (12.5 + 7)$
$TSA = \frac{22}{7} \times 3.5 \times 19.5$
Since $\frac{3.5}{7} = 0.5$:
$TSA = 22 \times 0.5 \times 19.5$
$TSA = 11 \times 19.5$
$TSA = 214.5$ cm$^2$

Final Answer: The total surface area of the toy is 214.5 cm$^2$.