Question 8

We need to find the total surface area of the remaining solid after hollowing out a cone from a cylinder.

Step 1 — Find the dimensions

Let's write down the given dimensions for the cylinder. The height of the cylinder is $h = \mathbf{2.4 \text{ cm}}$. The diameter of the cylinder is $\mathbf{1.4 \text{ cm}}$. So, the radius of the cylinder is half of the diameter.

$r = \frac{1.4}{2}$

$r = \mathbf{0.7 \text{ cm}}$

The conical cavity has the same height and diameter. So, for the cone, its height is $h = 2.4\text{ cm}$ and its radius is $r = 0.7\text{ cm}$.

We need to find the slant height ($l$) of the cone. 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:

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

$l = \sqrt{(2.4)^2 + (0.7)^2}$

$l = \sqrt{5.76 + 0.49}$

$l = \sqrt{6.25}$

$\boxed{l = 2.5 \text{ cm}}$

Step 2 — Calculate the total surface area

The total surface area of the remaining solid is made of three parts:

• the curved surface of the cylinder (the outer wall)
• the curved surface of the conical cavity (the sloping inner surface of the scooped-out cone)
• the base of the cylinder (the flat circle at the bottom)

We use these formulas:

• CSA of a cylinder $= 2\pi rh$
• CSA of a cone $= \pi rl$
• Area of a circular base $= \pi r^2$

Adding them gives the total surface area (TSA):

$\text{TSA} = 2\pi rh + \pi rl + \pi r^2$

Now we substitute the values $r = \mathbf{0.7 \text{ cm}}$, $h = \mathbf{2.4 \text{ cm}}$, and $l = \mathbf{2.5 \text{ cm}}$, using $\pi = \dfrac{22}{7}$:

$\text{TSA} = \left(2 \times \frac{22}{7} \times \mathbf{0.7} \times \mathbf{2.4}\right) + \left(\frac{22}{7} \times \mathbf{0.7} \times \mathbf{2.5}\right) + \left(\frac{22}{7} \times \mathbf{0.7} \times \mathbf{0.7}\right)$

Here $0.7 \div 7 = 0.1$, which makes each term simpler:

$\text{TSA} = (2 \times 22 \times 0.1 \times 2.4) + (22 \times 0.1 \times 2.5) + (22 \times 0.1 \times 0.7)$

$\text{TSA} = (4.4 \times 2.4) + (2.2 \times 2.5) + (2.2 \times 0.7)$

$\text{TSA} = 10.56 + 5.50 + 1.54$

$\boxed{\text{TSA} = 17.60 \text{ cm}^2}$

We need to round this to the nearest cm². The digit after the decimal point is $\mathbf{6}$, which is greater than $\mathbf{5}$. So, we round up the whole number part.

$\boxed{18 \text{ cm}^2}$

Answer

The total surface area of the remaining solid is $\mathbf{18 \text{ cm}^2}$.