Let's tackle this problem step by step.

a) Curved Surface Area of the Frustum

The curved surface area $A$ of a frustum is given by the formula:

$A = \pi (r_1 + r_2) l$

Where:

• $r_1$ is the radius of the larger base (8 cm),
• $r_2$ is the radius of the smaller base (2 cm),
• $l$ is the slant height (which we need to calculate).

Step 1: Calculate the Slant Height $l$

Using the Pythagorean theorem, we can find the slant height $l$. The vertical height of the frustum is 15 cm, and the difference in the radii of the two bases is $r_1 - r_2 = 8 - 2 = 6$ cm.

$l = \sqrt{(15^2 + 6^2)} = \sqrt{225 + 36} = \sqrt{261}$

Approximating:

$l \approx 16.155 cm$

Step 2: Calculate the Curved Surface Area

Now, substitute the values of $r_1$, $r_2$, and $l$ into the formula for the curved surface area:

$A = \pi (8 + 2) \times 16.155 = \pi \times 10 \times 16.155 = 161.55 \pi \, \text{cm}^2$

Thus, the curved surface area of the frustum is:

$A \approx 161.55 \pi \, \text{cm}^2$

b) Total Surface Area of the Frustum

The total surface area $A_{\text{total}}$ of the frustum includes the curved surface area and the areas of both circular bases.

The area of a circle is given by $\pi r^2$. Thus, the areas of the two circular bases are:

• Larger base: $\pi \times 8^2 = 64 \pi \, \text{cm}^2$
• Smaller base: $\pi \times 2^2 = 4 \pi \, \text{cm}^2$

Step 3: Calculate the Total Surface Area

The total surface area is the sum of the curved surface area and the areas of both bases:

$A_{\text{total}} = 161.55 \pi + 64 \pi + 4 \pi = 229.55 \pi \, \text{cm}^2$

Thus, the total surface area is:

$A_{\text{total}} \approx 229.55 \pi \, \text{cm}^2$

Would you like more details on any step or further clarifications?

Here are 5 related questions to deepen your understanding:

1. How would the formulas change if the height of the frustum was different?
2. How can we calculate the volume of this frustum?
3. What is the significance of the slant height in surface area calculations?
4. How do you find the surface area of a cone (without the frustum)?
5. How would we calculate the surface area if the frustum had different units?

Tip: Always check if the slant height or vertical height is given in problems involving frustums. It affects the approach!