Let's solve the questions step by step based on the image provided.

(a) Show that $l = 8$:

We know the total surface area (TSA) of the cone is given as:

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

Here:

• $r = 6 \, \text{cm}$ (radius)
• $l$ = slant height (to be determined)
• TSA = $84\pi \, \text{cm}^2$

Substituting the values into the formula:

$84\pi = \pi (6)(l) + \pi (6^2)$

Factor out $\pi$:

$84 = 6l + 36$

Simplify:

$6l = 84 - 36$ $6l = 48$

Divide by 6:

$l = 8 \, \text{cm}$

Thus, $l = 8$ is proven.

(b) Calculate the volume of the cone:

The volume of a cone is given by:

$V = \frac{1}{3} \pi r^2 h$

Here:

• $r = 6 \, \text{cm}$
• $h$ = height (to be calculated using Pythagoras’ theorem)
• $l = 8 \, \text{cm}$ (slant height)

From the Pythagoras theorem:

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

Substitute $l = 8$ and $r = 6$:

$8^2 = 6^2 + h^2$ $64 = 36 + h^2$ $h^2 = 64 - 36$ $h^2 = 28 \implies h = \sqrt{28} = 2\sqrt{7} \, \text{cm}$

Now, substitute $r = 6$ and $h = 2\sqrt{7}$ into the volume formula:

$V = \frac{1}{3} \pi (6^2) (2\sqrt{7})$ $V = \frac{1}{3} \pi (36)(2\sqrt{7})$ $V = \frac{1}{3} \cdot 72\sqrt{7} \pi$ $V = 24\sqrt{7} \pi \, \text{cm}^3$

Thus, the volume is $24\sqrt{7} \pi \, \text{cm}^3$.

(c) Find the radius of a similar cone with TSA = $47.25\pi \, \text{cm}^2$:

For a similar cone, the total surface area formula is the same:

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

Here:

• TSA = $47.25\pi$
• Let $r$ be the unknown radius.
• Since the cone is similar, the ratio of $l$ to $r$ remains the same as the original cone.

In the original cone:

$\frac{l}{r} = \frac{8}{6} = \frac{4}{3}$

So, for the new cone:

$l = \frac{4}{3}r$

Substitute into the TSA formula:

$47.25\pi = \pi r \left(\frac{4}{3}r\right) + \pi r^2$ $47.25 = \frac{4}{3}r^2 + r^2$ $47.25 = \frac{7}{3}r^2$

Multiply through by 3:

$141.75 = 7r^2$

Solve for $r^2$:

$r^2 = \frac{141.75}{7} = 20.25$

$r = \sqrt{20.25} = 4.5 \, \text{cm}$

Thus, the radius of the new cone is $4.5 \, \text{cm}$.

Summary of Answers:

(a) $l = 8 \, \text{cm}$ (proven)
(b) Volume = $24\sqrt{7} \pi \, \text{cm}^3$
(c) Radius of the similar cone = $4.5 \, \text{cm}$

Would you like me to explain any of the steps in more detail? 😊

Expanding Questions:

1. What is the relationship between slant height and height in a cone?
2. How does similarity affect geometric properties of cones?
3. Can we derive the volume formula for a cone starting from basic principles?
4. What other methods can be used to prove slant height in a cone?
5. How does scaling a cone affect its surface area and volume?

Tip:

Always double-check units in geometry problems to ensure consistent calculations.