Let’s solve the given problem step by step based on the image provided.

Problem Summary:

The height of a right circular cone $A$ is $8 \, \text{cm}$ and its base area is $225 \pi \, \text{cm}^2$. The problem has three parts:

(a) Find the base radius of cone $A$.

(b) Find the total surface area of cone $A$ in terms of $\pi$.

(c) Cone $B$ is similar to cone $A$, and the height of cone $B$ is $12 \, \text{cm}$. Find the total surface area of cone $B$.

Step-by-Step Solution:

(a) Find the base radius of cone $A$:

The base area of a circle is given by: $\text{Base Area} = \pi r^2$ Here, the base area is $225 \pi \, \text{cm}^2$. Substituting: $\pi r^2 = 225 \pi$ Cancel $\pi$ on both sides: $r^2 = 225$ Take the square root: $r = \sqrt{225} = 15 \, \text{cm}$ Thus, the base radius of cone $A$ is: $r = 15 \, \text{cm}$

(b) Find the total surface area of cone $A$ in terms of $\pi$:

The total surface area of a cone is given by: $\text{Total Surface Area} = \pi r^2 + \pi r l$ where $r$ is the radius and $l$ is the slant height. To find $l$, we use the Pythagoras theorem: $l = \sqrt{r^2 + h^2}$ Substitute $r = 15 \, \text{cm}$ and $h = 8 \, \text{cm}$: $l = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17 \, \text{cm}$ Now calculate the total surface area: $\text{Total Surface Area} = \pi r^2 + \pi r l = \pi (15^2) + \pi (15)(17)$ $\text{Total Surface Area} = 225 \pi + 255 \pi = 480 \pi \, \text{cm}^2$ Thus, the total surface area of cone $A$ is: $480 \pi \, \text{cm}^2$

(c) Find the total surface area of cone $B$:

Cone $B$ is similar to cone $A$. For similar cones, the ratio of their surface areas is the square of the ratio of their corresponding dimensions (e.g., height or radius).

The height of cone $B$ is $12 \, \text{cm}$, and the height of cone $A$ is $8 \, \text{cm}$. Thus, the ratio of their heights is: $\frac{\text{Height of } B}{\text{Height of } A} = \frac{12}{8} = \frac{3}{2}$ The ratio of the surface areas is: $\left( \frac{3}{2} \right)^2 = \frac{9}{4}$ The total surface area of cone $A$ is $480 \pi \, \text{cm}^2$. Using the ratio, the total surface area of cone $B$ is: $\text{Surface Area of } B = \frac{9}{4} \times 480 \pi = 1080 \pi \, \text{cm}^2$ Thus, the total surface area of cone $B$ is: $1080 \pi \, \text{cm}^2$

Final Answers:

1. (a): The base radius of cone $A$ is $15 \, \text{cm}$.
2. (b): The total surface area of cone $A$ is $480 \pi \, \text{cm}^2$.
3. (c): The total surface area of cone $B$ is $1080 \pi \, \text{cm}^2$.

Would you like me to explain any of these steps further?

Here are 5 related questions for practice:

1. What is the volume of cone $A$?
2. If cone $B$ has a radius proportional to cone $A$, what is its radius?
3. How does the slant height of cone $B$ compare to that of cone $A$?
4. What is the curved surface area (CSA) of cone $A$?
5. How would the total surface area change if cone $B$'s height were doubled?

Tip: Always verify dimensions and units carefully when working with geometry!