curved surface area of a cone $$ =\pi r l $$, where $$ r $$ is the radius and $$ l $$ is the slant height. A frustum is formed by removing a small cone from a similar, larger cone, as shown below. a) Calculate the length $$ x $$. b) Work out the total surface area of the frustum. Give your answer in terms

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It appears that the problem you've provided references a diagram or additional information that isn't included in your message. To accurately calculate the length $$ x $$ and determine the total surface area of the frustum, specific measurements or relationships between the dimensions of the larger and smaller cones are necessary.

However, I can provide you with a general approach to solving such problems involving frustums of cones. Let's break down both parts:

### **a) Calculating the Length $$ x $$**

**Understanding the Frustum:**
A frustum is created by removing a smaller, similar cone from a larger cone. Let's denote:
- $$ R $$ = Radius of the larger cone's base
- $$ r $$ = Radius of the smaller cone's base
- $$ H $$ = Height of the larger cone
- $$ h $$ = Height of the smaller cone
- $$ x $$ = Height of the frustum (the vertical height between the two circular bases)

**Using Similarity of Cones:**
Since the two cones are similar, the ratios of their corresponding dimensions are equal. Therefore:

$$
\frac{r}{R} = \frac{h}{H}
$$

Solving for $$ h $$:

$$
h = \frac{r}{R} \times H
$$

The height of the frustum $$ x $$ is the difference between the heights of the larger and smaller cones:

$$
x = H - h = H - \left(\frac{r}{R} \times H\right) = H \left(1 - \frac{r}{R}\right)
$$

**Expressed in Terms of Slant Heights:**
If you have the slant heights ($$ L $$ for the larger cone and $$ l $$ for the smaller cone), you can relate them using the Pythagorean theorem:

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

Given the similarity:

$$
\frac{l}{L} = \frac{r}{R} \quad \Rightarrow \quad l = \frac{r}{R} \times L
$$

You can substitute $$ h $$ from earlier into the equation for $$ l $$ to find $$ x $$ in terms of the slant heights if needed.

### **b) Total Surface Area of the Frustum**

The total surface area of a frustum includes the areas of the two circular bases and the curved (lateral) surface area.

**Components:**
1. **Top Base (Smaller Circle):**

$$
\text{Area} = \pi r^2
$$

2. **Bottom Base (Larger Circle):**

$$
\text{Area} = \pi R^2
$$

3. **Curved Surface Area:**

The curved surface area of the frustum can be found by subtracting the curved surface area of the smaller cone from that of the larger cone:

$$
\text{Curved Surface Area of Frustum} = \pi R L - \pi r l
$$

However, there's a more straightforward formula for the lateral surface area of a frustum:

$$
\text{Lateral Surface Area} = \pi (R + r) \times s
$$

where $$ s $$ is the slant height of the frustum. The slant height $$ s $$ can be related to the vertical height $$ x $$ and the radii $$ R $$ and $$ r $$:

$$
s = \sqrt{x^2 + (R - r)^2}
$$

**Total Surface Area:**

$$
\text{Total Surface Area} = \pi r^2 + \pi R^2 + \pi (R + r) s
$$

**Expressed in Terms of Given Variables:**

If you need the answer in terms of specific variables, substitute $$ s $$ with $$ \sqrt{x^2 + (R - r)^2} $$:

$$
\text{Total Surface Area} = \pi r^2 + \pi R^2 + \pi (R + r) \sqrt{x^2 + (R - r)^2}
$$

### **Summary**

- **Part a:** Without specific measurements, $$ x $$ can be expressed in terms of the heights and radii of the cones using similarity ratios.
  
- **Part b:** The total surface area of the frustum combines the areas of the two bases and the lateral surface area, which depends on the slant height of the frustum.

If you can provide additional details or specific measurements related to the problem, I can offer a more precise solution tailored to your needs.
To find the length $$ x $$ and the total surface area of the frustum, follow these steps: 1. **Calculate $$ x $$:** - Use the similarity of the cones to find $$ x $$ based on the radii and heights of the larger and smaller cones. 2. **Calculate the Total Surface Area:** - Add the areas of the two circular bases and the curved surface area of the frustum. - The curved surface area can be found using the slant height of the frustum. Without specific measurements, $$ x $$ and the total surface area can be expressed in terms of the given variables.

curved surface area of a cone ,
where is the radius and is the slant
height.
A frustum is formed by removing a
small cone from a similar, larger cone,
as shown below.
a) Calculate the length .
b) Work out the total surface area of
the frustum. Give your answer in terms

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curved surface area of a cone , where is the radius and is the slant height. A frustum is formed by removing a small cone from a similar, larger cone, as shown below. a) Calculate the length . b) Work out the total surface area of the frustum. Give your answer in terms