Question: a) Calculate the length $x$.
b) Work out the total surface area of the frustum. Give your answer in terms of $\pi$.
Graph/shape: A cone frustum is shown in the center; the top circular radius is $4$ cm, the slanted side of the small removed cone is $7$ cm, the full slanted side is $35$ cm, and the bottom radius is $x$.
1. **State the problem:**
We have a frustum of a cone formed by cutting a smaller cone from a larger cone. The top radius is $4$ cm, the slant height of the small removed cone is $7$ cm, the full slant height of the original cone is $35$ cm, and the bottom radius is $x$. We need to find $x$ and then calculate the total surface area of the frustum in terms of $\pi$.
2. **Find the length $x$ (bottom radius):**
The original cone and the smaller removed cone are similar triangles when viewed from the side.
The ratio of the radii equals the ratio of the slant heights:
$$\frac{4}{x} = \frac{7}{35}$$
3. **Solve for $x$:**
Cross multiply:
$$4 \times 35 = 7 \times x$$
$$140 = 7x$$
Divide both sides by 7:
$$\cancel{7}x = \frac{140}{\cancel{7}}$$
$$x = 20$$
So, the bottom radius $x$ is $20$ cm.
4. **Calculate the total surface area of the frustum:**
The total surface area $A$ of a frustum of a cone is given by:
$$A = \pi (r_1^2 + r_2^2) + \pi (r_1 + r_2) l$$
where $r_1$ and $r_2$ are the radii of the two circular ends, and $l$ is the slant height of the frustum.
5. **Find the slant height $l$ of the frustum:**
The slant height of the frustum is the difference between the full slant height and the small cone's slant height:
$$l = 35 - 7 = 28$$
6. **Substitute values:**
$$r_1 = 4, \quad r_2 = 20, \quad l = 28$$
Calculate the areas:
$$A = \pi (4^2 + 20^2) + \pi (4 + 20) \times 28$$
$$= \pi (16 + 400) + \pi (24) \times 28$$
$$= \pi (416) + \pi (672)$$
$$= \pi (416 + 672)$$
$$= \pi \times 1088$$
7. **Final answer:**
The length $x$ is $20$ cm.
The total surface area of the frustum is:
$$1088 \pi \text{ cm}^2$$