Question: curved surface area of a cone = $\pi rl$ where $r$ is the radius and $l$ is the slant height.
A frustum is shown below.
The total surface area of the frustum can be written in the form $2\pi x(ax + by)$, where $a$ and $b$ are numbers.
Work out the values of $a$ and $b$.
Graph/shape: bottom-right, a frustum of a cone is shown with the top radius labeled $x$ and the slanted height labeled $2y$ on the right side; the dashed lines indicate the removed cone above the frustum.
The total surface area of the frustum can be written in the form $2\pi x(ax + by)$, where $a$ and $b$ are numbers.
Work out the values of $a$ and $b$.
Not drawn accurately
Graph/shape: A frustum of a cone is shown in the center/bottom of the page, with top radius $x$, bottom radius $3x$, and slant height $2y$ on the right side; dashed lines extend above the frustum to a cone apex.
1. **State the problem:** We need to find the values of $a$ and $b$ such that the total surface area of the frustum can be expressed as $2\pi x(ax + by)$.
2. **Recall the formula for curved surface area of a cone:**
$$\text{Curved Surface Area} = \pi r l$$
where $r$ is the radius and $l$ is the slant height.
3. **Identify the parts of the frustum:**
- Top radius = $x$
- Bottom radius = $3x$
- Slant height of the frustum = $2y$
4. **Calculate the curved surface area of the frustum:**
The curved surface area of a frustum is the difference between the curved surface areas of the large cone and the smaller cone removed from the top.
- Large cone radius = $3x$
- Large cone slant height = $l$ (unknown for now)
- Small cone radius = $x$
- Small cone slant height = $l - 2y$ (since the frustum slant height is $2y$)
5. **Find $l$ using similar triangles:**
The height and radius scale proportionally.
Since the frustum slant height is $2y$, the difference in slant heights between the large and small cones is $2y$.
Using Pythagoras or similar triangles, the slant height of the large cone $l$ satisfies:
$$l^2 = (2y)^2 + (3x - x)^2 = 4y^2 + (2x)^2 = 4y^2 + 4x^2$$
So,
$$l = \sqrt{4x^2 + 4y^2} = 2\sqrt{x^2 + y^2}$$
6. **Calculate curved surface areas:**
- Large cone curved surface area:
$$\pi \times 3x \times l = \pi \times 3x \times 2\sqrt{x^2 + y^2} = 6\pi x \sqrt{x^2 + y^2}$$
- Small cone curved surface area:
$$\pi \times x \times (l - 2y) = \pi x (2\sqrt{x^2 + y^2} - 2y) = 2\pi x (\sqrt{x^2 + y^2} - y)$$
7. **Frustum curved surface area = large cone area - small cone area:**
$$6\pi x \sqrt{x^2 + y^2} - 2\pi x (\sqrt{x^2 + y^2} - y) = 6\pi x \sqrt{x^2 + y^2} - 2\pi x \sqrt{x^2 + y^2} + 2\pi x y = (6 - 2)\pi x \sqrt{x^2 + y^2} + 2\pi x y = 4\pi x \sqrt{x^2 + y^2} + 2\pi x y$$
8. **Express in the form $2\pi x(ax + by)$:**
We want to write:
$$4\pi x \sqrt{x^2 + y^2} + 2\pi x y = 2\pi x (2\sqrt{x^2 + y^2} + y)$$
Note that $\sqrt{x^2 + y^2}$ is not linear in $x$ and $y$, so we approximate or interpret the problem differently.
9. **Alternative approach (given the problem statement):**
The problem states the total surface area can be written as $2\pi x(ax + by)$.
Given the frustum has bottom radius $3x$, top radius $x$, and slant height $2y$, the curved surface area of the frustum is:
$$\pi (r_1 + r_2) l = \pi (3x + x) \times 2y = \pi \times 4x \times 2y = 8\pi x y$$
The total surface area includes the two circular areas:
- Bottom circle area: $\pi (3x)^2 = 9\pi x^2$
- Top circle area: $\pi x^2$
Total surface area:
$$8\pi x y + 9\pi x^2 + \pi x^2 = 8\pi x y + 10\pi x^2 = 2\pi x (5x + 4y)$$
10. **Therefore, the values are:**
$$a = 5, \quad b = 4$$