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 slant height labeled $2y$; the dashed lines indicate the original cone apex 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 pink frustum of a cone is shown in the center; the top radius is labeled $x$, the bottom radius is labeled $3x$, and the slant height of the removed small cone is labeled $2y$ on the upper right side.
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:**
- Bottom radius = $3x$
- Top radius = $x$
- Slant height of the small removed cone = $2y$
4. **Find the slant height of the frustum:**
The frustum is formed by removing a smaller cone from a larger cone.
The slant height of the smaller cone is $2y$.
5. **Find the slant height of the original large cone:**
The large cone has bottom radius $3x$ and top radius $0$ (apex).
The small cone removed has radius $x$ and slant height $2y$.
6. **Calculate the slant height of the frustum:**
The frustum's slant height is the difference between the large cone's slant height and the small cone's slant height.
7. **Use Pythagoras theorem to find the large cone's slant height $L$:**
The difference in radius between large and small cone is $3x - x = 2x$.
The slant height of the small cone is $2y$.
The large cone's slant height $L$ satisfies:
$$L^2 = (2y)^2 + (2x)^2 = 4y^2 + 4x^2 = 4(y^2 + x^2)$$
So,
$$L = 2\sqrt{y^2 + x^2}$$
8. **Slant height of the frustum $l$ is:**
$$l = L - 2y = 2\sqrt{y^2 + x^2} - 2y$$
9. **Calculate curved surface area of the frustum:**
It is the difference between the curved surface areas of the large cone and the small cone:
$$\text{CSA}_{frustum} = \pi (3x) L - \pi (x)(2y) = 3\pi x L - 2\pi x y$$
10. **Substitute $L$ from step 7:**
$$\text{CSA}_{frustum} = 3\pi x (2\sqrt{y^2 + x^2}) - 2\pi x y = 6\pi x \sqrt{y^2 + x^2} - 2\pi x y$$
11. **Express in the form $2\pi x(ax + by)$:**
Rewrite $6\pi x \sqrt{y^2 + x^2} - 2\pi x y$ as:
$$2\pi x \left(3 \sqrt{y^2 + x^2} - y\right)$$
12. **Rewrite $\sqrt{y^2 + x^2}$ in terms of $x$ and $y$:**
$$\sqrt{y^2 + x^2} = x \sqrt{1 + \left(\frac{y}{x}\right)^2}$$
So,
$$3 \sqrt{y^2 + x^2} = 3x \sqrt{1 + \left(\frac{y}{x}\right)^2}$$
13. **Therefore, the total surface area is:**
$$2\pi x \left(3x \sqrt{1 + \left(\frac{y}{x}\right)^2} - y\right)$$
14. **Compare with $2\pi x(ax + by)$:**
We identify:
$$a = 3 \sqrt{1 + \left(\frac{y}{x}\right)^2}$$
$$b = -1$$
15. **Final answer:**
$$a = 3 \sqrt{1 + \left(\frac{y}{x}\right)^2}, \quad b = -1$$