The diagram represents a solid cone - Edexcel - GCSE Maths - Question 19 - 2020 - Paper 2

The formula for the curved surface area of a cone is given by:

$A = \pi r l$

Where:

• $r$ is the radius
• $l$ is the slant height

Plugging in the values:

$A = \pi \times 10 \text{ cm} \times 25 \text{ cm}$

Thus,

$A = 250\pi \text{ cm}^2$

The smaller cone formed above the painted circle has a height of 15 cm (since the slant height of the original cone is 25 cm and the circle is 10 cm above the base). To find the radius of the smaller cone, we apply similar triangles:

$\frac{r_{small}}{h_{small}} = \frac{r_{big}}{h_{big}}$

Where:

• $r_{small}$ is the radius of the smaller cone.
• $h_{big} = 25 \text{ cm}$
• $h_{small} = 15 \text{ cm}$
• $r_{big} = 10 \text{ cm}$

Solving gives: $\frac{r_{small}}{15} = \frac{10}{25}$

This simplifies to: $r_{small} = \frac{10}{25} \times 15 = 6 \text{ cm}$

Now, we can find the curved surface area of the smaller cone using the same formula:

$A_{small} = \pi r_{small} l_{small}$

First, we need to find the slant height of the smaller cone using Pythagoras:

$l_{small} = \sqrt{(r_{small})^2 + (h_{small})^2} = \sqrt{(6)^2 + (15)^2}$

Calculating gives: $l_{small} = \sqrt{36 + 225} = \sqrt{261}$

Thus,

$A_{small} = \pi \times 6 \text{ cm} \times \sqrt{261}$

The area of the surface that is not painted grey is then: $Area_{not painted} = A - A_{small} = 250\pi - 6\pi\sqrt{261}$