The diagram shows two shaded shapes, A and B. Shape A is formed by removing a sector of a circle with radius $(3x - 1)$ cm from a sector of the circle with radius $(5 - 1)$ cm. Shape B is a circle of diameter $(2 - 2x)$ cm. The area of shape A is equal to the area of shape B. Find the value of $x$. You must show all your working.

GCSE Edexcel Maths Volume, Area & Surface Area: The diagram shows two shaded sha

The diagram shows two shaded shapes, A and B - Edexcel - GCSE Maths - Question 22 - 2020 - Paper 1

Question 22

The diagram shows two shaded shapes, A and B. Shape A is formed by removing a sector of a circle with radius $(3x - 1)$ cm from a sector of the circle with radius $... show full transcript

Worked Solution & Example Answer:The diagram shows two shaded shapes, A and B - Edexcel - GCSE Maths - Question 22 - 2020 - Paper 1

Step 1

Derive an algebraic expression for the area of A

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Answer

To find the area of shape A, we first need to calculate the area of the sector of the larger circle and subtract the area of the smaller sector:

For the larger circle with radius $4$ cm (since $5 - 1 = 4$ ):

The area of a sector is given by: $\text{Area} = \frac{1}{2} r^2 \theta$

For shape A, we can write: $\text{Area}_A = \frac{1}{2} (5 - 1)^2 \theta - \frac{1}{2} (3x - 1)^2 \theta$

Factoring out $heta$ , we obtain: $\text{Area}_A = \frac{\theta}{2} \left( 16 - (9x^2 - 6x + 1) \right) = \frac{\theta}{2} (15 - 9x^2 + 6x)$

Step 2

Derive an algebraic expression for the area of B

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Answer

For shape B which is a circle with diameter $(2 - 2x)$ cm, we first find the radius:

The area of circle B is: $\text{Area}_B = \pi r^2 = \pi (1 - x)^2 = \pi (1 - 2x + x^2)$

Step 3

Set the areas equal and rearrange

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Answer

$\frac{\theta}{2} (15 - 9x^2 + 6x) = \pi (1 - 2x + x^2)$

Assuming $heta \neq 0$ , we simplify this to: $15 - 9x^2 + 6x = \frac{2\pi}{\theta} (1 - 2x + x^2)$

Rearranging to form a quadratic equation gives: $-9x^2 + 6x - \frac{2\pi}{\theta} + \frac{2\pi}{\theta} 2x = 0$

Step 4

Factor or use the quadratic formula to find x

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Answer

Replacing the values, we simplify: $-9x^2 + (6 + 4\pi)x - \frac{2\pi}{\theta} = 0$

Using the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , we find: $x = \frac{-(6 + 4\pi) \pm \sqrt{(6 + 4\pi)^2 - 4(-9)(-\frac{2\pi}{\theta})}}{2(-9)}$

Thus we can find the value of $x$ .

The diagram shows two shaded shapes, A and B - Edexcel - GCSE Maths - Question 22 - 2020 - Paper 1

Worked Solution & Example Answer:The diagram shows two shaded shapes, A and B - Edexcel - GCSE Maths - Question 22 - 2020 - Paper 1

Derive an algebraic expression for the area of A

Derive an algebraic expression for the area of B

Set the areas equal and rearrange

Factor or use the quadratic formula to find x

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