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.

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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

Find the area of shape A

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Answer

To find the area of shape A, we need to calculate the area of the larger sector and then subtract the area of the smaller sector.

The area of a sector is calculated using the formula: $A = \frac{\theta}{360} \times \pi r^2$ where $\theta$ is the central angle in degrees.

For shape A, taking $\theta$ to be $45^{\circ}$ (as implied from the diagram context), the area of the larger sector (radius = $5 - 1$ ) and the smaller sector (radius = $3x - 1$ ) can be expressed as:

$A_{large} = \frac{45}{360} \times \pi (4)^2 = \frac{1}{8} \times 16\pi = 2\pi$

$A_{small} = \frac{45}{360} \times \pi (3x - 1)^2 = \frac{1}{8} \times \pi (3x - 1)^2$

Therefore, the area of shape A is: $A = A_{large} - A_{small} = 2\pi - \frac{\pi (3x - 1)^2}{8}$

Step 2

Find the area of shape B

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Answer

For shape B, the diameter is given as $(2 - 2x)$ cm, thus the radius $r$ is: $r = \frac{2 - 2x}{2} = 1 - x$

The area of shape B is calculated as: $A_{B} = \pi r^2 = \pi (1 - x)^2 = \pi (1 - 2x + x^2)$

Step 3

Set the areas equal and solve for x

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Answer

Setting the areas of shape A and shape B equal gives:

$2\pi - \frac{\pi (3x - 1)^2}{8} = \pi (1 - 2x + x^2)$

Dividing through by $\pi$ (assuming $\pi \neq 0$ ) yields:

$2 - \frac{(3x - 1)^2}{8} = 1 - 2x + x^2$

Multiply through by 8 to eliminate the fraction:

$16 - (3x - 1)^2 = 8 - 16x + 8x^2$

Expanding $(3x - 1)^2$ results in: $16 - (9x^2 - 6x + 1) = 8 - 16x + 8x^2$

This simplifies to: $16 - 9x^2 + 6x - 1 = 8 - 16x + 8x^2$

Rearranging the equation will give us a quadratic equation in the standard form. Once we do that, we can factor or use the quadratic formula to find the values 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

Find the area of shape A

Find the area of shape B

Set the areas equal and solve for x

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