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A square, with sides of length x cm, is inside a circle - Edexcel - GCSE Maths - Question 8 - 2017 - Paper 3

Question 8

A square, with sides of length x cm, is inside a circle. Each vertex of the square is on the circumference of the circle. The area of the circle is 49 cm². Work ou... show full transcript

Worked Solution & Example Answer:A square, with sides of length x cm, is inside a circle - Edexcel - GCSE Maths - Question 8 - 2017 - Paper 3

Step 1

The area of the circle is 49 cm²

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Answer

First, we can determine the radius of the circle. The area of a circle is given by the formula:

$A = heta^2$

where ( r ) is the radius. Given that the area is 49 cm², we can solve for ( r ):

$49 = \pi r^2$

Rearranging this gives:

$r^2 = \frac{49}{\pi}$

Calculating ( r ):

$r = \sqrt{\frac{49}{\pi}}$

Calculating this yields:

First compute ( \pi \approx 3.14 )
So, ( r^2 \approx \frac{49}{3.14} \approx 15.6 )
Thus, ( r \approx \sqrt{15.6} \approx 3.95 ) cm.

Step 2

Find x using the relationship with the square

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Answer

Each vertex of the square touches the circle. Therefore, the diameter of the circle is equal to the diagonal of the square. Using the relationship for the diagonal ( d ) of a square of side length ( x ):

$d = x\sqrt{2}$

The diameter of the circle is also:

$d = 2r = 2 \times 3.95 \approx 7.90 \text{ cm}$

Setting the two expressions for the diameter equal to each other gives:

$x\sqrt{2} = 7.90$

To find x, rearranging gives:

$x = \frac{7.90}{\sqrt{2}}$

Calculating ( x ):

First, compute ( \sqrt{2} \approx 1.414 )
So, ( x \approx \frac{7.90}{1.414} \approx 5.57 \text{ cm}$$.

Step 3

Give your answer correct to 3 significant figures

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Answer

Finally, rounding 5.57 to three significant figures, we find:

$x \approx 5.57 ext{ cm}$ .

A square, with sides of length x cm, is inside a circle - Edexcel - GCSE Maths - Question 8 - 2017 - Paper 3

Worked Solution & Example Answer:A square, with sides of length x cm, is inside a circle - Edexcel - GCSE Maths - Question 8 - 2017 - Paper 3

The area of the circle is 49 cm²

Find x using the relationship with the square

Give your answer correct to 3 significant figures

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