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The diagram shows a cube with edges of length $x$ cm and a sphere of radius 3 cm - Edexcel - GCSE Maths - Question 8 - 2021 - Paper 1

Question 8

The diagram shows a cube with edges of length $x$ cm and a sphere of radius 3 cm. The surface area of the cube is equal to the surface area of the sphere. Show tha... show full transcript

Worked Solution & Example Answer:The diagram shows a cube with edges of length $x$ cm and a sphere of radius 3 cm - Edexcel - GCSE Maths - Question 8 - 2021 - Paper 1

Step 1

The surface area of the cube

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Answer

The surface area of a cube can be calculated using the formula: $ext{Surface Area}_{ ext{cube}} = 6a^2$ where $a$ is the length of an edge of the cube. Substituting $a = x$ , we get: $ext{Surface Area}_{ ext{cube}} = 6x^2.$

Step 2

The surface area of the sphere

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Answer

The surface area of a sphere is given by: $ext{Surface Area}_{ ext{sphere}} = 4\pi r^2$ Here, $r = 3$ cm, thus: $ext{Surface Area}_{ ext{sphere}} = 4\pi (3)^2 = 36\pi.$

Step 3

Setting up the equation

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Answer

To find $x$ , equate the two surface areas: $6x^2 = 36\pi.$

Step 4

Solving for x

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Answer

Rearranging the equation gives us: $x^2 = 6\pi,$ thus, $x = \sqrt{6\pi}.$ Given that we want to show that $x = \sqrt{k}/2$ , we can set $k = 24\pi$ , leading to: $x = \frac{\sqrt{24\pi}}{2}.$ Therefore, $k$ is indeed a multiple of 24 which is an integer.

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