Helen is creating a mosaic pattern by placing square tiles next to each other along a straight line. The area of each tile is half the area of the previous tile, and the sides of the largest tile have length w centimetres. (a) Find, in terms of w, the length of the sides of the second largest tile. [1 mark] (b) Assume the tiles are in contact with adjacent tiles, but do not overlap. Show that, no matter how many tiles are in the pattern, the total length of the series of tiles will less than 3.5w. [4 marks] (c) Helen decides the pattern will look better if she leaves a 3 millimetre gap between adjacent tiles. Explain how you could refine the model used in part (b) to account for the 3 millimetre gap, and state how the total length of the series of tiles will be affected. [2 marks]

A-Level AQA Maths Pure Geometric Sequences & Series: Helen is creating a mosaic patte

Helen is creating a mosaic pattern by placing square tiles next to each other along a straight line - AQA - A-Level Maths Pure - Question 9 - 2018 - Paper 3

Question 9

Helen is creating a mosaic pattern by placing square tiles next to each other along a straight line. The area of each tile is half the area of the previous tile, an... show full transcript

Worked Solution & Example Answer:Helen is creating a mosaic pattern by placing square tiles next to each other along a straight line - AQA - A-Level Maths Pure - Question 9 - 2018 - Paper 3

Step 1

Find, in terms of w, the length of the sides of the second largest tile.

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Answer

The area of the first tile is given by the side length squared, which can be expressed as: $A_1 = w^2$ The area of the second tile is half that of the first tile: $A_2 = \frac{1}{2} A_1 = \frac{1}{2} w^2$ Since the area of a square is equal to the square of its side length, we set: $A_2 = s_2^2$ Thus: $s_2^2 = \frac{1}{2} w^2$ Taking the square root gives: $s_2 = w \cdot \frac{1}{\sqrt{2}}$

Step 2

Show that, no matter how many tiles are in the pattern, the total length of the series of tiles will be less than 3.5w.

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Answer

In this case, the lengths of the tiles form a geometric sequence where:

First term: $a = w$
Common ratio: $r = \frac{1}{\sqrt{2}}$ The formula for the sum $S_n$ of the first n terms of a geometric series is: $S_n = a \frac{1 - r^n}{1 - r}$ As $n$ approaches infinity, since $r < 1$ , we can find the sum: $S_\infty = \frac{w}{1 - \frac{1}{\sqrt{2}}}$ To simplify: $S_\infty = \frac{w}{\frac{\sqrt{2} - 1}{\sqrt{2}}} = \frac{w \sqrt{2}}{\sqrt{2} - 1}$ We can approximate $\sqrt{2}$ as approximately 1.414, leading to: $S_\infty < 3.5w$

Step 3

Explain how you could refine the model used in part (b) to account for the 3 millimetre gap.

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Answer

To refine the model accounting for a 3 mm gap between tiles, we would need to add the additional length for gaps between each tile. If there are n tiles, there will be n - 1 gaps. Hence, the total additional length required is: $3(n - 1) \text{ mm}$ Thus, the new total length of the tiles including gaps becomes: $S_{new} = S_n + 3(n - 1) \text{ mm}$ The total length will no longer have an upper limit since adding gaps will contribute a consistently increasing amount of length.

Helen is creating a mosaic pattern by placing square tiles next to each other along a straight line - AQA - A-Level Maths Pure - Question 9 - 2018 - Paper 3

Worked Solution & Example Answer:Helen is creating a mosaic pattern by placing square tiles next to each other along a straight line - AQA - A-Level Maths Pure - Question 9 - 2018 - Paper 3

Find, in terms of w, the length of the sides of the second largest tile.

Show that, no matter how many tiles are in the pattern, the total length of the series of tiles will be less than 3.5w.

Explain how you could refine the model used in part (b) to account for the 3 millimetre gap.

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