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. (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 be less than $3.5w$. (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.

A-Level AQA Maths Mechanics Quantities, Units & Modelling: 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 Mechanics - 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 Mechanics - 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 largest tile is $w^2$ . The area of the second largest tile is half of that, so:

$ext{Area of second tile} = \frac{w^2}{2}$

To find the side length of the second largest tile, we take the square root of the area:

$\text{Side length of second tile} = \sqrt{\frac{w^2}{2}} = \frac{w}{\sqrt{2}}$

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

The length of each tile forms a geometric sequence where:

The first term $a = w$
The common ratio $r = \frac{1}{\sqrt{2}}$

The sum $S$ of an infinite geometric series can be calculated using: $S = \frac{a}{1 - r}$

Substituting the known values: $S = \frac{w}{1 - \frac{1}{\sqrt{2}}} = \frac{w \sqrt{2}}{\sqrt{2} - 1}$

To simplify, we rationalize the denominator: $S = \frac{w \sqrt{2}(\sqrt{2} + 1)}{1} = w(\sqrt{2} + 1)$

Approximating $\sqrt{2} + 1 \approx 2.414$ , we find that the total length: $S < 3.5w$

This shows the total length of the tiles is less than $3.5w$ .

Step 3

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.

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Answer

To refine the model, we need to account for the additional gap of 3 mm between each tile. This means that for every tile there will be an additional $0.003m$ to the total length.

If there are $n$ tiles, the total additional length due to the gaps will be: $0.003n$

Thus, the new total length of the series of tiles will be: $S + 0.003n$

As the number of tiles increases, the total length will not have an upper limit and can exceed values calculated without the gaps.

Helen is creating a mosaic pattern by placing square tiles next to each other along a straight line - AQA - A-Level Maths Mechanics - 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 Mechanics - 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, and state how the total length of the series of tiles will be affected.

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