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 Pure Modelling with 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 largest tile has a side length of $w$ . The area of this tile is $w^2$ .

The area of the second largest tile is half of this, which means:

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

Since the tile is square, if we let $s$ be the side length of the second tile, then:

$s^2 = \frac{w^2}{2} \implies s = \frac{w}{\sqrt{2}}.$

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 sequence of side lengths of the tiles can be modeled as a geometric sequence where the first term $a = w$ and the common ratio $r = \frac{1}{\sqrt{2}}$ .

The sum $S_n$ of the first $n$ terms of a geometric series is given by:

$S_n = a \frac{1 - r^n}{1 - r}.$

As $n$ approaches infinity, the sum converges to:

$S = \frac{w}{1 - \frac{1}{\sqrt{2}}} = w \frac{\sqrt{2}}{\sqrt{2} - 1}.$

This approximates to $3.4w$ , thus proving that the total length 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 for a 3 mm gap between tiles, we need to account for the additional length introduced by these gaps.

Each gap adds 3 mm to the total length per tile. Therefore, if there are $n$ tiles, there will be $n - 1$ gaps. The additional length due to gaps would be:

$\text{Total Gap Length} = 3(n - 1) \text{ mm}.$

Thus, the total length of the series of tiles will now include this extra length and will exceed $3.5w$ . The new total length can be expressed as:

$\text{Total Length with Gaps} = S_n + 3(n - 1) \text{ mm}.$

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, and state how the total length of the series of tiles will be affected.

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