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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.
Answer
The area of each tile forms a geometric sequence where the largest tile has an area of and the second largest tile has an area of rac{w^2}{2}. Therefore, the side length of the second largest tile can be found by taking the square root of its area:
ext{Length of the second largest tile} = rac{w}{ ext{sqrt}(2)}
Step 2
Show that the total length of the series of tiles will be less than $3.5w$.
Answer
The side lengths of the tiles form a geometric sequence. The first tile's length is and the common ratio is r = rac{1}{ ext{sqrt}(2)}. The sum of an infinite geometric series can be expressed as:
S = \frac{a}{1 - r} = \frac{w}{1 - rac{1}{ ext{sqrt}(2)}} = \frac{w \cdot ext{sqrt}(2)}{\text{sqrt}(2) - 1}
Calculating this gives:
as long as we manipulate appropriately using relative estimates.
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.
Answer
The total length of the tiles will now need to account for the 3 millimetre gap between each adjacent tile. For every tile, an additional 3 mm is added to the total length. Therefore, if there are tiles:
This adjustment means that the total length will have no upper limit and will exceed due to the added gaps between the tiles.