A patio was to be laid in a design with one tile in the first row,two tiles in the second row,three tiles in the third row, and so on. Mr Tong had 60 tiles to use. How tiles should be placed in the bottom row to use the most tiles?
12
it is twelve because 60 divided by 5 is twelve so each row would have 1-5 tiles in it
To determine how many tiles should be placed in the bottom row to use the most tiles, we need to identify a pattern in the number of tiles in each row.
From the given information, we know that the number of tiles in each row follows the sequence: 1, 2, 3, ...
This sequence represents the sum of consecutive positive integers, which is also known as an arithmetic series. The sum of an arithmetic series can be found using the formula:
𝑆 = (𝑛/2) * (𝑎 + 𝑙)
Where 𝑆 is the sum of the series, 𝑛 is the number of terms in the series, 𝑎 is the first term, and 𝑙 is the last term.
In this case, the number of terms is equal to the number of rows, and the first term is 1. We need to find the value of 𝑙 (the last term) given that the sum of the series is 60 tiles.
Using the formula for the sum of an arithmetic series, we can rewrite it as:
60 = (𝑛/2) * (1 + 𝑙)
Now, we want to find the maximum number of tiles in the bottom row, which means we want to maximize the value of 𝑙.
To do this, we can try different values for 𝑛 (the number of rows) and calculate the corresponding value of 𝑙. We will select the value of 𝑛 that gives us the highest value of 𝑙.
Let's start by trying different values for 𝑛 and calculating 𝑙:
For 𝑛 = 1:
60 = (1/2) * (1 + 𝑙)
120 = 1 + 𝑙
𝑙 = 119
For 𝑛 = 2:
60 = (2/2) * (1 + 𝑙)
60 = 1 + 𝑙
𝑙 = 59
For 𝑛 = 3:
60 = (3/2) * (1 + 𝑙)
40 = 1 + 𝑙
𝑙 = 39
For 𝑛 = 4:
60 = (4/2) * (1 + 𝑙)
30 = 1 + 𝑙
𝑙 = 29
We can continue this process for other values of 𝑛, but we can observe that as 𝑛 increases, the value of 𝑙 decreases. Therefore, to maximize the number of tiles in the bottom row, we need to have the smallest possible 𝑛.
In this case, 𝑛 = 4 gives us the highest value for 𝑙, which is 29 tiles.
So, to use the most tiles, we should place 29 tiles in the bottom row.