Final answer:
The value of n can be found by setting up an equation that represents the number of squares of different sizes on the grid. By solving the equation, we find that n is approximately 10.167, which rounds up to 11. Therefore, the answer is a) 10.
Step-by-step explanation:
To find the value of n, we need to consider the number of squares that can be formed on a rectangular grid of points. We know that there are 100 squares of three different sizes. Let's analyze each size separately:
- The smallest size square has side length 1 and there are a total of 5 x (n-1) of them.
- The medium size square has side length 2 and there are a total of 4 x (n-2) of them.
- The largest size square has side length 3 and there are a total of 3 x (n-3) of them.
So, the total number of squares is given by:
(5 x (n-1)) + (4 x (n-2)) + (3 x (n-3)) = 100
We can simplify this equation and solve for n:
5n + 4n + 3n - (5 + 8 + 9) = 100
12n - 22 = 100
12n = 122
n = 122/12
n = 10.167
Therefore, the value of n is approximately 10.167. Since n represents the number of columns in the grid, we need to round it up to the nearest whole number, which is 11. Therefore, the answer is option a) 10.