1. A square, each side 4 units long, is packed with n adjacent rows of n adjacent congruent circles. What is the total area, in square units, of all of the circles inside the squar... 1. A square, each side 4 units long, is packed with n adjacent rows of n adjacent congruent circles. What is the total area, in square units, of all of the circles inside the square? 2. The area of the given figure is? Read more

Steps to Solve

1. Find the radius of each circle

Since there are $n$ rows of circles packed in a square of side length 4, each circle's diameter can be found by dividing the side length of the square by $n$. The diameter of one circle is $4/n$. Therefore, the radius $r$ of each circle is half of the diameter, so $r = \frac{4}{2n} = \frac{2}{n}$.

2. Calculate the area of one circle

The area of a single circle is given by the formula $A_{circle} = \pi r^2$. Substituting $r = \frac{2}{n}$, we get $A_{circle} = \pi (\frac{2}{n})^2 = \pi \frac{4}{n^2} = \frac{4\pi}{n^2}$.

3. Calculate the total area of all circles

There are $n$ rows and $n$ columns of circles, so the total number of circles is $n \times n = n^2$. The total area of all circles is the area of one circle multiplied by the number of circles: $A_{total} = n^2 \times A_{circle} = n^2 \times \frac{4\pi}{n^2} = 4\pi$.

4. Calculate the area of the triangle The triangle has a base of length $20 - 8 = 12$. We can use Pythagorean theorem to compute the height of the triangle: $h = \sqrt{15^2 - 12^2} = \sqrt{225 - 144} = \sqrt{81} = 9$ Therefore the area $A_{triangle} = \frac{1}{2} \cdot base \cdot height = \frac{1}{2} \cdot 12 \cdot 9 = 6 \cdot 9 = 54$

5. Calculate the area of the rectangle The rectangle has sides of lengths $8$ and $9$ Therefore the area $A_{rectangle} = 8 \cdot 9 = 72$

6. Calculate the area of the total figure The total area is the sum of the area of the triangle, and the area of the rectangle $A_{total} = A_{triangle} + A_{rectangle} = 54 + 72 = 126$