To find the total area of all five squares, we'll follow these steps:
1. Determine the side lengths of the squares:
- The largest square has a side length of 27 units.
- Each subsequent square has a side length that is two-thirds the side length of the previous square.
- Second square's side length:
[tex]\[
\text{Side length of second square} = 27 \times \frac{2}{3} = 18 \, \text{units}
\][/tex]
- Third square's side length:
[tex]\[
\text{Side length of third square} = 18 \times \frac{2}{3} = 12 \, \text{units}
\][/tex]
- Fourth square's side length:
[tex]\[
\text{Side length of fourth square} = 12 \times \frac{2}{3} = 8 \, \text{units}
\][/tex]
- Fifth square's side length:
[tex]\[
\text{Side length of fifth square} = 8 \times \frac{2}{3} = 5.33 \, \text{units} \quad (\text{approximately})
\][/tex]
2. Calculate the area of each square:
- The area of a square is given by the side length squared:
- Area of the largest square:
[tex]\[
\text{Area of largest square} = 27^2 = 729 \, \text{square units}
\][/tex]
- Area of the second square:
[tex]\[
\text{Area of second square} = 18^2 = 324 \, \text{square units}
\][/tex]
- Area of the third square:
[tex]\[
\text{Area of third square} = 12^2 = 144 \, \text{square units}
\][/tex]
- Area of the fourth square:
[tex]\[
\text{Area of fourth square} = 8^2 = 64 \, \text{square units} \quad (\text{approximately})
\][/tex]
- Area of the fifth square:
[tex]\[
\text{Area of fifth square} = 5.33^2 = 28.44 \, \text{square units} \quad (\text{approximately})
\][/tex]
3. Sum the areas to find the total area:
- Add the areas of all five squares:
[tex]\[
\text{Total area} = 729 + 324 + 144 + 64 + 28.44 = 1289.44 \, \text{square units} \quad (\text{approximately})
\][/tex]
The total area of all five squares is 1289.44 square units.
