To solve this problem, we can use the concept of ratios and proportions.

Given:

• The ratio of the dimensions of Rectangle A to Rectangle B is $3:5$.
• The area of Rectangle B is $80 \, \text{cm}^2$.

Let's denote the area of Rectangle A as $A_A$ and the area of Rectangle B as $A_B = 80 \, \text{cm}^2$.

Since the rectangles are similar and their dimensions are in the ratio $3:5$, the ratio of their areas will be the square of the ratio of their corresponding sides. Thus, the ratio of the areas of Rectangle A to Rectangle B is:

$\left(\frac{3}{5}\right)^2 = \frac{9}{25}$

So, the area of Rectangle A to the area of Rectangle B can be expressed as:

$\frac{A_A}{A_B} = \frac{9}{25}$

Substituting the given area of Rectangle B:

$\frac{A_A}{80} = \frac{9}{25}$

Now, solve for $A_A$:

$A_A = \frac{9}{25} \times 80 = \frac{720}{25} = 28.8 \, \text{cm}^2$

Thus, the area of Rectangle A is $28.8 \, \text{cm}^2$.

Would you like more details or have any questions? Here are five related questions you might find interesting:

1. How would the solution change if the ratio was 4:5 instead of 3:5?
2. If the perimeter of Rectangle B is 36 cm, what is the perimeter of Rectangle A?
3. What is the ratio of the perimeters of Rectangle A to Rectangle B?
4. If Rectangle A has a width of 4 cm, what is its length?
5. How would the area of Rectangle A change if the ratio of their sides were squared?

Tip: When dealing with similar figures, remember that the ratio of their areas is the square of the ratio of their corresponding sides.