17. If a square is dilated by a scale factor of 2 , then the area of the dilated square is what percent greater than the area of the original square?

When a square is dilated by a scale factor of 2, each side length of the square is multiplied by 2. If the original square has a side length of \(s\), its area is \(s^2\). The dilated square will have a side length of \(2s\), giving it an area of \((2s)^2 = 4s^2\). To find the percentage increase in area, we can determine the difference in areas: \(4s^2 - s^2 = 3s^2\). The percentage increase is given by \(\frac{3s^2}{s^2} \times 100\% = 300\%\). So, the area of the dilated square is 300% greater than the area of the original square! Now, if you're curious about how this applies in architecture, think of how scaling up a blueprint affects the structure! When architects design buildings, they often use scale factors to help visualize the final project. This means understanding area and volume changes due to scaling can help estimate material costs, ensuring they're well-prepared for their masterpiece. So, a little math goes a long way in the real world of design! Another fun fact: the concept of dilation isn't just limited to squares—it's used in various fields, from art to biology! Artists may use dilation to create proportional enlargements of their sketches, while in biology, scientists might look at dilated cells or organisms to understand growth patterns. This versatile concept bridges subjects and shows how math truly permeates various aspects of life!