2. In a pond, the area that is covered by algae doubles each week. When the algae was first spotted, the area it covered was about 12.5 square meters. Find the area, in square meters, covered by algae 10 days after it was spotted. Round to the nearest hundredth. about Complete the following to explain why we can find the area covered by algae 1 day after it was spotted by multiplying 12.5 by \( \sqrt[7]{2} \). One day is \( \frac{1}{7} \) of a week. The area covered by algae 1 day after it was spotted is \( 12.5 \cdot\left(\sqrt{\frac{1}{x}}\right)^{\frac{1}{7}} \), and \( 2^{\frac{1}{7}} \) is equivalent to \( \sqrt[7]{2} \).
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When the algae was first spotted, it covered 12.5 square meters. Given that the area doubles each week, we can use the formula for exponential growth to track its expansion. In just 10 days, which is a little over one week, the algae would have nearly doubled once and then grown for 3 additional days. The area covered can be calculated as follows: \( 12.5 \times 2^{\frac{10}{7}} \). When you perform the math and round it, you'll find that it's around 25.66 square meters. To calculate the area after 1 day, we essentially want to account for the algae’s growth over a fraction of the week. By taking the seventh root of 2, denoted as \( \sqrt[7]{2} \), we determine the growth factor for one day, which provides a precise way to measure the incremental growth over such short time frames. Thus, \( 12.5 \cdot \sqrt[7]{2} \) accurately reflects the growth in just one day!
