1) Two sizes of seaweed plants are growing in a cove. Due to changing water temperatures, the larger seaweed plants are thriving-while the smaller plants are dying out. Originally, there were 85 large seaweed plants and 700 small seaweed plants. The large seaweed population is increasing by \( 5 \% \) per year, while the small seaweed population is decreasing by \( 7 \% \) per year. How many of each type of seaweed will there be in 30 years?

To solve this problem, we will use the concept of exponential growth and decay. Let's denote: - \( L \) as the number of large seaweed plants - \( S \) as the number of small seaweed plants Initially: - \( L = 85 \) - \( S = 700 \) The large seaweed plants are increasing by \( 5\% \) per year, so the growth rate is \( 0.05 \). The small seaweed plants are decreasing by \( 7\% \) per year, so the decay rate is \( 0.07 \). We can use the formula for exponential growth and decay to find the number of each type of seaweed after 30 years: For large seaweed plants: \[ L_{\text{after 30 years}} = L_{\text{initial}} \times (1 + \text{growth rate})^{30} \] For small seaweed plants: \[ S_{\text{after 30 years}} = S_{\text{initial}} \times (1 - \text{decay rate})^{30} \] Let's calculate the number of each type of seaweed after 30 years. Calculate the value by following steps: - step0: Calculate: \(85\left(1+0.05\right)^{30}\) - step1: Add the numbers: \(85\times 1.05^{30}\) - step2: Convert the expressions: \(85\left(\frac{21}{20}\right)^{30}\) - step3: Simplify: \(85\times \frac{21^{30}}{20^{30}}\) - step4: Rewrite the expression: \(5\times 17\times \frac{21^{30}}{20^{30}}\) - step5: Rewrite the expression: \(5\times 17\times \frac{21^{30}}{5^{30}\times 4^{30}}\) - step6: Reduce the numbers: \(17\times \frac{21^{30}}{5^{29}\times 4^{30}}\) - step7: Multiply the fractions: \(\frac{17\times 21^{30}}{5^{29}\times 4^{30}}\) Calculate or simplify the expression \( 700 * (1 - 0.07)^30 \). Calculate the value by following steps: - step0: Calculate: \(700\left(1-0.07\right)^{30}\) - step1: Subtract the numbers: \(700\times 0.93^{30}\) - step2: Convert the expressions: \(700\left(\frac{93}{100}\right)^{30}\) - step3: Simplify: \(700\times \frac{93^{30}}{100^{30}}\) - step4: Rewrite the expression: \(100\times 7\times \frac{93^{30}}{100^{30}}\) - step5: Reduce the numbers: \(7\times \frac{93^{30}}{100^{29}}\) - step6: Multiply: \(\frac{7\times 93^{30}}{100^{29}}\) After 30 years, there will be approximately 367 large seaweed plants and 79 small seaweed plants.