This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's how to find the time it takes for the population to reach 9 times its original size: Step 1: Start with the given equation. The population growth is modeled by the equation: 3^t/5 = 9 Step 2: Express both sides with the same base. We know that 9 can be written as 3^2. 3^t/5 = 3^2 Step 3: Equate the exponents. Since the bases are the same, the exponents must be equal: (t)/(5) = 2 Step 4: Solve for t. Multiply both sides by 5: t = 2 × 5 t = 10 Alternatively, using logarithms as requested: Step 1: Start with the given equation. 3^t/5 = 9 Step 2: Take the logarithm of both sides. Using the common logarithm (base 10): (3^t/5) = (9) Step 3: Apply the logarithm property (a^b) = b (a). (t)/(5) (3) = (9) Step 4: Isolate t. t = (5 (9))/((3)) Step 5: Simplify the logarithmic expression. Since 9 = 3^2, we can write (9) as 2 (3): t = (5 · (2 (3)))/((3)) t = 5 × 2 t = 10 It takes 10 years for the population to reach 9 times its original size. The time taken is 10 years. Last free one today — make it count tomorrow, or type /upgrade for unlimited.