A large open meadow near your city is shrinking in size because developers have started building new homes there. The relationship between A, the area of the meadow, in hectares, and t, the elapsed time, in months, since the construction began is modeled by the following function. A=1562.5*10^(-0.1 t) How many months of construction will there be before the area of the meadow decreases to 500 hectares? Give an exact answer expressed as a base-10 logarithm. months

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A large open meadow near your city is shrinking in size because developers have started building new homes there. The relationship between  A, the area of the meadow, in hectares, and  t, the elapsed time, in months, since the construction began is modeled by the following function.  A=1562.5*10^(-0.1 t) How many months of construction will there be before the area of the meadow decreases to 500 hectares? Give an exact answer expressed as a base-10 logarithm. months

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Problem and Given Function: Understand the problem and write down the given function. We are given the function A = 1562.5 * 10^(-0.1t), where A is the area of the meadow in hectares and t is the time in months since construction began. We need to find the value of t when A is 500 hectares.Equation Setup: Set up the equation with the given area. We need to solve for t when A = 500 hectares. So, we set up the equation: 500 = 1562.5 * 10^(-0.1t).Isolate Exponential Term: Divide both sides of the equation by 1562.5 to isolate the exponential term. 500 / 1562.5 = 10^(-0.1t)Calculate Left Side: Calculate the left side of the equation. 500 / 1562.5 = 0.32 (approximately) So, 0.32 = 10^(-0.1t).Take Base-10 Logarithm: Take the base-10 logarithm of both sides to solve for t. log(0.32) = log(10^(-0.1t))Property of Logarithms: Use the property of logarithms that log(b^x) = x * log(b). log(0.32) = -0.1t * log(10) Since log(10) is 1, we have: log(0.32) = -0.1tSolve for t: Divide both sides by -0.1 to solve for t. t = log(0.32) / -0.1

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