The value of Vishal's car is depreciating exponentially. The relationship between V, the value of his car, in dollars, and t, the elapsed time, in years, since he purchased the car is modeled by the following equation. V=22,500*10^(-(t)/( 12)) How many years after purchase will Vishal's car be worth $10,000 ? Give an exact answer expressed as a base-10 logarithm. years

Question

The value of Vishal's car is depreciating exponentially. The relationship between  V, the value of his car, in dollars, and  t, the elapsed time, in years, since he purchased the car is modeled by the following equation.  V=22,500*10^(-(t)/( 12)) How many years after purchase will Vishal's car be worth  $10,000 ? Give an exact answer expressed as a base-10 logarithm. years

Bytelearn · Accepted Answer

Set up equation: Set up the equation with the given value of the car. We are given the equation V = 22,500 * 10^(-(t)/(12)) and we want to find the time t when V = $10,000. So, we set up the equation 10,000 = 22,500 * 10^(-(t)/(12)).Divide sides: Divide both sides of the equation by 22,500 to isolate the exponential term. 10,000 / 22,500 = 10^(-(t)/(12)) This simplifies to 4/9 = 10^(-(t)/(12)).Convert to logarithmic form: Convert the equation to logarithmic form to solve for t. We can take the base-10 logarithm of both sides to get log(4/9) = log(10^(-(t)/(12))). Using the property of logarithms that log(b^x) = x * log(b), we get log(4/9) = -(t)/(12) * log(10). Since log(10) is 1, this simplifies to log(4/9) = -(t)/(12).Solve for t: Solve for t. To isolate t, we multiply both sides by -12: -12 * log(4/9) = t.Calculate exact value: Calculate the exact value of t. t = -12 * log(4/9). This is the exact answer expressed as a base-10 logarithm.

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