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During one decade, the price of silver decreased at a rate that was proportional to the price of silver at that time.The price for an ounce of silver was initially, and it was after years.What was the price for an ounce of silver after years?Choose answer:(A) (B) (C)
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Q. During one decade, the price of silver decreased at a rate that was proportional to the price of silver at that time.The price for an ounce of silver was initially, and it was after years.What was the price for an ounce of silver after years?Choose answer:(A) (B) (C)
- Exponential Decay Formula: The problem states that the decrease in the price of silver is proportional to its current price, which suggests that we are dealing with exponential decay. The general formula for exponential decay is , where is the price at time , is the initial price, is the decay constant, and is the base of the natural logarithm.
- Initial Price and Time Given: We are given that the initial price is \$\(22\) and the price after \(7\) years is \$\(5\).\(50\). We can use these two pieces of information to find the decay constant \(k\). We have \(P(7) = 22 \cdot e^{-7k} = 5.50\).
- Solving for Decay Constant: To find \(k\), we need to solve the equation \(22 \cdot e^{-7k} = 5.50\). We start by dividing both sides by \(22\) to isolate \(e^{-7k}\) on one side of the equation: \(e^{-7k} = \frac{5.50}{22}\).
- Calculating Decay Constant: Calculating the right side of the equation gives us \(e^{-7k} = 0.25\). Now we can take the natural logarithm of both sides to solve for \(-7k\): \(\ln(e^{-7k}) = \ln(0.25)\).
- Finding Price After \(5\) Years: The natural logarithm of \(e\) to any power is just that power, so we have \(-7k = \ln(0.25)\). Now we can solve for \(k\) by dividing both sides by \(-7\): \(k = \frac{\ln(0.25)}{-7}\).
- Calculating Price After \(5\) Years: Calculating \(k\) gives us \(k \approx \ln(0.25) / -7 \approx 0.25\). Now that we have the decay constant, we can use it to find the price after \(5\) years using the formula \(P(5) = 22 \times e^{-5k}\).
- Error in Calculation: Substituting the values into the formula gives us \(P(5) = 22 \times e^{(-5 \times 0.25)}\). Now we calculate \(e^{(-5 \times 0.25)}\) and then multiply by \(22\) to find \(P(5)\).
- Error in Calculation: Substituting the values into the formula gives us \(P(5) = 22 \times e^{(-5 \times 0.25)}\). Now we calculate \(e^{(-5 \times 0.25)}\) and then multiply by \(22\) to find \(P(5)\).Calculating \(e^{(-5 \times 0.25)}\) gives us approximately \(e^{(-1.25)}\). Multiplying this by \(22\) should give us the price after \(5\) years.
- Error in Calculation: Substituting the values into the formula gives us \(P(5) = 22 \times e^{(-5 \times 0.25)}\). Now we calculate \(e^{(-5 \times 0.25)}\) and then multiply by \(22\) to find \(P(5)\).Calculating \(e^{(-5 \times 0.25)}\) gives us approximately \(e^{(-1.25)}\). Multiplying this by \(22\) should give us the price after \(5\) years.After calculating, we find that \(P(5) \approx 22 \times e^{(-1.25)} \approx 22 \times 0.287 \approx \$(6.31)\). However, this is not one of the options provided, which means there must be a mistake in our calculations.
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