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The value of a computer is decreasing at a rate that is proportional at any time to the value of the computer at that time.
The computer is worth 
$850 initially, and it is worth 
$306 after 2 years.
How much will the computer be worth after 5 years?
Choose 1 answer:
(A) 
$0
(B) 
$66
(C) 
$79

The value of a computer is decreasing at a rate that is proportional at any time to the value of the computer at that time.\newlineThe computer is worth $850 \$ 850 initially, and it is worth $306 \$ 306 after 22 years.\newlineHow much will the computer be worth after 55 years?\newlineChoose 11 answer:\newline(A) $0 \$ 0 \newline(B) $66 \$ 66 \newline(C) $79 \$ 79

Full solution

Q. The value of a computer is decreasing at a rate that is proportional at any time to the value of the computer at that time.\newlineThe computer is worth $850 \$ 850 initially, and it is worth $306 \$ 306 after 22 years.\newlineHow much will the computer be worth after 55 years?\newlineChoose 11 answer:\newline(A) $0 \$ 0 \newline(B) $66 \$ 66 \newline(C) $79 \$ 79
  1. Exponential Decay Model: We are given that the value of the computer decreases proportionally to its current value, which suggests that the depreciation follows an exponential decay model. The general formula for exponential decay is:\newlineV(t)=V0e(kt) V(t) = V_0 \cdot e^{(-kt)} \newlinewhere:\newline- V(t) V(t) is the value of the computer at time t t ,\newline- V0 V_0 is the initial value of the computer,\newline- k k is the decay constant,\newline- t t is the time in years,\newline- e e is the base of the natural logarithm.\newlineWe need to find the decay constant k k using the initial value and the value after 22 years.
  2. Initial Values: First, let's plug in the values we know into the exponential decay formula to find k k . We know that V_0 = $850 and V(2) = $306 . So, we have:\newline306=850e(2k) 306 = 850 \cdot e^{(-2k)} \newlineNow, we need to solve for k k .
  3. Solving for k: To solve for k k , we first divide both sides of the equation by 850850:\newline306850=e(2k) \frac{306}{850} = e^{(-2k)}
  4. Calculating k: Next, we take the natural logarithm of both sides to get rid of the exponential:\newlineln(306850)=ln(e(2k)) \ln\left(\frac{306}{850}\right) = \ln\left(e^{(-2k)}\right) \newlineln(306850)=2kln(e) \ln\left(\frac{306}{850}\right) = -2k \cdot \ln(e) \newlineSince ln(e)=1 \ln(e) = 1 , we can simplify this to:\newlineln(306850)=2k \ln\left(\frac{306}{850}\right) = -2k
  5. Finding Value after 55 Years: Now we solve for k k by dividing both sides by 2-2:\newlinek=ln(306850)2 k = \frac{-\ln\left(\frac{306}{850}\right)}{2} \newlineLet's calculate the value of k k .
  6. Finding Value after 55 Years: Now we solve for k k by dividing both sides by 2-2:\newlinek=ln(306850)2 k = \frac{-\ln\left(\frac{306}{850}\right)}{2} \newlineLet's calculate the value of k k .Using a calculator, we find:\newlinek=ln(306850)2ln(0.36)2(1.02165)20.510825 k = \frac{-\ln\left(\frac{306}{850}\right)}{2} \approx \frac{-\ln(0.36)}{2} \approx \frac{-(-1.02165)}{2} \approx 0.510825
  7. Finding Value after 55 Years: Now we solve for k k by dividing both sides by 2-2:\newlinek=ln(306850)2 k = \frac{-\ln\left(\frac{306}{850}\right)}{2} \newlineLet's calculate the value of k k .Using a calculator, we find:\newlinek=ln(306850)2ln(0.36)2(1.02165)20.510825 k = \frac{-\ln\left(\frac{306}{850}\right)}{2} \approx \frac{-\ln(0.36)}{2} \approx \frac{-(-1.02165)}{2} \approx 0.510825 Now that we have the decay constant k k , we can use it to find the value of the computer after 55 years. We plug k k , V0 V_0 , and t=5 t = 5 into the exponential decay formula:\newlineV(5)=850e(0.5108255) V(5) = 850 \cdot e^{(-0.510825 \cdot 5)} \newlineLet's calculate V(5) V(5) .
  8. Finding Value after 55 Years: Now we solve for k k by dividing both sides by 2-2:\newlinek=ln(306850)2 k = \frac{-\ln\left(\frac{306}{850}\right)}{2} \newlineLet's calculate the value of k k .Using a calculator, we find:\newlinek=ln(306850)2ln(0.36)2(1.02165)20.510825 k = \frac{-\ln\left(\frac{306}{850}\right)}{2} \approx \frac{-\ln(0.36)}{2} \approx \frac{-(-1.02165)}{2} \approx 0.510825 Now that we have the decay constant k k , we can use it to find the value of the computer after 55 years. We plug k k , V0 V_0 , and t=5 t = 5 into the exponential decay formula:\newlineV(5)=850e(0.5108255) V(5) = 850 \cdot e^{(-0.510825 \cdot 5)} \newlineLet's calculate V(5) V(5) .Using a calculator, we find:\newlineV(5)=850e(0.5108255)850e(2.554125)8500.077665.96 V(5) = 850 \cdot e^{(-0.510825 \cdot 5)} \approx 850 \cdot e^{(-2.554125)} \approx 850 \cdot 0.0776 \approx 65.96
  9. Finding Value after 55 Years: Now we solve for k k by dividing both sides by 2-2:\newlinek=ln(306850)2 k = \frac{-\ln\left(\frac{306}{850}\right)}{2} \newlineLet's calculate the value of k k .Using a calculator, we find:\newlinek=ln(306850)2ln(0.36)2(1.02165)20.510825 k = \frac{-\ln\left(\frac{306}{850}\right)}{2} \approx \frac{-\ln(0.36)}{2} \approx \frac{-(-1.02165)}{2} \approx 0.510825 Now that we have the decay constant k k , we can use it to find the value of the computer after 55 years. We plug k k , V0 V_0 , and t=5 t = 5 into the exponential decay formula:\newlineV(5)=850e(0.5108255) V(5) = 850 \cdot e^{(-0.510825 \cdot 5)} \newlineLet's calculate V(5) V(5) .Using a calculator, we find:\newlineV(5)=850e(0.5108255)850e(2.554125)8500.077665.96 V(5) = 850 \cdot e^{(-0.510825 \cdot 5)} \approx 850 \cdot e^{(-2.554125)} \approx 850 \cdot 0.0776 \approx 65.96 The value 65.96 65.96 is close to $66 , which is one of the given answer choices. Since the value of the computer cannot be exactly $66 due to rounding during calculations, we choose the closest answer, which is $66 .

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