Please answer both #1 and #2. Thank you! A certain piece of machinery was purcha
ID: 3343212 • Letter: P
Question
Please answer both #1 and #2. Thank you!
A certain piece of machinery was purchased 4 yr ago by Garland Mills for $500,000. Its present resale value is $340,000. Assuming that the machine's resale value decreases exponentially, what will it be 5 yr from now? (Round your answer to the nearest dollar.) $ Wood deposits recovered from an archaeological site contain 23% of the C-14 they originally contained. How long ago did the tree from which the wood was obtained die' (Round your answer to the nearest whole number.) yrExplanation / Answer
a) Sine the price decreases exponentially, we can model this with:
A = P*e^(kt).
Since the initial value is $470k, we have P = 500000 and so the equation becomes:
A = 500000e^(kt).
Since the value 4 years after purchasing it is $340k, we have:
340000 = 500000*e^(4k)
==> e^(4k) = 34/50
==> 4k = ln(34/50)
==> k = -0.0964
Thus, the price after t years is 500000e^(-0.0964*t) and so the price 5 years later (9 years after buying) is:
A = 500000e^(-0.0964*9)] %u2248 $209949.6
b) Carbon 14 has a half-life of 5730 years. One half-life is the amount of time needed to reduce the amount of radioactivity by a factor of two.
How many factors of two does it take to reduce the amount of radioactivity to 34% of the initial amount?
(1/2) ^ N = 23/100
Take logs of both sides:
log(1/2) * N = log (23/100)
-.3010 * N = -0.638
Divide both sides by -0.3010:
N = 2.12
So the wood C-14 in the wood has been decaying for 2.12 half-lives, or
2.12 * 5730 = 12150.5 years