A painting purchased in 1998 for $200,000 is estimated to be worth [tex]v(t)=200,000 e^{t / 10}[/tex] dollars after [tex]t[/tex] years. At what rate will the painting be appreciating in 2007?
In 2007, the painting will be appreciating at $ [ ] per year.
(Round to the nearest dollar as needed.)
Answer (2)
In 2007, the painting appreciates at approximately $49,192 per year. This is calculated by finding the derivative of the value function and evaluating it at 9 years after the purchase. The derivative shows how quickly the value of the painting is increasing over time.
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Find the derivative of the value function: v ′ ( t ) = 20000 e t /10 .
Determine the time elapsed: t = 2007 − 1998 = 9 years.
Evaluate the derivative at t = 9 : v ′ ( 9 ) = 20000 e 9/10 .
Calculate the rate of appreciation and round to the nearest dollar: 49192 .
Explanation
Problem Analysis The problem asks us to find the rate at which a painting is appreciating in 2007, given that it was purchased in 1998 for 200 , 000 an d i t s v a l u e a f t er t ye a rs i s g i v e nb y t h e f u n c t i o n v(t) = 200,000e^{t/10}$. The rate of appreciation is the derivative of the value function with respect to time.
Finding the Derivative First, we need to find the derivative of the value function v ( t ) = 200 , 000 e t /10 with respect to t . Using the chain rule, we have:
v ′ ( t ) = d t d ( 200 , 000 e t /10 ) = 200 , 000 d t d ( e t /10 ) = 200 , 000 ⋅ 10 1 e t /10 = 20 , 000 e t /10 .
This derivative, v ′ ( t ) , gives us the rate of appreciation at any time t .
Evaluating the Derivative at t=9 We want to find the rate of appreciation in 2007. Since the painting was purchased in 1998, the time elapsed between 1998 and 2007 is 2007 − 1998 = 9 years. Therefore, we need to evaluate the derivative at t = 9 :
v ′ ( 9 ) = 20 , 000 e 9/10 .
Calculating the Rate of Appreciation Now, we calculate the value of v ′ ( 9 ) :
v ′ ( 9 ) = 20 , 000 e 9/10 ≈ 20 , 000 × 2.4596 ≈ 49192.46
Rounding this to the nearest dollar, we get $49,192.
Final Answer Therefore, in 2007, the painting will be appreciating at approximately 49192 dollars per year.
Examples
Understanding rates of change is crucial in finance. For example, when analyzing investments like stocks or real estate, calculating the rate of appreciation (or depreciation) helps investors make informed decisions about when to buy or sell assets. This concept is also used in economics to model economic growth, inflation rates, and other key indicators, providing valuable insights for policymakers and businesses.
