a painting purchased in 1998 for $250,000 is estimated to be worth $v(t)=250,000e^{t/8}$ dollars after t…

a painting purchased in 1998 for $250,000 is estimated to be worth $v(t)=250,000e^{t/8}$ dollars after t years. at what rate will the painting be appreciating in 2007?\nin 2007, the painting will be appreciating at $ per year.\n(round to the nearest dollar as needed.)

a painting purchased in 1998 for $250,000 is estimated to be worth $v(t)=250,000e^{t/8}$ dollars after t years. at what rate will the painting be appreciating in 2007?\nin 2007, the painting will be appreciating at $ per year.\n(round to the nearest dollar as needed.)

Answer

Explanation:

Step1: Find the value of ( t )

The time ( t ) from 1998 to 2007 is ( t = 2007 - 1998=9) years.

Step2: Differentiate the function ( v(t)=250000e^{t/8} )

Using the chain - rule, if ( y = e^{u}) and ( u=\frac{t}{8}), then (\frac{dy}{dt}=\frac{dy}{du}\cdot\frac{du}{dt}). The derivative of (y = e^{u}) with respect to (u) is (e^{u}), and the derivative of (u=\frac{t}{8}) with respect to (t) is (\frac{1}{8}). So (v^\prime(t)=250000\times\frac{1}{8}e^{t/8}).

Step3: Substitute ( t = 9 ) into ( v^\prime(t) )

(v^\prime(9)=250000\times\frac{1}{8}e^{9/8}). First, calculate (e^{9/8}\approx e^{1.125}\approx3.08). Then (v^\prime(9)=250000\times\frac{1}{8}\times3.08). (250000\times\frac{1}{8}=31250). (31250\times3.08 = 96250).

Answer:

(96250)