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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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BuyFindarrow_forward

Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

Forestry

The value of a tract of timber is

V ( t ) = 100 , 000 e 0.8 t

where t is the time in years, with t = 0 corresponding to 2010.

If money earns interest continuously at 10%, then the present value of the timber at any time t is

A ( t ) = V ( t ) e 0.10 t .

Find the year in which the timber should be harvested to maximize the present value function.

images

To determine

To calculate: The year in which timber has be harvested in order to maximize the present value function.

Explanation

Given:

The value of a tract of timber is V(t)=100000e0.8t where t is the time in years, with t=0 corresponding to 2010.

If money earns interest continuously at 10%, then the present value of the timber at any time t is,

A(t)=V(t)e0.10t.

Formula used:

ddxuv=udvdx+vdudx

ddx(xn)=nxn1

ddxex=ex

Calculation:

Let us consider the expression, A(t)=V(t)e0.10t.

By putting V(t)=1000000e0.8t in above equation, we get

A(t)=V(t)e0.10t=100000e0.8t×e0.10t

Now, we have to differentiate the function with respect to t to maximize the function and put in equal to 0. Therefore,

dA(t)dt=d(100000e0

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