This problem has been slightly modified from the original appearing in Chapter 2

Question

This problem has been slightly modified from the original appearing in Chapter 2 of Adler's Modeling the Dynamics of Life. Suppose that the mass M(t)| of an insect (in grams) and the volume V(t)|(in cubic centimeters) are known functions of time (in days). (We're only going to work with positive values of t|.)Let M(t) = 4 + t2^| and V(t) = 1 + 4t|. Write a formula for the density as a function of time: Compute the derivative of the density: What is the positive critical point? t =| What is the density when the derivative is zero? Is the density increasing or decreasing after the time when the derivative is zero?

Explanation / Answer

mass M(t)=(4+t2),volume V(t)=(1+4t)

density =mass/volume

density =(4+t2)/(1+4t)

derivative of density =[(0+2t)(1+4t) -(4+t2)(0+4)]/(1+4t)2

derivative of density =[(2t)(1+4t) -(4+t2)(4)]/(1+4t)2

derivative of density =[2t+8t2 -16-4t2)]/(1+4t)2

derivative of density =[2t+4t2 -16]/(1+4t)2

for critical point derivative is zero

[2t+4t2 -16]/(1+4t)2=0

[2t+4t2 -16]=0

2t2+t-8=0

t=[-1+((1)2-4(2)(-8))]/(2*2)

t=[-1+65]/4

t=1.766

density =(4+1.7662)/(1+4*1.766)

density =0.883

deivative >0 for t>1.766

so density is increasing

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