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4. The annualized cost to put foam insulation in your attic is 2000 + 650t and the annual energy cost of heat loss ( and cooling loss) is 19000 -------------- t where t is the insulation thickness. What is the most economical thickness?

Answer :

letmeanswer

Answer:

Economical thickness is 3.823 units

Explanation:

Insulation Annual Cost=2000+650t

Energy cost of heat loss=19000*(t) ^ (-0.5)

Total annual Cost=2000+650t+19000t ^ (-0.5)

We need to find optimum value of thickness such that Total annual cost is minimum. Therefore, we have dC/dt=0 and d^C/dt^2>0

Equation 1) ...........dC/dt=650-9500t^-1.5

Equation 2) ...........d^C/dt^2=4750t^-1.5

For 1) we have 650-9500t^(-1.5)=0

650/9500=t^-0.5

190/13=t^0.5

t=3.823 units

Hence, if we check for obtained value of "t" we have second order condition met then

d^2C/dt^2=4750*(3.823)^(-1.5)=635.4584551>0 by the way for any value of "t">0 we will suffice second order condition

Hence, economical thickness is 3.823 units

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