An insured's roof cost $4,000 when installed 5 years ago. It has been damaged by hail and must be replaced. The new roof will cost $6,000 at today’s prices. If the roof has been depreciating at $200 per year and the insured’s policy is written on the actual cash value(ACV), how much will the policy pay toward the insured's new roof?

Answer:ACV=$4,500Step-by-step explanation:We have that the actual cash value (ACV) is defined as:[tex]ACV=\dfrac{R\times(E-C)}{E}[/tex]Where:[tex]ACV =[/tex] actual cash value [tex]R =[/tex] replacement cost or purchase price of the item [tex]E =[/tex] expected life of the item [tex]C =[/tex] current life of the itemThen we have R=$6,000, C=5years, and to find the expected life of the item we can use the depreciating of the roof, then if the roof is depreciating $200 each year we just need to divide $4,000 by $200 to find the expected life of the roof:[tex]\dfrac{4,000}{200}=20[/tex]Then the espected life of the roof is 20 years, with this result we have all the data, then:[tex]ACV=\dfrac{\$6,000\times (20-5)}{20}=\dfrac{\$6,000\times (15)}{20}=\dfrac{\$90,000}{20}=\$4,500[/tex]Then the ACV is $4,500