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The cost of a High Definition television now averages $700, but the cost is decreasing about 15% per year. In how many years will the cost be under $200?

Sagot :

Answer: 7.7 years

Step-by-step explanation:

Given

The cost of HD TV is $700 and it is decreasing at the rate of 15% per year

After one year it becomes

[tex]\Rightarrow 700-700\times 15\%\\\Rightarrow 700(1-0.15)[/tex]

after 2 year it becomes

[tex]\Rightarrow 700(1-0.15)-700(1-0.15)\times 15\%\\\Rightarrow 700(1-0.15)(1-0.15)\\\Rightarrow 700(1-0.15)^2[/tex]

Suppose, after x years it becomes $200

[tex]\Rightarrow 200=700(1-0.15)^x\\\\\Rightarrow \dfrac{2}{7}=0.85^x\\\\\text{Taking log both sides}\\\\\Rightarrow x=\dfrac{\ln (\frac{2}{7})}{\ln (0.85)}\\\\\Rightarrow x=7.7\ \text{years}[/tex]

Thus, it takes 7.7 years for the cost to come under $200.

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