A computer loses 30% of its value each year.a) write formula for the value of the computer after n years.b) How many years will it be before the value of the computer falls below 10% of its...

A computer loses 30% of its value each year.
a) write formula for the value of the computer after n years.
b) How many years will it be before the value of the computer falls below 10% of its original value?

Expert Answers
ishpiro eNotes educator| Certified Educator

If a computer loses 30% of its value each year, then every year the value of the computer is 70% of its original value. That is, if original value of the computer is $100, then in one year it will lose 30%, or $30, in value, and its new value will be $70.

To express this in general, denote the original value of the computer by A. Then after one year, its value will be 0.7A. The same will happen the following year, so the value of the computer after two years will be (0.7)*(0.7A)=(0.7)^2 * A, after three years it will be (0.7)*(0.7)^2 * A = (0.7)^3 * A, and so on. So it can be seen that after n years the value of the computer A(n) will be (0.7)^n * A.

So the formula for the value of the computer after n years is

A(n) = (0.7)^n * A

If after n years the value of the computer falls below 10% of the original, then A(n) is less than 0.1A. To find n for which A(n) equals 0.1A, solve the equation

0.1A = (0.7)^n*A

(0.7)^n = 0.1

This is an exponential equation and the solution is a logarithm with the base 0.7 of 0.1:

n = 6.46

So after 7 years, the value of the computer will fall below 10% of its original value.

 

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