# 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?

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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 0.7*A*. 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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