Answer:
It will take approximately 13 years for the calculator to be $2
Explanation:
We need to set up a depreciation equation for the cost of the calculator per year.
Let the initial cost be I, the present value be P and the depreciation percentage be r%
Thus;
P = I( 1 - r%)^t
r% is same as r/100
For the two years initially;
75 =I( 1 - r/100)^2•••••••••(i)
For the 8 years, we have the equation as
10 = I(1 - r/100)^8 ••••••••(ii)
Now, simply divide the second equation by the first directly; we have;
10/75 = (1 - r/100)^8 / (1 - r/100)^2
2/15 = (1-r/100)^6
Take the log of both sides
ln(2/15) = ln(1-r/100)^6
ln(2/15) = 6 ln (1-r/100)
-2.015 = 6 ln (1 - r/100)
divide both sides by 6
-0.34 = ln (1-r/100)
1-r/100 = e^(-0.34)
1-r/100 = 0.715
Multiply through by 100
100 - r = 71.5
r = 100-71.5
r = 28.5%
Let’s now calculate the initial value;
P = I( 1 - r/100)^t
Let’s use the two years time frame
75 = I( 1 - 28.5/100)^2
75 = I(0.715)^2
I = 75/(0.715)^2
I = 146.71 which is approximately $147
Now, the time it will take for the calculator to be $2
P = I(1-r/100)^t
2 = 147 ( 1 - r/100)^t
divide both sides by 147
0.014 = (0.715)^t
ln 0.014 = ln (0.715)^t
t ln 0.715 = ln 0.014
t = ln 0.014/ln 0.715
t = 12.72
t is approximately 13 years