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Sagot :
Logarithms
Answer
It will decrease to half its original value after 7 years.
Explanation
The value of the investment decreases by 10% per year. This means that it will remain at 90% of its previous year's value.
The 90% of any value is found by multiplying the percentage divided by 100:
[tex]\frac{90}{100}=0.9[/tex]Step 1- writing an equation of the situation
We have that after 1 year the value of the investment will be given by the 90% of $1000. This is:
[tex]1000\cdot0.9[/tex]After 2 years, it will be the 90% of the last value:
[tex]1000\cdot0.9\cdot0.9=1000\cdot0.9^2[/tex]After 3 years, it will be the 90% of the last value:
[tex]1000\cdot0.9^2\cdot0.9=1000\cdot0.9^3[/tex]After 4 years, it will be the 90% of the last value:
[tex]1000\cdot0.9^3\cdot0.9=1000\cdot0.9^4[/tex]...
we can see a relation between the exponent of 0.9 and the years that have passed.
After n years, it will be
[tex]1000\cdot0.9^n[/tex]We want to find when the value decreases to half its original value. Since
1000/2 = 500
then, we want to find the number of the year n when the value if 500:
[tex]\begin{gathered} 1000\cdot0.9^n=500 \\ \downarrow \\ n=\text{?} \end{gathered}[/tex]Step 2 - solving the equation for n
In order to find the year when the value is at half the original, we must solve the equation for n.
First, we take 1000 to the right side:
[tex]\begin{gathered} 1000\cdot0.9^n=500 \\ \downarrow\text{ dividing both sides by 1000} \\ \frac{1000\cdot0.9^n}{1000}=\frac{500}{1000} \\ 0.9^n=\frac{500}{1000} \\ \downarrow\text{ since }\frac{500}{1000}=0.5 \\ 0.9^n=0.5 \end{gathered}[/tex]We want to "leave n alone" on one side of the equation, in order to do it we use logarithm.
Let's remember the relation between logarithms and exponentials:
[tex]b^c=a\rightleftarrows\log _ba=c[/tex]Then, in this case:
[tex]0.9^n=0.5\rightleftarrows\log _{0.9}0.5=n[/tex]Using the calculator, we have:
[tex]\log _{0.9}0.5=6.5788[/tex]If we round the answer we will have that
[tex]\log _{0.9}0.5=6.5788\approx6.6\approx7[/tex]Then, n = 7
Therefore, after 7 years the investment decreases to half its original value.
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