Answer:
approximately 10.46 yrs
Step-by-step explanation:
To find out how many years it will take for Julian's investment to triple, we can use the formula for compound interest:
�
=
�
×
(
1
+
�
)
�
A=P×(1+r)
t
Where:
�
A is the amount of money accumulated after
�
t years, including interest.
�
P is the principal amount (the initial amount of money).
�
r is the annual interest rate (in decimal form).
�
t is the time the money is invested for, in years.
In this case, Julian invests $3000, and three years later his investment is worth $4,125.35. We need to find
�
t when the investment triples, so we set
�
A to
3
�
3P:
3
�
=
�
×
(
1
+
�
)
�
3P=P×(1+r)
t
Now, let's plug in the given values:
3
×
3000
=
3000
×
(
1
+
�
)
�
3×3000=3000×(1+r)
t
Simplify:
9000
=
3000
×
(
1
+
�
)
�
9000=3000×(1+r)
t
Divide both sides by 3000:
3
=
(
1
+
�
)
�
3=(1+r)
t
To solve for
�
t, we'll use logarithms. Taking the natural logarithm (ln) of both sides:
ln
(
3
)
=
ln
[
(
1
+
�
)
�
]
ln(3)=ln[(1+r)
t
]
By the properties of logarithms, we can bring the exponent
�
t down:
ln
(
3
)
=
�
×
ln
(
1
+
�
)
ln(3)=t×ln(1+r)
Now, we can solve for
�
t:
�
=
ln
(
3
)
ln
(
1
+
�
)
t=
ln(1+r)
ln(3)
Given the investment triples in value over
�
t years, we know
�
>
0
r>0 (since the investment is growing), and thus
1
+
�
>
1
1+r>1, which means the logarithm is defined. Now, we need to calculate
�
r using the initial and final values of the investment:
4125.35
3000
=
(
1
+
�
)
3
3000
4125.35
=(1+r)
3
4125.35
3000
=
(
1
+
�
)
3
3000
4125.35
=(1+r)
3
1.375117
=
(
1
+
�
)
3
1.375117=(1+r)
3
Taking the cube root of both sides:
1.375117
3
=
1
+
�
3
1.375117
=1+r
1.11285
=
1
+
�
1.11285=1+r
�
=
1.11285
−
1
r=1.11285−1
�
≈
0.11285
r≈0.11285
Now, plug
�
r into the equation for
�
t:
�
=
ln
(
3
)
ln
(
1
+
0.11285
)
t=
ln(1+0.11285)
ln(3)
�
=
ln
(
3
)
ln
(
1.11285
)
t=
ln(1.11285)
ln(3)
�
≈
1.09861228867
0.10509205413
t≈
0.10509205413
1.09861228867
�
≈
10.4586
t≈10.4586
So, it will take approximately 10.46 years for Julian's investment to triple.
