Answer :
Answer: 6.75%
Step-by-step explanation:
To find the annual interest rate compounded continuously, we can use the formula for continuous compounding:
=
×
A=P×e
rt
Where:
A is the amount of money accumulated after n years, including interest.
P is the principal amount (the initial amount of money).
r is the annual interest rate (in decimal).
t is the time the money is invested for in years.
e is the base of the natural logarithm, approximately equal to 2.71828.
Given:
P = $3,000
A = $6,300
=
11
t=11 years
We need to solve for
r:
6300
=
3000
×
11
6300=3000×e
11r
Divide both sides by 3000:
11
=
6300
3000
e
11r
=
3000
6300
11
=
2.1
e
11r
=2.1
Now, take the natural logarithm of both sides:
11
=
ln
(
2.1
)
11r=ln(2.1)
=
ln
(
2.1
)
11
r=
11
ln(2.1)
Now, we can calculate the value of
r:
≈
0.741937
11
≈
0.06745
r≈
11
0.741937
≈0.06745
To express this as a percentage, we multiply by 100:
≈
0.06745
×
100
%
≈
6.75
%
r≈0.06745×100%≈6.75%
So, Dale would reach his goal with an annual interest rate of approximately 6.75%
