Hannah invests $3850 dollars at an annual rate of 6% compounded continuously, according to the formula A=Pe^rt, where A is the amount, P is the principal, e=2.718, r is the rate of interest, and t is the time, in years.
(a) Determine, to the nearest dollar the amount of money she will have after 5 years.
(b) Determine how many years, to the nearest year, it will take for her investment to have a value of $10,000.
r = .06
A = 3850 * e^(.06*5)
A = $ 5197
b)
10,000 = 3850 * e^(.06 t)
ln ( 2.597) = .06 t
t = 16
To determine the amount of money Hannah will have after 5 years, we can use the formula A=Pe^rt.
(a) To calculate the amount, we need to plug in the given values into the formula. The principal (P) is $3850, the annual interest rate (r) is 6%, and the time (t) is 5 years.
First, let's convert the interest rate (r) to a decimal by dividing it by 100: 6%/100 = 0.06.
Now we can plug the values into the formula:
A = 3850 * e^(0.06 * 5)
The value of e is approximately 2.718 (base of the natural logarithm).
A = 3850 * 2.718^(0.06 * 5)
Using a calculator or a computer program to calculate 2.718^(0.06 * 5), we get approximately 1.3479.
A = 3850 * 1.3479
Now we can multiply to find the amount:
A ≈ $5189.61
Therefore, to the nearest dollar, Hannah will have approximately $5,190 after 5 years.
(b) To determine how many years it will take for Hannah's investment to reach a value of $10,000, we can rearrange the formula A=Pe^rt to solve for t.
Divide both sides of the equation by P:
A/P = e^rt
To isolate t, take the natural logarithm of both sides:
ln(A/P) = rt
Divide both sides by r:
ln(A/P)/r = t
Now we can plug in the given values to calculate t:
t = ln(10000/3850) / 0.06
Using a calculator or a computer program, we can calculate ln(10000/3850) / 0.06:
t ≈ 8.64
Therefore, it will take approximately 8.64 years (rounded to the nearest year) for Hannah's investment to reach a value of $10,000.
To solve these problems, we will use the formula for continuous compound interest:
A = Pe^(rt)
where A is the amount after time t, P is the principal, e is the base of the natural logarithm (approximately 2.718), r is the rate of interest, and t is the time in years.
(a) To determine the amount of money Hannah will have after 5 years, we can substitute the given values into the formula.
P = $3850 (the principal)
r = 6% = 0.06 (the annual interest rate)
t = 5 years
Using the formula A = Pe^(rt):
A = 3850e^(0.06 * 5)
To calculate this, we need to find the value of e^(0.06 * 5).
e^(0.06 * 5) ≈ 1.340477
A ≈ 3850 * 1.340477 ≈ $5,157.94
Therefore, Hannah will have approximately $5,157.94 after 5 years.
(b) To determine how many years it will take for her investment to reach $10,000, we can rearrange the formula:
A = Pe^(rt)
Dividing both sides by P:
A/P = e^(rt)
Substituting the given values:
A = $10,000
P = $3850
r = 6% = 0.06
We need to solve for t. Taking the natural logarithm of both sides, we get:
ln(A/P) = rt
Dividing both sides by r:
ln(A/P) / r = t
Using ln(A/P) / r ≈ 0.067658, we can calculate:
t ≈ 0.067658 / 0.06
t ≈ 1.12763
Therefore, it will take approximately 1.128 years (to the nearest year) for Hannah's investment to have a value of $10,000.