Alfred is saving up money for a down payment on a house. He currently has $3908, but knows he can get a loan at a lower interest rate if he can put down $4861. If he invests the $3908 in an account that earns 4.8% annually, compounded continuously, how long will it take Alfred to accumulate the $4861? Round your answer to two decimal places, if necessary.

Question

Alfred is saving up money for a down payment on a house. He currently has $3908, but knows he can get a loan at a lower interest rate if he can put down $4861. If he invests the $3908 in an account that earns 4.8% annually, compounded continuously, how long will it take Alfred to accumulate the $4861? Round your answer to two decimal places, if necessary.

Answer 1

3908*e^.048t = 4861

t = 4.55 years

Answer 2

Well, Alfred is trying to be a serious adult and save up for a house. Good for him! Let's see how long it will take him to accumulate that extra amount he needs.

To solve this problem, we can use the continuous compound interest formula:

A = P * e^(rt)

Where:
A is the desired amount ($4861),
P is the initial investment ($3908),
r is the interest rate (4.8% or 0.048),
t is the time we're trying to find.

Plugging in the values, we get:

4861 = 3908 * e^(0.048t)

Now, to solve for t, let's get rid of that pesky e. We'll take the natural logarithm of both sides:

ln(4861) = ln(3908 * e^(0.048t))

Using the properties of logarithms, we can simplify this to:

ln(4861) = ln(3908) + 0.048t

Now, isolate t:

0.048t = ln(4861) - ln(3908)

Finally, divide both sides by 0.048:

t = (ln(4861) - ln(3908)) / 0.048

After plugging this into a calculator, we find that t ≈ 2.29

So, it will take Alfred approximately 2.29 years to accumulate the additional amount he needs for the down payment on the house.

Well, that wasn't very funny, was it? I guess saving for a house isn't exactly "clowning" around. But hey, at least Alfred will have a cozy place to call "clown" sweet home!

Answer 3

To find out how long it will take Alfred to accumulate $4861, we can use the continuous compound interest formula:

A = P * e^(rt)

Where:
A is the future value (amount Alfred wants to accumulate)
P is the present value (initial amount Alfred has)
e is the base of the natural logarithm (approximately 2.71828)
r is the annual interest rate (4.8% or 0.048)
t is the time in years (what we need to solve for)

Given:
P = $3908
A = $4861
r = 0.048

We can rearrange the formula to solve for t:

t = ln(A/P) / r

Substituting the given values:

t = ln(4861/3908) / 0.048

t = ln(1.244654) / 0.048

Using a calculator, we find:

t ≈ 0.2147 years

So, it will take approximately 0.21 years for Alfred to accumulate $4861.

Answer 4

To find out how long it will take Alfred to accumulate the required amount of $4861, we need to use the compound interest formula:

A = P * e^(rt)

Where:
A = the final amount
P = the initial amount
r = interest rate per year (in decimal form)
t = time in years
e = Euler's number, approximately 2.71828

In this case, Alfred has $3908 initially and needs to accumulate $4861. The interest rate is 4.8% or 0.048 in decimal form.

Let's plug in the values and solve for t:

4861 = 3908 * e^(0.048t)

Now, we can divide both sides by 3908:

4861/3908 = e^(0.048t)

Simplifying further:

1.2440116 = e^(0.048t)

To find t, we need to take the natural logarithm of both sides:

ln(1.2440116) = ln(e^(0.048t))

Using the property of logarithms:

ln(1.2440116) = 0.048t * ln(e)

Since ln(e) equals 1, we can simplify further:

ln(1.2440116) = 0.048t

Now, divide both sides by 0.048:

t = ln(1.2440116) / 0.048

Using a calculator, we find that ln(1.2440116) is approximately 0.21892, so:

t = 0.21892 / 0.048

t ≈ 4.56 years

Therefore, it will take approximately 4.56 years for Alfred to accumulate the required amount of $4861.