How do I solve these problems?

Complete the table for a savings account in which interest is compounded continuously.

1. Initial Investment: $1000
Annual % Rate: 3.5%
Time to Double: ?
Amount After 10 Years: ?

2. Initial Investment: $750
Annual % Rate: ?
Time to Double: 7 3/4 yr
Amount After 10 Years?

3. Initial Investment: $500
Annual % Rate: ?
Time to Double: ?
Amount After 10 Years: $1505.00

4. Initial Investment: ?
Annual % Rate: 4.5%
Time to Double: ?
Amount After 10 Years: $10,000.00

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To solve these problems, we need to use the formula for compound interest, which is given by:

A = P * e^(rt)

where:
A = the amount after time t
P = the initial investment
r = the annual interest rate (as a decimal)
t = the time in years
e = the mathematical constant approximately equal to 2.71828

Let's go through each problem step by step:

1. To find the time it takes to double the initial investment, we can use the formula:

2P = P * e^(rt)

Dividing both sides of the equation by P, we get:

2 = e^(rt)

To solve for t, we take the natural logarithm of both sides:

ln(2) = rt

Rearranging the equation to solve for t, we have:

t = ln(2) / r

Plugging in the given values, we get:

t = ln(2) / 0.035

And we can evaluate this expression to find the time it takes to double the investment.

To find the amount after 10 years, we can use the same formula with the given values:

A = 1000 * e^(0.035 * 10)

Evaluate this expression to find the amount after 10 years.

2. In this problem, we are given the time it takes to double the investment, which is 7 3/4 years. We can use the same formula as before, but this time we need to solve for the interest rate, r.

2P = P * e^(rt)

Dividing both sides by P and rearranging the equation, we get:

2 = e^(7.75r)

To find the interest rate, we can take the natural logarithm of both sides and solve for r:

ln(2) = 7.75r

r = ln(2) / 7.75

Plugging in this value, we can calculate the interest rate.

To find the amount after 10 years, we use the formula:

A = 750 * e^(r * 10)

Evaluate this expression to find the amount after 10 years.

3. In this problem, we are given the amount after 10 years, which is $1505. To find the interest rate, we can rearrange the compound interest formula:

A = P * e^(rt)

to solve for r:

r = ln(A/P) / t

Plugging in the given values, we can find the interest rate.

To find the time it takes to double the investment, we can rearrange the formula again to solve for t:

t = ln(2) / r

Evaluate this expression to find the time to double the investment.

4. In this problem, we are given the amount after 10 years, which is $10,000. Let's assume the initial investment is P. We can use the compound interest formula to solve for P:

A = P * e^(rt)

10,000 = P * e^(0.045 * 10)

Dividing both sides by e^(0.045 * 10) and solving for P, we get:

P = 10,000 / e^(0.045 * 10)

Evaluate this expression to find the initial investment.