Compound Interest A man invests 6500 in an account that pays 6% interest per year, compounded

step 1 Okay, so for Part A, looking to find the amount after two years. Now, we're told that we're comp

Compound Interest A man invests 6500 in an account that...

Key Concepts

Compound Interest

Compound interest is the process of earning interest on both the initial principal and the accumulated interest from previous periods. This concept plays a crucial role in financial mathematics by demonstrating how investments or loans can grow over time as interest is computed on an increasing base.

Continuous Compounding

Continuous compounding is a method of calculating interest where the frequency of compounding is increased to an infinite number of periods per year. Instead of calculating interest at discrete intervals, continuous compounding uses the natural exponential function to model growth, leading to the formula A = Pe^(rt), where A is the amount, P is the principal, r is the rate, and t is time.

Exponential Growth

Exponential growth is a process in which a quantity increases by a constant multiplicative rate over equal time intervals. In the context of interest, this implies that the amount in an investment grows exponentially over time, reflecting the effects of compound interest, especially under conditions of continuous compounding.

Natural Logarithm

The natural logarithm is the logarithm to the base 'e' and is instrumental in solving equations involving exponential functions. When determining time periods in continuously compounded interest problems, taking the natural logarithm of both sides of the equation allows for the exponent containing the time variable to be isolated, thereby facilitating the solution.

Transcript

00:01 Okay, so for part a, looking to find the amount after two years.

00:05 Now, we're told that we're compounded continuously.

00:09 So i want to use the formula for continuous compound interest.

00:12 So that's going to be a of t, the amount after two years is equal to, i think i think of it, as pert.

00:20 So we have the principle p times this number e raised to the rt power.

00:28 So where we have that the principle is going to be equal to a given as $6 ,500, so $6 ,500, our interest rate is going to be 6%.

00:43 So as a decimal, that's 0 .06, right? and our time, t is, well, in two years.

00:54 So our t is going to be, well, two years or t is equal to two.

00:58 Okay? just go ahead and put these values.

01:00 Use into our formula.

01:02 So we have that a of t, the amount after t years, is going to be equal to our initial amount or our principal, which is 6500 times e raised to the rt.

01:15 So e raised to the 0 .06 times 2.

01:22 Okay? so the calculator here, write 0 .06 times 2.

01:26 That's 0 .12.

01:28 So we have that a of t, is equal to 6 ,500 times e to the 0 .1 to power.

01:37 So, right, you're going to take e to this power first, and then take that value and then multiply it by 6500.

01:43 So e to this power should give us like 1 .1 .1275.

01:47 Okay, leave that in your calculator and then multiply that by 6500.

01:52 And we should see here that a .f .t is equal to 7 ,328 ,000.

02:00 Um point like seven two nine so in around that for money right so seven two nine it rounds with seven three so therefore the amount after two years is seven thousand three hundred twenty eight dollars and seventy three cents okay then for part b we're looking to find out how long it will take for the amount to be eight thousand dollars so again we're using right we're using our continuous compound interest formula.

02:34 So again, a of t is equal to p times e raised to the rt power.

02:41 Okay.

02:42 So here, right, our principle is still 6 ,500.

02:48 But now, right, we want to set our function equal the amount we want to be 8 ,000.

02:55 Okay? so we go ahead and we take our function and we set it equal.

03:04 To 8 ,000.

03:05 So we have 8 ,000 is equal to that principle, so 6 ,500 times e to the rt.

03:14 So again, r is 0 .06.

03:17 And now t, we're finding out when, right? when are we going to have $8 ,000? so now we're solving for t, right? so what we could do here, we want to basically undo e to a power.

03:30 We want to use a natural log...

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