Compound Interest. Suppose that 750 is invested at 7 %...

Compound Interest. Suppose that $\$ 750$ is invested at $7 \%$ interest, compounded semiannually. a) Find the function for the amount to which the investment grows after $t$ years. b) Graph the function. c) Find the amount of money in the account at $t=1,6,10,15,$ and 25 years. d) When will the amount of money in the account reach $\$ 3000 ?$

Compound Interest. Suppose that $\$ 750$ is invested at $7 \%$ interest, compounded semiannually.
a) Find the function for the amount to which the investment grows after $t$ years.
b) Graph the function.
c) Find the amount of money in the account at $t=1,6,10,15,$ and 25 years.
d) When will the amount of money in the account reach $\$ 3000 ?$

Key Concepts

Graphing Exponential Functions

Graphing exponential functions provides a visual representation of how an investment grows over time. By plotting the compound interest function, one can observe the characteristic exponential curve, where the rate of increase accelerates as time progresses, reflecting the continuous impact of compounded growth.

Solving Exponential Equations

Solving exponential equations is a critical mathematical process used to determine variables such as time or interest rate in growth models. In compound interest problems, logarithms are typically used to solve for the time it takes for an investment to reach a specific value by isolating the exponent in the compound interest formula.

Compound Interest

Compound interest is a financial concept where the interest accrued on an investment or loan is reinvested or added to the principal so that, in subsequent periods, interest is earned on both the initial principal and the accumulated interest. This results in an exponential growth of the investment over time as interest is effectively earning interest.

Exponential Growth

Exponential growth describes processes that increase at rates proportional to their current value, leading to rapid increases over time. In the context of compound interest, the investment grows exponentially because the accumulated interest compounds at regular intervals, causing the total amount to increase at a constant percentage rate.

Compounding Frequency

Compounding frequency refers to the number of times per year that the accrued interest is calculated and added to the principal. The frequency of compounding (such as annually, semiannually, quarterly, or monthly) affects the effective interest rate and, consequently, the total growth of the investment over time.

Transcript

00:01 All righty.

00:01 So we're talking about compound interest some more, and we need to, first of all, given the information, write the equation in terms of amount and the time and years.

00:11 So we've got our a is the amount with is equal to the principal, which we're told the investment amount is $750.

00:21 Then we have parentheses 1 plus the rate, which is 7%.

00:25 So that's going to be 0 .07 as a decimal compounded semi -annually, which is two times a year.

00:30 And that's going to be raised to the 2t power.

00:33 So to get that a little bit more sufficient, we're going to have 750 times 1 .035 to the 2t power.

00:46 And then from there, we're asked to graphs.

00:48 So i'm going to go to decimus .com, which is a free graphing calculator website, so we can kind of see the graph.

00:54 So we're going to type in the equation, y equals 750 parentheses.

01:02 1 .035 raised 2.

01:05 So you press shift 6 to the two.

01:07 Instead of t, you're going to use x.

01:10 So that way we're using it in terms of an x and y.

01:13 So y represents our amount.

01:15 X represents our time and ears.

01:18 And notice we don't see anything.

01:20 That's because we need to go and kind of get this from $5 .50.

01:22 Now, it does say in the question that we need to keep in mind the amount of years...

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