For each of the scenarios given in Exercises 1-6,

step 1 This question is about compounding continuously. That means the amount is being compounded ideal

Key Concepts

Doubling Time

Doubling time is the period required for an investment or any exponentially growing quantity to double in size. Within the framework of exponential growth and continuous compounding, the doubling time can be determined by solving the equation 2P = Pe^(rt). This leads to t = ln(2)/r. Understanding doubling time provides a tangible measure of how quickly an investment grows, making it a practical tool in financial planning.

Average Rate of Change

The average rate of change is a measure that describes how a quantity changes on average over a specific interval. In financial contexts, it can represent the average increase in the amount of an account over a period, calculated as the difference in the amount at two points divided by the time interval between them. This concept helps in understanding the growth dynamics of an investment in practical terms by quantifying the incremental changes over distinct time intervals.

Continuously Compounded Interest

This concept refers to the process by which interest is added to the principal continuously, rather than at discrete time intervals. The formula A = Pe^(rt) expresses the amount A in the account after time t, where P is the initial principal, r is the annual interest rate, and e is the base of the natural logarithm. Understanding this formula is essential for solving problems that involve investments or loans where interest is compounded continuously.

Exponential Growth

Exponential growth describes the way in which quantities increase over time at a rate proportional to their current value. In the context of continuously compounded interest, the growth of the investment is modeled by an exponential function, which captures how the investment accumulates value over time. This concept is widely applicable in finance, population dynamics, and many other fields where growth processes are involved.

Transcript

00:01 This question is about compounding continuously.

00:03 That means the amount is being compounded ideally every moment.

00:09 Okay, so the investment amount instead of in the normal case, it is being competent every monthly or annually or quarterly.

00:16 So here the thing is the equation that we have that we have used for this kind of problems is a of t is equal to p .e to the power of r.

00:24 Okay.

00:27 This is the equation that you need to use for this and the principle amount that is we has $500 and the rate of interest is 0 .75 % but when it comes to this equation that is a of t is equal to p e to the part of r t is the amount that you're expecting on the end of t years and yeah t is in express in years that is the time period and p is the principal amount is the number e that is oiless number and r is the rate of interest but rate of interest should be expressed in decimals and oaken percentage so here it's given in percentage so you need to divided by 100 in order to use this in equation that means that will be 0 .0075.

01:05 So what we are asked to find is that we need to find the equation or formula or we need to express the amount that is a in terms of in as a function of t so that's what we are supposed to do we just need to substitute these values in this equation and that's all that's enough that is sorry i'll use the black is equal to a of t is equal to p that is 500 multiplied by e to the power of rt r is 0 .0075t.

01:35 This is the function that represents this particular account.

01:39 Then b, that is how many years it will take to reach, sorry, how much amount will be there in the account at the end of five years, 10 years, 30 years and 35 years.

01:50 Okay.

01:51 It is first of all, a of 5 is 500 multiplied by e to the bar of 0 .007 5 multiplied by 5.

01:59 This is what a of five at the end of five years will have in the account that is 500 multiplied by e to the power of 0 .075 multiplied by 5 there will be 519 519 .1 this will be the amount that you have to need round it up to the nearest cent so that's why it became 511 sorry 19 .11 now at the end of 10 years this particular thing this account will have 1 075 multiplied by 10 that will be 538 38 .94 this must be the amount that will be remaining at the end of 10 years now at the end of 30 years the end of 30 years of 30 is equal to 500 multiplied by each to the power of 0 .0075 multiplied by 30 that will be so that will be 626 .6 .6 .26 .16.

03:18 This must be the amount at the end of 30 years.

03:21 And when it comes to 35 years, that is a of 35 is equal to 500 multiplied by 8 to the power of 0 .0075 multiplied by 35 multiplied by 35 will be 650, 650, 650.

03:39 T09 dollars this much will be the amount remaining at the end of 35 years okay this is this is what you can expect in that account if the interest rate continues to be like this and in the third part what we have to do is that determine how long it will take for the initial investment to double okay so that means a of t is equal to $1 ,000 right means p p is 500 that means a after you will be $1000 and we need to find the time taken to reach this target so that means 1000 is equal to 500 multiplied by by e to the power of rt right r is 0 .0075 multiplied by t we need to find t so here dividing both sides by 500 we'll get each to the power of 0 .0075 t is equal to two now we need to apply in natural logarithm to solve this because we have exponent power here but we'll use natural logarithm you are free to use any kind of any logarithm but here we'll use natural logarithm because it makes the whole thing easier because we have e here right so natural log of e to the power of 0 .0075 t is equal to natural log of 2 and applying properties of logarithm we'll get 0 .0075t natural log of e which is 1 so we don't need to write right is equal to natural log of 2 is 0 .693 yeah 6931 6931 that means t is equal to 0 .6931 divided by 0 .0075.

05:27 It is 16931 divided by 0 .0075 is 92 .931 divided by 0 .0075 is 92 .131.

05:39 4133 years which means prouting it off we'll get 92 years so that means in 92 years we can expect the amount to double okay so that is what this means now moving on to the third question that is sorry fourth question that is find and interpret the average rate of change of amount in the account from the end of fourth year to 50 year and that is at the end of 40 to 50 we'll have different amounts right so we need to we need to find the average rate of change from 40 to 50 to the rate of change from 34 to 30 50 that means we need to find the amounts at the end of 40 50 50 right so amount at the end of 40th year 4th year is 500 multiplied by e to the power of 100 multiplied by e to the power of 0 .0075 multiplied by 4.

06:47 Right? this is the amount that will be the end of 40.

06:51 That will be 512...

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