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Homework: MML hw 3 Ch 4 Exponential & Logarithmic Score: 30 of 32 (28 complete) 5.2.7-BEA company invests 520,000 in CD Ihat eatns 7%0 compounded continuously H...

Question

Homework: MML hw 3 Ch 4 Exponential & Logarithmic Score: 30 of 32 (28 complete) 5.2.7-BEA company invests 520,000 in CD Ihat eatns 7%0 compounded continuously How lonq will it take for the account to be worth 530,000?

Homework: MML hw 3 Ch 4 Exponential & Logarithmic Score: 30 of 32 (28 complete) 5.2.7-BE A company invests 520,000 in CD Ihat eatns 7%0 compounded continuously How lonq will it take for the account to be worth 530,000?



Answers

Use an exponential model and a graphing calculator a estimate the answer in each problem. If Jean invests $\$ 2300$ in a retirement account with a $6 \%$ interest rate compounded annually, how long will it take until Jean's account has a balance of $\$ 4150 ?$

Has asked us to write an exponential expression for the given values. And when you're dealing with continuous compounding interest, the formula used for that is P E to the R T, where P is the principal. So looking at part A that's given us $70 he is the mathematical constant, which is equal to approximately 2.71 to 8. Are is the interest rate, and in this question we're giving the interest rates and percent form, so we need to convert that to decimal form. Ah, and he is time measured here in this question that is, in years. So if we look at part a forgiven the principle as $70 e as this number here, the interest rate is 4%. That we change that to point 04 and T is three in years. So the exponential expression for the values given in part a would be 70 times e to the 700.4 times three. Um, and this question does not say we need to get exact value for that. We put that into a calculator, give you the exact value, and for part b very similarly. Ah, the principal was given as 6 90 times e to the 5% so 50.5 times two, which is the years.

Mhm. But again, probably when you use an exponential model and graphing calculator to estimate the answer. So mhm. We want to see us um we're going to triple the investment this time. So if we have the amount is equal to three times the amount of actually uh huh meetings are key. We know we want the final amount to be three times our initial amount. So when we divide by a not on both sides, take the natural log, get rid of this E. We're left with Artie and we know that the interest rate are is going to be 575 15 0.575 So we're going to take the natural log of three divided by this right here. So we end up seeing it going to take about 19 years for us to triple our money if we compounded continuously.

Today using log rhythms and the graphing calculator. We're going to find the time required for each initial amount to be at least equal to the final amount which is $8,000 deposit at 3% compounded quarterly to reach at least $23,000. And so what we're going to do is we're going to use our formula for compound quarterly which is going to be a equals P. Which is your principal Which is your starting amount times one plus your rate divided by your and which is how many times is being compounded to the N. T. Power. And so in order to do this we have to look through the question and plug in our values. And so if it tells you that we are starting with 8000 and the final amount is 23,000 we're gonna plug 23,000 and for a okay We're going to plug our 8000 in for P. Which is our starting amount which is then going to get multiplied By 1-plus hour rate. And so if we look back at our question we look at our rate is 3% compounded quarterly. And so we convert our percentage of 3% 2.03 Which were then going to divide by four since it's being compounded quarterly. And that scene four is going to be on your exponent to the four T. Power. And so T. Is what we're going to solve for. And so what we wanna do is want to isolate everything away from the t. And so by doing that we're going to divide by 8000 on both sides. When we divide by 8000 those will reduce to one and we'll do 23,000 Divided by 8000. And so we take our calculator and we do 23,000 And we divide by 8000 And we're gonna get 2.875. And so 2.875 is now going to be equal to this value to the fourty power. So we can combine using Pandas and figure out what is inside of the parentheses. And so I would do .03 We're gonna divide that by four and then add one to that. So we're just combining our terms here and we're going to get one point 0075. And we're going to take that to the four T. Power. And so when we talk about using logarithms what we're going to do is in order for this to work we're going to have to bring our exponent down to the same level so that we can then solve for T. And so what we do is we use log and so we bring the fourty which always goes in front of the law algorithm, explain. It goes in front of the algorithm. Then you're gonna get log of one 0075. Mhm. And just like anything else? We have to balance it. So whatever we do on one side of the law algorithm we have to also do on the other side. And so therefore we're going to also put a log on the left side in front of the two points 875. And so now we're able to solve for t. By dividing everything by getting rid of the T. And so we're going to do here is we're going to divide by log of 1.0075 on both sides. So that these reduce And we'll do the same thing on this side, we'll get log of one 0075. Uh huh. And so finally you're going to take your calculator and you're going to divide log Of 2.875. Close the parentheses, so you know you're not making a mistake And then you're gonna multiplied by log of 1.0075. And so what you're gonna get is 141.33. Yeah. And that's going to equal what's left over, which is 14. We do the inverse operation of multiplication, which is the vision. And so now we're just gonna divide everything by four. And so we're gonna take our calculator, we're gonna divide by four And we're going to get 35.33. We check our question, it doesn't tell you that you need to around to a certain amount, it just says find the required time. And so therefore you could state that it's 35 .33 years, which is going to equal it's, etc. And that's how you find how much time it would take to go from $8,000 To $23,000 at a 3% interest rate compounded quarterly. Thanks everybody.

The solution to the question # 99 is here We are given with a schools $2 25 housing Our is close to 3.2%. Or we can write it as 0.032 and his four and t. Will be 11 divided by four. That is 2.75. So from the formula of compound interest which is his schools to be into one plus R. By an to the poverty in to win. Let us substitute the values. We will get a squirrels too is provided to us. That is 25 1000. That april's to pee into one plus 0.32 Divided by four. to the power 2.7 five in 24. So solving this you will get your P. To be as 20- 90 2.24. And so this would be the solution to the question. Thank you.


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