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For how long must $\$ 1700$ be left on deposit at $6 \%$ compounded continuously to reach a total accumulation of $\$ 3200 ?$

$$10 y 6 m 16 d$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

Campbell University

McMaster University

Baylor University

Lectures

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For how long must $\$ 2200…

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For how long must $\$ 700$…

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One thousand dollars is …

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as soon as you see the phrase compounded continuously, that means that we're doing the perp formula A equals P E to the R T. So as we're looking at this, the 1700 the problem represents the P, the principal, the 32 hundreds, the ending amount. So that's the A value and we're going to solve for how much time. And so we're gonna put the rate in here. But it's very important that you put the rate as a decimal. So just remember, 6%. The word percent means out of 100 per out of sent 100 which is 0.6 I hope that makes sense. Eso Anyway, to solve this problem, then you've got to get t by itself. So the first step is to divide 1700 over, and you could simplify that to, like, 32/17. Um, you know that is equal. So then, to get rid of that, you take the natural log of both sides, and, uh so I guess I I'll go ahead and just write the natural log, be there is equal 2.0 60. I hope that makes sense because this is the power rule that you can bring that in front on the natural geography is just one. That's why they cancel out. So then your last step is to divide both sides by 0.6 Now, I'm confident that every teacher let you use a calculator on this. So that's what I'm doing of 32/17, divided by 0.6 And I got a value for T to be 10.542 And this unit is in years also takes about 10.5 years. Now, as faras units go are you know, breaking it down to the day. What you could do is truncate that number. So you just look at the decimals of subtract 10 off and then multiplied by 12 because there's 12 months in a year s O. That would take me to six months, because I get 6.505 so six months and it truncate again. So take away that six and then I think you should multiply by 30 was about 30. Um, So take this 0.505 and multiply by 30 and you get a number. That's 15 point something. 135 But I think I'll round that up to 16 because that 15 days you don't quite have enough. You gotta wait till that 16th day. So this is the answer out. Right down. Um, and your teacher might disagree with me on the days, but that's about right. It

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