A house was valued at 125000 in the year 1985 the value...

A house was valued at $125,000 in the year 1985. The value appreciated to $145,000 by the year 2006. A) If the value is growing exponentially, what was the annual growth rate between 1985 and 2006? $r = \text{_____}$ Round the growth rate to 4 decimal places. B) What is the correct answer to part A written in percentage form? $r = \text{_____}$ %. C) Assume that the house value continues to grow by the same percentage. What will the value equal in the year 2010? value = $ 150,000 Round to the nearest thousand dollars.

Transcript

00:01 Hi there, so for this problem, we are told that a house was value.

00:05 So this is the value, let's label this as the value p at zero, which corresponds to the year 1992.

00:13 That will be 105 ,000.

00:16 Okay.

00:17 So that was the initial value for this.

00:19 And in the year 2004, that will be 10 years later, so that will be then this after 10 years, is $150 ,000.

00:29 So this was at the year 1992.

00:35 And well, sorry, it's 12 years later, sorry, because that is the difference between 2004 in 1992 because, yes, that will be 12.

00:45 12 years later in 2004.

00:51 So, okay, now, for part a of this problem, the question is, if the value is grown exponentially, what was the annual growth rate between 1992? in 2004.

01:04 Okay.

01:06 So what we need to do in this case is to determine that annual growth rate.

01:19 So we use the following expression that the amount at any given time is equal to the initial value times 1 plus the rate elevated to the times t.

01:31 And we need to solve for the rate.

01:32 So first we can substitute the values in here.

01:35 Already we know that this initial value is 105 ,000, and this value we're going to use the other condition.

01:43 Okay, so that will be 150 ,000 equals to 105 ,000, this times one plus the rate are elevated to 12.

01:55 Now we can pass this to divide to the other side.

01:57 We can cancel these three zeros with this.

02:01 We have 150 divided by 105.

02:05 Then we can apply the neparian logarithm in both sides of this.

02:08 So we will obtain 12 times the neparian logarithm of 1 plus the rate art.

02:17 Okay.

02:19 Well, we can do it in another way.

02:22 So what we can do is we can elevate both sides of this at 1 divided by 12...