so ive been stuck on this for hours its precalculus

Question

A house was valued at $125,000 in the year 1985. The value appreciated to $150,000 by the year 2007

.A) What was the annual growth rate between 1985 and 2007?  = ___ 

Round the growth rate to 4 decimal places.

B) What is the correct answer to part A written in percentage form? =___ %

.C) Assume that the house value continues to grow by the same percentage. What will the value equal in the year 2012 ? value = $  Round to the nearest thousand dollars.

Answer 1

This is compound annual interest or appreciation and the formula is:

(A)

A= P(1+r)^t

Where:

A=final amount =$150,000

P=principal or the starting amount = $125,000

r=annual growth rate = ?

t=time in years = 2007-1985 = 22 years

Apply the formula:

150,000 = 125,000 (1+ r)²²

Divide both sides by 125,000:

1.2 = (1+r)²²

Next is to get rid of the exponent 22 by raising another exponent on both sides by 1/22 to create power of a power.

(1.2)^1/22 = ((1+r)²²)^1/22

In power of a power property of exponents, we multiply exponents. (22)(1/22) = 1

(1.2)^1/22 = (1+r)¹

(1.2)^1/22 = 1+r

(1.2)^1/22 -1= r

r = 0.00832177858111

r ≈ 0.8322%

(B) You can start from 2007 to 2012.

Using the same formula, the following values are:

P=150,000

A =?

r=0.8322% = 0.008322

t= 2012-2007 = 5 years.

You should know how to do this now and don't forget to round off your answer as instructed.

Answer 2

Hello Katie!

An exponential growth function can be modeled by the form y = a * (1+r)^x where a is the initial value at year 0, and r is the annual growth rate. In this case, x is years since 1985, and y is the value of the house.

The value of the house in 1985 is 125000, so a = 125000. Our function now looks like this: y = 125000 * (1+r)^x.

To find the annual rate of growth r, we can plug in the other point we were given. In 2007, the value of the house is 150000. 2007 is 22 years since 1985, so this point is (22, 150000). Plugging this in gives:

150000 = 125000 * (1+r)²². We can then divide both sides by 125000 to get:

1.2 = (1+r)²². Using a calculator to take the 22nd root of both sides leaves us with:

1.0083 = 1+r. Subtracting 1 from both sides to isolate r leaves us with:

0.0083 = r. This is the answer to part A.

For part B, to represent this as a percent, we multiply r by 100 (because a percent is out of 100 or divided by). Written as a percent, the annual growth rate is 0.83%.

For part C, remember that our model now looks like this: y = 125000 * (1.0083)^x. We are asked to find the value of the house in 2012 assuming the value still grows according to this formula. 2012 is 27 years since 1985, so we can plug in 27 for x in our model and we need to find y.

y = 125000 * (1.0083)²⁷. Simplifying using a calculator gives that approximately y = 156,346.

The question asks for the answer to be rounded to the nearest thousand dollars, so our answer is that the house is estimated to be worth 156,000 in the year 2012.

Good luck on your homework!

Another Katie