Question

A two-bedroom house in Seattle was worth $400,000 in 2005. Its appreciation rate is 3.5% each year.

a. How much is the house worth in 2015?
b. When will it be worth $800, 000?
c. In Jacksonville, houses are depreciating at 2% per year. If a house is worth $200, 000 now, how much value
will it have lost in 10 years?

Answer 1

Answers:

(a) $564,239.50

(b) 2026

(c) $36,585.44

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Step-by-step explanation:

Part (a)

One template for exponential equations would be

y = a*b^x

Another template is

y = a(1+r)^x

which is a bit more descriptive. I'll use the second template.

• a = 400000 = starting value
• r = 0.035 = growth rate in decimal form

r is positive for exponential growth or appreciation.

The equation y = a(1+r)^x updates to y = 400000*(1+0.035)^x and then simplifies to y = 400000*(1.035)^x

From here we plug in x = 10 because the year 2015 is ten years after 2005.

So,

y = 400000*(1.035)^x

y = 400000*(1.035)^10

y = 564239.504248449

y = 564239.50

The house is worth about $564,239.50 in the year 2015.

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Part (b)

We work the process of part (a) in reverse.

This time we know what y is and we want to solve for x.

Use logarithms to isolate the exponent.

y = 400000*(1.035)^x

800000 = 400000(1.035)^x

800000/400000 = (1.035)^x

2 = (1.035)^x

Log(2) = Log( (1.035)^x )

Log(2) = x*Log( 1.035 )

x = Log(2)/Log(1.035)

x = 20.1487916840008

If x = 20, then,

y = 400000*(1.035)^x

y = 400000*(1.035)^20

y = 795,915.545386338

y = 795,915.55

We're short of the goal of $800,000.

If x = 21, then

y = 400000*(1.035)^x

y = 400000*(1.035)^21

y = 823,772.589474859

y = 823,772.59

We've gone over the goal.

Somewhere between x = 20 and x = 21 is when the house will be worth exactly $800,000.

It's better to side with x = 21 since x = 20 comes up short.

21 years after 2005 is 2005+21 = 2026

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Part (c)

Go back to the template: y = a(1+r)^x

This time we have

• a = 200,000
• r = -0.02

The r value is negative to indicate exponential decay or depreciation.

y = a(1+r)^x

y = 200000(1+(-0.02))^x

y = 200000(1-0.02)^x

y = 200000(0.98)^x

The 0.98 means the house keeps 98% of its value from year to year.

Plug in x = 10.

y = 200000(0.98)^x

y = 200000(0.98)^10

y = 163,414.56137751

y = 163,414.56

The house starts off at $200,000 and ten years later it's $163,414.56

Subtract the values to determine how much home value was lost.

200,000 - 163,414.56 = 36,585.44

The home value will have lost $36,585.44 over the 10 year period.