In the year 1985, a house was valued at 110,000 . By the...

In the year $1985,$ a house was valued at $\$ 110,000 .$ By the year $2005,$ the valueciated to $\$ 145,000$ . What was the annual growth rate between 1985 and 2005$?$ Assume that the valued to grow by the same percentage. What was the value of the house in the year 2010$?$

Key Concepts

Compound Interest

Compound interest refers to the process where an investment's earnings, from either capital gains or interest, are reinvested to generate additional earnings over time. This concept is based on the principle of exponential growth, where the value increases at a constant percentage rate per period, and previous gains help to boost future increases.

Annual Growth Rate

The annual growth rate is the fixed percentage increase applied to an investment or asset each year, calculated in such a way that when compounded over a number of periods, it leads to the observed total growth. It is determined by the formula: Growth Rate = (Final Value / Initial Value)^(1/Number of Years) - 1, reflecting the idea that the percent increase is applied consistently over time.

Extrapolation of Future Values

Extrapolating future values involves applying the calculated constant annual growth rate to estimate the value at a future point beyond the known timeframe. This is based on the compound interest formula, ensuring that the pattern of growth continues steadily according to the rate computed.

Time Periods in Compound Growth

Time periods are the individual intervals (usually years) over which the growth is compounded. They are crucial in determining the exponent in the compound interest formula, as they represent how often the growth rate is applied and directly affect the overall calculation of future values.

Transcript

00:03 All right, so we have a house in 1985 that was valued at $110 ,000.

00:12 And in 2005, it was valued at, or valucated, at $44 ,000.

00:20 What was the annual growth rate between 1985 and 2005 and then also figure out the value in 2010? so we know that we're going to be using the exponential graph or exponential equation a times b to the since we're doing the population and time and stuff not population just time actually we're going to do t not x for the exponent and we have two values we have 1985 in 2005 so let's say let's said 1985 at the beginning so it'll be the times equal to zero parts so we know that at nineteen eighty five the initial value would have been a hundred and ten thousand dollars so and we would do a hundred and ten thousand times b to the well it's no time is passed like it's been value today basically so it would be to the zero and that's equal to just a hundred and ten thousand and then we have y is equal to hundred so we have the base rate or the base number we still the raid but now 20 years has passed.

01:36 We know this is actually equal to a number.

01:38 This is equal to $145 ,000.

01:46 Now actually we could just solve for b right now.

01:50 So after doing some calculations, we can divide both by 110 ,000 both sides and then take 1 out of 20.

01:57 So actually let me let me draw out the steps for that, just for you to understand.

02:01 So we're going to divide both sides by 110 ,000.

02:08 That'll give us, let's see, i'll do my calculator in my phone, 110 ,000.

02:15 It's going to be 1 .318 is equal to b to the 20...