From the graph the median price paid for a home in 1968 is about $20000 and in 1993 is about $100000. Use these data points to construct a linear and an exponential model to represent the growth in median price from 1968 to 1993.

Question

From the graph the median price paid for a home in 1968 is about $20000 and in 1993 is about $100000. Use these data points to construct a linear and an exponential model to represent the growth in median price from 1968 to 1993.

for p(t) linear I got 20000+3200t
but for the exponential I get something wrong and weird like 20000*0.97^t/25
Help, Please & Thank You

Answer 1

looks like you considered

1968 as t = 0 , that's fine

so you would two ordered pairs (0,20000) and (25,100000)
I agree with your linear equation, but you should state it as
p(t) = 20000 + 3200t, where t is the number of years after 1968

for the exponential:
let p(t) = 20000 e^(kt), where t is the number of years after 1968
so for (25,100000)
100000 = 20000 e^(25k)
5 = e^ 25k
take ln of both sides
ln 5 = ln (e^ 25k)
ln 5 = 25k ln e, but ln e = 1
25k = ln 5
k = ln 5/25

p(t) = 20000 e^(ln5/25 t)
or p(t) = 2000 e^ (.0643775 t)

You really could have used any base, I have no idea why you chose .97

Answer 2

the "half-life" is 25 years.

(1/2)^(1/25) = 0.97265

Answer 3

To construct a linear model for the growth in median price from 1968 to 1993, you have correctly used the formula p(t) = 20000 + 3200t, where t represents the number of years since 1968. The constant term 20000 represents the median price in 1968, and the coefficient of t, 3200, represents the rate of increase in the median price per year.

Now let's construct an exponential model using the given data points. An exponential model is of the form p(t) = a * b^t, where a and b are constants to be determined.

To find the exponential model, we need to find the value of b. Let's use the data point from 1993, where t = 25 (since it's 25 years after 1968) and p(t) = 100000:
100000 = a * b^25

Now, we can use the other data point from 1968, where t = 0 and p(t) = 20000:
20000 = a * b^0
20000 = a * 1
a = 20000

So, now we have:
100000 = 20000 * b^25

To solve for b, we divide both sides of the equation by 20000:
5 = b^25

To isolate b, we take the 25th root of both sides:
b = ∛5 ≈ 1.083

Therefore, the exponential model for the growth in median price from 1968 to 1993 is:
p(t) = 20000 * (1.083)^t

Please note that the values of a and b may vary slightly depending on the accuracy of calculations, but this is the general process to construct an exponential model based on the given data points.