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
done, check back to your last post of this
To construct a linear model, we can use the slope-intercept form of a linear equation: y = mx + b, where y is the dependent variable (in this case, the median price), x is the independent variable (time), m is the slope of the line, and b is the y-intercept (the median price at time 0).
Given that the median price in 1968 is about $20,000, we can use the point-slope form of a linear equation to determine the slope (m):
m = (y2 - y1) / (x2 - x1)
m = (100000 - 20000) / (1993 - 1968)
m ≈ 80000 / 25
m ≈ 3200
Now, we substitute the values of the slope and one of the points into the slope-intercept equation: y = mx + b
Using the point (1968, 20000), we can solve for b:
20000 = 3200 * 1968 + b
20000 = 6273600 + b
b = -6253600
Therefore, the linear model for the growth in median price from 1968 to 1993 is:
p(t) = 3200t - 6253600
Now, let's move on to the exponential model. In an exponential model, the general form is:
p(t) = a * r^t, where p(t) is the median price at time t, a is the initial value (the median price at t = 0), r is the growth rate, and t is time.
Given the median price in 1968 (about $20,000) and 1993 (about $100,000), we can determine the growth rate (r):
100000 = 20000 * r^(1993-1968)
5 = r^25
Taking the 25th root of both sides, we find:
r ≈ 5^(1/25)
Now we substitute the initial value (a ≈ 20000) and the growth rate into the exponential model equation:
p(t) ≈ 20000 * (5^(1/25))^t
p(t) ≈ 20000 * 5^(t/25)
Therefore, the exponential model for the growth in median price from 1968 to 1993 is:
p(t) ≈ 20000 * 5^(t/25)
I hope this helps! Let me know if you have any further questions.