Final answer:
The equation for the national healthcare expenditures in the United States, H (in billions of dollars), t years since 2000, assuming a constant growth rate based on the expenditures for 2017 and projected for 2024 is H = 261.21t - 846.00.
Step-by-step explanation:
To write an equation for the national healthcare expenditures in the United States, H (in billions of dollars), t years since 2000, with the given data for 2017 and 2024, we use the two points (17, 3584.6) and (24, 5413.1), representing the years and expenditures respectively.
First we find the slope of the line connecting these points which represents the rate of growth per year:
Slope, m = (H2 - H1) / (t2 - t1) = (5413.1 - 3584.6) / (24 - 17) = 1828.5 / 7 = 261.21429...
However, we need to round the slope to two decimal places:
Rounded slope = 261.21
Next, we find the y-intercept by using the slope and one of the points:
H = mt + b
3584.6 = 261.21(17) + b
3584.6 = 4430.57 + b
b = -845.97
Round b to two decimal places we get:
b = -845.97 ≈ -846.00
Now the equation of the line is:
H = 261.21t - 846.00
Since we need the values with respect to the year 2000, we'll convert our slope back to billions (divide by 1000) and adjust the intercept accordingly:
H = 0.261t - 0.846 (in trillions)
To convert to billions, multiply the values by 1000:
H = 261.21t - 846.00 (in billions)
This equation represents the national healthcare expenditures growing at a constant rate from the year 2000.