Question

A medical clinic is reducing the number of incoming patients by giving vaccines before flu season. During week 5 of flu season, the clinic saw 90 patients. In week 10 of flu season, the clinic saw 60 patients. Assume the reduction in the number of patients each week is linear. Write an equation in function form to show the number of patients seen each week at the clinic.

Answer 1

Answer: -6x+120

Step-by-step explanation: To write an equation in function form that represents the number of patients seen each week at the clinic, we need to consider the given information. We are told that the reduction in the number of patients each week is linear. Let's use the given information to find the slope (rate of change) and the y-intercept of the linear equation. The clinic saw 90 patients in week 5 and 60 patients in week 10. To find the slope, we can use the formula: slope = (y2 - y1) / (x2 - x1) Using the points (5, 90) and (10, 60), we can calculate the slope: slope = (60 - 90) / (10 - 5) = -30 / 5 = -6 Now that we have the slope, we need to find the y-intercept. We can choose any point on the line to do this. Let's use the point (5, 90). The equation of a line in slope-intercept form is given by: y = mx + b Where m is the slope and b is the y-intercept. Substituting the values we have: 90 = -6(5) + b Simplifying: 90 = -30 + b Adding 30 to both sides: 90 + 30 = b b = 120 Now we have the slope (-6) and the y-intercept (120). Substituting these values into the equation of a line, we get: f(x) = -6x + 120 So, the correct equation in function form to show the number of patients seen each week at the clinic is: f(x) = -6x + 120

Answer 2

`f(x) = -6x +120`

Explanation:

Let's call y the number of patients treated each week

Let's call x the week number.

If the reduction in the number of patients each week is linear then the equation that models this situation will have the following form:

`y = mx + b`

Where m is the slope of the equation and b is the intercept with the x-axis.

If we know two points on the line then we can find the values of m and b.

We know that During week 5 of flu season, the clinic saw 90 patients, then we have the point:

(5, 90)

We know that In week 10 of flu season, the clinic saw 60 patients, then we have the point:

(10, 60).

Then we can find m and b using the followings formulas:

`m=(y_2-y_1)/(x_2-x_1)` and
`b=y_1-mx_1`

In this case:
`(x_1, y_1) = (5, 90)` and
`(x_2, y_2) = (10, 60)`

Then:

`m=(60-90)/(10-5)`

`m=-6`

And

`b=90-(-6)(5)`

`b=120`

Finally the function that shows the number of patients seen each week at the clinic is:

`f(x) = -6x +120`