Question

Find a formula for the linear equation graphed below. p = f(h) =

Answer 1

The formula for the linear equation graphed is f(h) = 400h + 17000

How to determine the equation of the line

From the question, we have the following parameters that can be used in our computation:

The graph

A linear equation is represented as

p = f(h) = mh + c

Where

c = y when h = 0; i.e. the y-intercept

m = slope i.e. the rate of change

Also, we have the points on the graph

(5, 19000) and (10, 21000)

The slope (m) is calculated as

`m = (y_2 - y_1)/(x_2 - x_1)`

So, we have

`m = (21000 - 19000)/(10 - 5)`

`m = (2000)/(5)`

m = 400

Recall that

f(h) = mh + c

So, we have

f(h) = 400h + c

Using the point (10, 21000), we have

400 * 10 + c = 21000

4000 + c = 21000

Collect the like terms

c = 21000 - 4000

Evaluate

c = 17000

Recall that

f(h) = 400h + c

So, we have

f(h) = 400h + 17000

Hence, the equation is f(h) = 400h + 17000

Answer 2

Given

Points (5,190000), (10,210000) and (20,250000)

Solve for the slope of the line

To solve for the slope of the line, recall the formula

`\begin{gathered} m=(y_2-y_1)/(x_2-x_1) \\ \text{where} \\ (x_1,y_1)\text{ and }(x_2,y_2)\text{ are any two points in the graph} \end{gathered}`

For this case we will use (5,190000) and (10,210000)

`\begin{gathered} (x_1,y_1)=\mleft(5,190000\mright) \\ (x_2,y_2)=\mleft(10,210000\mright) \\ \\ \text{Substitute to the slope formula} \\ m=(y_2-y_1)/(x_2-x_1) \\ m=(210000-190000)/(10-5) \\ m=(20000)/(5) \\ m=4000 \end{gathered}`

Now that we have m = 4,000, we will use this to solve for the y-intercept.

Recall the slope-intercept form of a linear equation

`\begin{gathered} y=mx+b \\ \text{where} \\ m\text{ is the slope} \\ b\text{ is the y-intercept} \end{gathered}`

We will use point (10,210000), in this case

`\begin{gathered} \text{Substitute the following} \\ (x,y)=\mleft(10,210000\mright) \\ m=4000 \\ \\ \text{THEN} \\ y=mx+b \\ 210000=40000+b \\ 210000=40000+b \\ 250000-40000=b \\ b=170000 \end{gathered}`

Summarizing everything, the equation of the line is

`p=f(h)=4000h+170000`