Chapter 5.6, Problem 26PPE

To determine

To frame: an equation of a bike path which is parallel to the given main street and passing through park entrance.

Answer to Problem 26PPE

The required equation of bike path is, y = 2x + 4

Explanation of Solution

Given Information

The given graph represents a main street and a bike path needs to be constructed, which will be parallel to the main street.

So, firstly, we will have to find out the equation of the main street from the given graph.

Then, we will set up an equation, parallel to the main street and passing through the co-ordinates of the park entrance.

Formula Used

y = mx + C

Where, ‘ x’ and ‘ y’ are variables and ‘ m’ is the slope of the equation, ‘ c’ being the y intercept respectively.

And Slope = $\frac{Dy}{Dx}=\frac{\text{change in }y}{\text{change in }x}$

Graphical Approach

  

Calculation

Finding equation of main street:

It passes through, $\left(\frac{3}{2},-3\right)$

So, y -intercept = −3

Slope = $\frac{Dy}{Dx}=\frac{\text{change in }y}{\text{change in }x}$

  $=\frac{3}{\frac{3}{2}}=2$

So, Slope = m = 2

Equation of main street ⇒ y = mx + C

  ⇒y = 2x − 3

Now, the equation of bike path will be parallel to main street,

So, slope of bike path will also be, m = 2

Conclusion

Now, we are required to frame the equation of bike path

Hence, equation of bike path,

y = mx + C ; where m = 2

⇒y = 2x + C

But we know this line will pass through park entrance (0, 4)

Putting, (0, 4) in y = 2x + C

we get ⇒ 4 = 2 × 0 + C ( C = 4

Therefore, the required equation of bike path is, y = 2x + 4