Q. 4.32 - Elementary Statistics

Question

Road Grade. The grade of a road is defined as the distance it rises (or falls) to the distance it runs horizontally, usually expressed as a percentage. Consider a road with a positive grade, $g$. Suppose that you begin driving on that road at an altitude $a_0$

a. Find the linear equation that expresses the altitude, $a$, when you have driven a distance, $d$, along the road. (Hint: Draw a graph and apply the Pythagorean Theorem

b. Identify and interpret the $y$-intercept and slope of the linear equation in part (a).

c. Apply your results in parts (a) and (b) to a road with a $5\%$ grade and an initial altitude of $1 mile$. Express your answer for the slope to four decimal places.

d. For the road in part (c). what altitude will you reach after driving $10 miles$ along the road?

e. For the road in part (c), how far along the road must you drive to reach an altitude of $3 miles$?

Step-by-Step Solution

Answer

(a) The required linear equation is, $a=a_0+gd$

(b) The $y$-intercept is, $a_0$ and the slope is, $g$

(c) The required answer is, $a=1+0.05d$

(d) The required altitude is, $1.5 miles$

(e) The distance to be driven is, $40 miles$

Part (a) Step 1: Given information

We are given that, 

The road has a positive grade, $g$

And we begin driving at an altitude of $a_0$

Part (a) Step 2: Simplify

We need to indicate altitude after driving distance $d$ on the road.

Grade $g$ represents a fraction of the height of each unit. When we drive a distance of $d$, the height thus rises by $gd$.

However, the first height is $a_0$ so the total height after the rise is $a_0+gd$.

Hence the equation is, $a=a_0+gd$

Here, $a$ is the altitude after covering some distance

Part (b) Step 1: Given information

We need to find out the $y$-intercept and slope of the equation.

Part (b) Step 2: Simplify

From part (a)

The $y$-intercept is the constant in the equation i.e. $a_0$

And the slope is the coefficient of $d$, i.e., $g$

Part (c) Step 1: Given information

We need to find the value of altitude i.e. $a$

Part (c) Step 2: Simplify

As the values are, 

   $$\begin{gathered} a_0=1 \\ g=5\%=0.05 \end{gathered}$$

So, from part (a) and above values we get, 

      $a=a_0+gd=1+0.05d$

Part (d) Given information

We need to find the value  $a$ at a distance $d=10 miles$ and other conditions same as that of part (c)

Part (d) Step 2: simplify

From part (c) we get 

      $a=1+0.05d=1+\left(0.05\times 10\right)=1+0.5=1.5$

Hence the reached altitude is, $1.5 miles$

Part (e) Step 1: Given information

We need to find the distance drove.

Part (e) Step 2: Simplify

From part (c)

     $a=a_0+gd=1+0.05d$

As $a=3$

So, 

     $$\begin{gathered} 3=1+0.05d \\ 2=0.05d \\ d=40 \end{gathered}$$

Therefore, distance drove is $d=40 miles$