Problem 82 - A Graphical Approach to Precalculus with Limits

Question

Braking Distance See Exercise \(81 .\) If a car is traveling 50 mph downhill, then its braking distance on wet pavement is given by $$ D(x)=\frac{2500}{30(0.3+x)} $$ where \(x<0\) for a downhill grade. (a) Evaluate \(D(-0.1)\) and interpret the result. (b) What happens to braking distance as the downhill grade becomes steeper? (c) The graph of \(D\) has a vertical asymptote at \(x=-0.3\) Give the physical significance of this asymptote. (d) Estimate the grade associated with a braking distance of 350 feet.

Step-by-Step Solution

Answer

(a) 416.67 feet; longer distance on steeper slopes; (c) grade is too steep to brake; (d) grade is approximately -0.062.

Step 1: Evaluating D(-0.1)

To find \( D(-0.1) \), substitute \( x = -0.1 \) into the formula.\[ D(-0.1) = \frac{2500}{30(0.3 + (-0.1))} = \frac{2500}{30 \times 0.2} = \frac{2500}{6} = 416.67 \text{ feet}\] This calculation shows that the braking distance on a \(-0.1\) downhill grade is approximately 416.67 feet.

Step 2: Investigating the effect of downhill grade

Examine \( D(x) = \frac{2500}{30(0.3 + x)} \) as \( x \) decreases further (becomes more negative). Since the denominator \( 30(0.3 + x) \) decreases, \( D(x) \) increases. Thus, the braking distance becomes longer as the downhill grade becomes steeper.

Step 3: Understanding the vertical asymptote

The vertical asymptote at \( x = -0.3 \) occurs because \( D(x) \) becomes undefined (denominator zero) at this point. Physically, this means a grade of \(-0.3\) is too steep to safely brake, since the calculated distance approaches infinity, indicating a potential risk for the inability to stop the car effectively.

Step 4: Estimating the grade for a given braking distance

We need to find \( x \) for which \( D(x) = 350 \). Set up the equation:\[ 350 = \frac{2500}{30(0.3 + x)} \] Solve for \( x \):\[ 350 \times 30(0.3 + x) = 2500 \] \[ 10500(0.3 + x) = 2500 \] \[ 3150 + 10500x = 2500 \] \[ 10500x = 2500 - 3150 \] \[ 10500x = -650 \] \[ x = \frac{-650}{10500} = -0.062 \] Therefore, the grade associated with a braking distance of 350 feet is approximately \(-0.062\).

Key Concepts

Downhill Grade and its Effects on Braking Distance

When considering driving on a downhill slope, the angle or steepness of this slope is crucial for braking calculations. This is referred to as the downhill grade. In this scenario, `x` represents the grade, with negative values indicating a descent. A steeper downhill grade (or a more negative `x`) results in a longer braking distance.

• Braking distance increases as the steepness of the slope increases.
• A greater slope requires a longer distance to stop the vehicle, especially on wet pavement.

This occurs because gravity assists the car in maintaining its speed or even increasing it, which demands a greater force to come to a stop.

Understanding Vertical Asymptote in Braking Distance Function

The concept of a vertical asymptote arises when a function approaches infinity, implying a dramatic change or practical impossibility. In the braking distance formula \[ D(x) = \frac{2500}{30(0.3 + x)} \]we notice a vertical asymptote at \(x = -0.3\). This occurs because the denominator becomes zero, making the function's output infinite:

• Mathematically, the function becomes undefined at \(-0.3\).
• Physically, this points out a condition where the car cannot brake effectively.

This result underscores that a grade of \(-0.3\) is unmanageable, as the vehicle would theoretically need an infinite distance to stop safely.

Evaluating the Braking Distance Function

Function evaluation plays a crucial role in understanding how different grades affect braking distance. To evaluate the function at a specific grade, substitute the value of \(x\) into \(D(x)\). For example, evaluating at \(x = -0.1\) gives:\[D(-0.1) = \frac{2500}{30(0.3 + (-0.1))} = \frac{2500}{6} = 416.67 \text{ feet}\]This calculation indicates that for a \(-0.1\) downhill grade, the braking distance is approximately 416.67 feet.

• Monitor the safety implications of these results, as longer distances can signal increased risks.
• Know that precise function evaluation ensures effective decision-making based on different scenarios.

Graphical Interpretation and Estimation of Grades

Graphical interpretation involves understanding the function behavior visually, which helps to estimate outcomes such as braking distances for specific grades. By interpreting the graph of \(D(x)\), you can determine the grade that corresponds to a particular distance. For instance, to find the grade where the braking distance is 350 feet, solve:\[ 350 = \frac{2500}{30(0.3 + x)} \]This equation simplifies to \(x \approx -0.062\), which means:

• The grade is approximately \(-0.062\) for a braking distance of 350 feet.
• Graphically analyzing this helps visualize how the distance changes with varying slopes.

Graphical interpretation not only aids in quick estimations but also in comprehending the underlying math concepts more intuitively.