Problem 81 - A Graphical Approach to College Algebra

Question

Solve each problem. The grade \(x\) of a hill is a measure of its steepness. For example, if a road rises 10 feet for every 100 feet of horizontal distance, then it has an uphill grade of \(x=\frac{10}{100},\) or \(10 \% .\) The braking (or stopping) distance \(D\) for a car traveling at 50 mph on a wet, uphill grade is given by $$D(x)=\frac{2500}{30(0.3+x)}$$ (GRAPH CANT COPY) (a) Evaluate \(D(0.05)\) and interpret the result. (b) Describe what happens to braking distance as the hill becomes steeper. (c) Estimate the grade associated with a braking distance of 220 feet.

Step-by-Step Solution

Answer

(a) 238.10 feet; (b) Braking distance decreases as hill gets steeper; (c) Grade is approximately 7.88%.

Step 1: Evaluate D(0.05)

Substitute \(x = 0.05\) into the function \(D(x) = \frac{2500}{30(0.3 + x)}\). We have:\[ D(0.05) = \frac{2500}{30(0.3 + 0.05)} = \frac{2500}{30(0.35)} = \frac{2500}{10.5} \approx 238.10 \text{ feet} \]The braking distance for a road with a 5% uphill grade is approximately 238.10 feet.

Step 2: Analyze Braking Distance as Hill Becomes Steeper

As the uphill grade \(x\) increases, the value \(0.3 + x\) in the denominator of \(D(x) = \frac{2500}{30(0.3 + x)}\) increases. Thus, \(D(x)\) will decrease, indicating that the braking distance becomes shorter as the hill becomes steeper.

Step 3: Estimate Grade for D = 220

Set \(D(x) = 220\) and solve for \(x\):\[220 = \frac{2500}{30(0.3 + x)}\]Multiply both sides by \(30(0.3 + x)\):\[220 \times 30(0.3 + x) = 2500\]Solving for \(x\):\[6600(0.3 + x) = 2500\]\[1980 + 6600x = 2500\]\[6600x = 2500 - 1980\]\[6600x = 520\]\[x = \frac{520}{6600} \approx 0.0788\]The grade associated with a braking distance of 220 feet is approximately 7.88%.

Key Concepts

Braking Distance

The braking distance is the stretch a vehicle travels from when the brakes are applied to when it fully stops. This is crucial to understand because in real-life situations, knowing the braking distance can prevent accidents.
In this context, the algebraic function given by \( D(x) = \frac{2500}{30(0.3 + x)} \) helps calculate the braking distance for different uphill grades. This means the steeper the hill or incline (captured by the variable \( x \)), the shorter the braking distance becomes.
It's important because if you are aware that a car traveling at 50 mph has a shorter stopping distance going uphill, drivers can adjust their speed and distance accordingly to drive safely.
When substituting \( x = 0.05 \), we find that the braking distance \( D \) is approximately 238.10 feet on a 5% uphill grade. This value helps drivers anticipate the space required to stop, particularly in wet conditions.

Hill Grade Calculation

Hill grade calculation is the process of determining the steepness of a hill. It is defined as the ratio of vertical rise to horizontal distance, often expressed as a percentage.
For example, a 10% grade means the road rises 10 feet for every 100 feet of horizontal distance. This concept is crucial, especially in transportation, as it affects vehicles' performance and safety.
To estimate a grade given a particular braking distance (for example, 220 feet), set the equation \( D(x) = 220 \) and solve for \( x \). During calculations:

• Multiply both sides by the factor in the denominator (\(30(0.3 + x)\)),
• Simplify and rearrange the resulting equation,
• Solve for \( x \) to find the hill grade.

In this case, the grade comes out to approximately 7.88%.
Understanding and being able to compute the hill grade helps in planning travel routes and designing safe and efficient roadways.

Function Interpretation

Function interpretation involves understanding how a mathematical function represents or describes real-world scenarios. Here, the function \( D(x) = \frac{2500}{30(0.3 + x)} \) describes how braking distance changes with a hill's grade.
This function tells us that:

• The braking distance \( D \) decreases as the grade \( x \) increases.
• At a certain point, having too steep a hill significantly shortens the stopping distance.

This interpretation offers meaningful insights into how physics and mathematics combine to affect real-world driving conditions.
By analyzing the behavior of this function, particularly how modifications in \( x \) affect \( D \), drivers and engineers can make informed decisions to improve road safety and vehicle handling on different inclines.