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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 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 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.