Problem 64 - Algebra A Combined Function

Question

Police departments find it very useful to be able to approximate driving speeds in skidding accidents. If the road surface is wet concrete, the function \(S(x)=\sqrt{10.5 x}\) is used, where \(S(x)\) is the speed of the car in miles per hour and \(x\) is the distance skidded in feet. Find how fast a car was moving if it skidded 280 feet on wet concrete.

Step-by-Step Solution

Answer

The car was moving at approximately 54.22 miles per hour.

Step 1: Identify Given Information

In the problem, you are given the skid distance \(x = 280\) feet. The formula to calculate the speed \(S(x)\) on wet concrete is \(S(x) = \sqrt{10.5 \times x}\). We need to find the speed \(S(x)\).

Step 2: Plug the Given Value into the Formula

Replace \(x\) in the formula \(S(x) = \sqrt{10.5 \times x}\) with the given skid distance, which is 280 feet. This gives us \(S(280) = \sqrt{10.5 \times 280}\).

Step 3: Calculate the Expression Inside the Square Root

First, calculate the product inside the square root: \(10.5 \times 280\). This results in 2940. So, we have \(S(280) = \sqrt{2940}\).

Step 4: Calculate the Square Root

Find the square root of 2940. Using a calculator, \(\sqrt{2940} \approx 54.22\).

Step 5: State the Final Answer

The calculated speed is approximately 54.22 miles per hour. This is how fast the car was moving when it skidded 280 feet on wet concrete.

Key Concepts

Quadratic Function

Quadratic functions are a vital concept in algebra, representing polynomial equations of degree two. These equations often take the form:

• \( f(x) = ax^2 + bx + c \)

The graph of a quadratic function is a parabola, which could open upwards or downwards, depending on the sign of the coefficient \( a \).Understanding quadratic functions helps us solve various real-world problems, including physics calculations like determining speeds from skidding distances.In our given problem, the relationship appears through a formula derived to approximate speed using distance and a constant.This function does not appear in the classical quadratic form but still employs one of the core characteristics, which involves squaring, evident in the use of the square root.

Square Root Calculation

The concept of square roots is crucial in many mathematical applications. Calculating a square root involves finding a number which, when multiplied by itself, results in the given number:

• The square root of a number \( y \) is written as \( \sqrt{y} \)

In our context, after computing the product \( 10.5 \times x \), which equals 2940, we need to take the square root to determine the speed \( S(x) \).Using a calculator, you find that \( \sqrt{2940} \approx 54.22 \). This step converts the skid distance into meaningful information, such as car speed. This presents a typical application of square root calculations in physics and engineering contexts.

Problem Solving Steps

Solving algebraic problems often involves a structured process. Here are the typical steps followed in the exercise, illustrated to find \( S(x) \):

• Step 1: Identify and note down the given information.
• Step 2: Plug these values into the formula you have at hand.
• Step 3: Compute any necessary intermediate expressions.
• Step 4: Simplify the expressions, such as calculating the square root.
• Step 5: Present the solution clearly.

This systematic approach not only simplifies solving complex equations but also checks your progress and ensures no vital steps are skipped.

Physics and Algebra

Combining physics and algebra allows us to solve practical problems efficiently. In physics, equations are often used to describe real-world phenomena. Algebra helps rearrange, simplify, and solve these equations. In our exercise, the blend enables the calculation of speed based on skid distance. Physics provides the context (e.g., acceleration, friction levels), while algebra does the heavy lifting with calculations:

• Using specific formulas like \( S(x) = \sqrt{10.5 \times x} \).
• Interpreting the results to make practical conclusions such as determining speed.

This synthesis allows learners to see beyond numbers and appreciate the dynamic relationships underlying many scientific laws.