Problem 48 - Algebra 2

Question

Civil engineers are designing a section of road that is going to dip below sea level. The road’s curve can be modeled by the equation \(y=0.00005 x^{2}-0.06 x,\) where \(x\) is the horizontal distance in feet between the points where the road is at sea level and \(y\) is the elevation (a positive value being above sea level and a negative being below). The engineers want to put stop signs at the locations where the elevation of the road is equal to sea level. At what horizontal distances will they place the stop signs?

Step-by-Step Solution

Answer

Place stop signs at distances of 0 feet and 1200 feet.

Step 1: Set the Equation to Zero

Since the engineers need stop signs at points where the road meets sea level, we need to set the elevation equation to zero. Thus, we solve:\[ 0.00005x^2 - 0.06x = 0 \]

Step 2: Factor the Equation

Factor out the common term in the equation:\[ x(0.00005x - 0.06) = 0 \]

Step 3: Solve Each Factor

Set each factor of the equation to zero:1. \( x = 0 \) gives one solution.2. \( 0.00005x - 0.06 = 0 \) solve for \( x \).

Step 4: Solve the Second Factor

To solve \( 0.00005x - 0.06 = 0 \), add \( 0.06 \) to both sides:\[ 0.00005x = 0.06 \]Now, divide by \( 0.00005 \):\[ x = \frac{0.06}{0.00005} = 1200 \]

Step 5: Identify Horizontal Distances

From the solutions \( x = 0 \) and \( x = 1200 \), these are the horizontal distances where the road meets sea level. Hence, stop signs should be placed at these points.

Key Concepts

Factoring

Factoring is a key technique in solving quadratic equations. It allows us to break down complex equations into simpler ones that are easier to handle.
In the exercise, we have the equation \(0.00005x^2 - 0.06x = 0\). This is a quadratic equation, which takes the general form \(ax^2 + bx + c = 0\). Here, the goal is to get the equation in the form where it can be expressed as a product of two factors.
One approach is to factor out the greatest common factor from each term. We can see that \(x\) is a common factor. By factoring out \(x\) from each term in this equation, we get:

• \(x(0.00005x - 0.06) = 0\)

Factoring helps isolate the possible solutions by setting each factor to zero individually. This simplifies the process of finding where a quadratic equation equals zero, helping us solve for variable \(x\).

Roots of Equations

Finding the roots of an equation means determining the values of \(x\) for which the equation equals zero. These roots are where the graph of the equation crosses the x-axis.
In the given exercise, the roots are found after factoring the quadratic equation into two parts: \(x\) and \(0.00005x - 0.06\). Once we have factored the equation, we set each factor equal to zero to find the roots:

• \(x = 0\)
• \(0.00005x - 0.06 = 0\)

Solving each of these, we get:

• One root is directly \(x = 0\).
• The other root requires solving \(0.00005x - 0.06 = 0\).

To solve the second equation, we add \(0.06\) to both sides and divide by \(0.00005\), resulting in \(x = 1200\). These roots tell us at which points the road will meet the sea level.

Solving Equations

Solving equations is a systematic process to find unknown values that satisfy an equation's criteria. For quadratic equations, solving typically starts with setting the equation to zero, as we need to find when the equation equals sea level (elevation = 0 in this case).
Once set to zero, as in \(0.00005x^2 - 0.06x = 0\), the next step is factoring, which simplifies the problem by breaking it into more manageable pieces. After factoring, we solve each factor separately, as these represent potential solutions for the variable.
This sequential approach was applied in the exercise using:

• Factor the equation: \(x(0.00005x - 0.06) = 0\).
• Set each part to zero: \(x = 0\) and \(0.00005x - 0.06 = 0\).
• Discuss and derive the solutions: solve \(x = 0\) and rearrange \(0.00005x = 0.06\) to find \(x = 1200\).

This method shows how quadratic equations can be systematically addressed to find practical solutions, such as where to place stop signs on the road.