Question

PROBLEM SOLVING The range (in miles) of a broadcast signal from a radio tower is bounded by a circle given by the equation $$ x^2+y^2=1620 $$ A straight highway can be modeled by the equation $$ y=-\frac{1}{3} x+30 \text {. } $$ For what lengths of the highway are cars able to receive the broadcast signal?

Short answer

The broadcast signal is received on the portion of the highway that corresponds to the absolute difference between the two \(x\)-values calculated in Step 3.

Step-by-step solution

Set up the equations

Given that the circle and the line cross at the points where the broadcast signal starts and ceases to be received, one should substitute \(y\) from the equation of the line into the equation of the circle to find the \(x\) values. So the equation \(x^2+y^2=1620\) is replaced with \(x^2+( -\frac{1}{3}x + 30 )^2 = 1620\)

Simplify and find the x values

Next, expand the brackets and simplify to form a quadratic equation. Then, solve the quadratic equation to find the \(x\) values. The equation simplifies to \(\frac{10}{9}x^2 - 20x + 900 = 1620\), further simplifying to \(\frac{10}{9}x^2 - 20x - 720 = 0\). Solving this equation will give two \(x\) values which represent the distances on the highway where the signal starts and ends.

Solve the quadratic equation

Use the quadratic formula \(x=\frac{-b ± \sqrt{b^2 - 4ac}}{2a}\) to find the \(x\) values. For our equation, \(a= \frac{10}{9}\), \(b=-20\), and \(c=-720\). Calculating these, we get two \(x\) values.

Calculate the Distance

The lengths of the highway that will receive the signal will be the absolute difference between the two \(x\) values (since \(x\) is distance and must be positive). To calculate this, we subtract the smaller \(x\) value from the larger one.

Key concepts

Systems of Equations

In real life, issues often involve more than one variable and are represented by systems of equations. Essentially, a system of equations is a set of two or more equations that share two or more unknowns. To find the point of intersection between a circle, which represents the range of a signal, and a straight highway, we need to use a system of equations. One represents the circle, and the other the linear path of the highway.

In the exercise, cars can receive the broadcast signal where the linear equation of the highway intersects with the circular broadcast range. Understanding how to combine these equations and solve them simultaneously is crucial. Substituting values from one equation into the other helps us find the precise points of intersection. This method of substitution is especially useful when one equation is linear (like the highway) and the other is quadratic (like the broadcast signal's range).

Quadratic Formula

When a system of equations results in a quadratic equation, a common method for finding the solution is using the quadratic formula. This formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is derived from the standard form of a quadratic equation, \(ax^2 + bx + c = 0\). It provides the roots of the equation, which represent the x-values for the points of intersection in our problem.

By inputting the coefficients (a, b, and c) from our simplified quadratic equation into the formula, we receive two solutions. These solutions correspond to the distances on the highway where the signal reaches the cars. It's an essential tool for cracking problems involving curves and allows us to find the precise points where roads—or, indeed, any linear paths—cross those curves.

Quadratic Functions

A quadratic function takes the form \(f(x) = ax^2 + bx + c\), and its graph is a parabola. In our exercise, the circle equation and its transformation into a quadratic equation is a practical example of how quadratic functions model real-world scenarios.

The parabolic nature of the function is significant as it implies that the broadcast signal has a maximum range and then diminishes. Changes in the quadratic function's coefficients will alter the parabola's shape and, therefore, the range and shape of the broadcast signal. In analyzing the quadratic function derived from the system, we can discern the effects of these coefficients on the distance over which the signal is received on the highway. Understanding quadratic functions is critical for predicting and optimizing outcomes in countless real-world applications, such as signal broadcasting, projectile motion, and even economics.