Question

Use the cross section of the radio telescope dish shown below. The cross section of the telescope's dish can be modeled by the polynomial function $$ y=\frac{167}{500^{2}}(x+500)(x-500) $$ where \(x\) and \(y\) are measured in feet, and the center of the dish is where \(x=0\) Use the model to find the coordinates of the center of the dish.

Short answer

The x-coordinate of the center of the dish is 0. The y-coordinate is found by substituting \(x=0\) into the given polynomial equation.

Step-by-step solution

Identify the x-coordinate of the center

The question already gives us that the center of the dish is where \(x=0\). Hence, the x-coordinate of the center is 0.

Find the y-coordinate of the center

The y-coordinate of the center can be found by substituting the value of \(x=0\) into the given equation of the function. This will provide us with the corresponding y value. Let's do the substitution: \[ y = \frac{167}{500^{2}}(0+500)(0-500) \]

Compute the y-coordinate

After substituting the values, the resulting computation will be \[ y = \frac{167}{500^{2}} * 500 * -500\]. After performing the computation, this will give us the y-coordinate of the center of the dish.

Key concepts

Coordinate System

The coordinate system is a fundamental framework used for plotting points, lines, and even more complex shapes like our polynomial function representing a radio telescope dish.

Think of the coordinate system as a map of the x and y axes, which intersect at a point called the origin, designated as (0,0). Every point on this map can be identified using a pair of numbers, the x-coordinate and the y-coordinate, which signify horizontal and vertical positions, respectively. In our exercise, the model provided defines the relationship between horizontal (x) and vertical (y) distances for various points on the curve of the radio telescope dish. Understanding this system is crucial as it helps us in locating the center of the dish, which, for most objects of symmetry, lies at the origin, where both coordinates are zero.

Substitution Method

The substitution method is a powerful tool frequently employed to solve equations or systems of equations and to find particular values for functions. By replacing a variable with its equivalent value, we simplify complex expressions and, as a result, find the solutions we're looking for.

In the context of our telescope dish problem, we used substitution to find the y-coordinate of the center. We substituted the x-value of 0 into the function, simplifying it greatly. The process follows a straightforward formula: locate the variable in your equation, replace it with the given value, and simplify. This method is integral to mathematics and plays a vital role in solving a variety of problems beyond our exercise, including those that involve quadratic equations.

Quadratic Equations

Quadratic equations form a central part of algebra and are recognizable by their highest power of x being a square. The standard form of a quadratic equation is given as \( ax^2 + bx + c = 0 \), where \( a \) cannot be zero.

Our radio telescope's cross-section is represented by a polynomial that resembles a quadratic equation once expanded. When solving quadratic equations, various methods including factoring, completing the square, and the quadratic formula are used. While we didn't need to solve the quadratic equation to find the y-coordinate at the center of the dish, understanding these equations is essential when working with polynomial functions. The quadratic aspect comes into play when investigating the function further, such as finding x-intercepts or the maximum height of the dish.