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 \(x=0\) Explain how to use the algebraic model to find the width of the dish.

Short answer

The width of the dish is 1000 feet.

Step-by-step solution

Understand the Polynomial Function

The polynomial function used to model the dish's cross section is \(y=\frac{167}{500^{2}}(x+500)(x-500)\). Here, \(x\) represents the horizontal distance from the center point of the dish, while \(y\) denotes the vertical distance on the surface of the dish at any given point \(x\).

Find the Width of the Dish

The width of the dish corresponds to the distance between the two points on the \(x\)-axis where the cross-section (represented by \(y\)) is equal to zero. So, solve the equation \(y=\frac{167}{500^{2}}(x+500)(x-500) = 0\) for \(x\). This gives two solutions, corresponding to the where the cross section meets the \(x\)-axis.

Solve for \(x\)

The equation \(y=0\) yields \(x = -500\) and \(x = 500\) when solved. These points represent the left and right edges of the dish viewed in the cross section.

Calculate the Width

To find the width of the dish, subtract the smaller \(x\)-coordinate from the larger. In this case, \(500 - (-500) = 1000\).

Key concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve a variable raised to whole-number exponents and multiplied by coefficients. In the context of our exercise, the polynomial function \(y=\frac{167}{500^{2}}(x+500)(x-500)\) models the shape of the radio telescope's dish. Here's an easy breakdown of its components:

- The highest exponent in a polynomial indicates its degree, which tells us about the shape of the graph it produces.
- In this function, the degree is 2 because the highest exponent on \(x\) is 2 (due to the multiplication of \(x+500\) and \(x-500\)), making it a quadratic function.
- The coefficients, in this case \(\frac{167}{500^2}\), influence the steepness and size of the parabola that represents the dish.
- The factors \(x+500\) and \(x-500\) help us to determine the width of the dish by setting \(y=0\) and solving for \(x\), which is a standard method of finding the roots of polynomial functions.

Understanding these functions is crucial because they are used to model real world phenomena ranging from physics to economics, and in this particular instance, the precise shape of a radio telescope dish.

Solving Equations

Solving equations is a fundamental part of algebra that involves finding the values for variables that make the equation true. There are different strategies depending on the type of the equation and its complexity:

- For polynomial equations, such as \(y=\frac{167}{500^{2}}(x+500)(x-500)\), the solution often involves factoring or using the quadratic formula.

• To factor, look for values that satisfy the equation set to zero.
• The quadratic formula can be used if factoring is complicated, but it isn't necessary here since the equation is already factored.

- Once the polynomial is factored, setting \(y=0\) and solving the remaining binomials for \(x\) will give you the roots of the equation, representing the points where the graph intersects the \(x\)-axis.
- If we replace \(y\) with zero in our given model, we effectively find the points on the \(x\)-axis that correspond to the edges of the telescope dish.

By mastering equation-solving techniques, students can solve a wide range of mathematical and applied science problems, providing them with critical tools for academic and professional success.

Horizontal and Vertical Distances

Horizontal and vertical distances are measurements across the \(x\)-axis and the \(y\)-axis, respectively, in a Cartesian coordinate system. These measurements are often used to describe positions and dimensions in a two-dimensional space:

- Horizontal distance in our exercise is the measure from the center of the dish \(x=0\) to its edge along the \(x\)-axis.
- Vertical distance would be the measurement from the \(x\)-axis, up or down, to a point on the graph of the function, representing the depth of the dish at any given horizontal point.
- To find the total width of the telescope's dish, we use the horizontal distances obtained by solving our polynomial equation for \(x\) when \(y=0\), which are the roots of the function.

• The distance from the left edge \(x=-500\) to the center \(x=0\), and from the center to the right edge \(x=500\) combine to give us the total width.
• By understanding this concept, we can apply it to problems involving the design and analysis of structures and the behavior of objects in physics.

The study of how these distances affect the dynamics of systems is a key aspect of disciplines like geometry, physics, and engineering.