Problem 47 - Algebra 1: Concepts and Skills

Question

The cross section of the telescope’s dish can be modeled by the polynomial function $$y=\frac{14}{41^{2}}(x+41)(x-41)$$ where \(x\) and \(y\) are measured in feet, and the center of the dish is at \(x=0\) Use the model to find the coordinates of the center of the dish.

Step-by-Step Solution

Answer

The coordinates of the center of the dish are (0, -14/41).

Step 1: Identify the function

The polynomial function representing the cross section of the telescope’s dish is \(y=\frac{14}{41^{2}}(x+41)(x-41)\)

Step 2: Substitute \(x = 0\)

We need to substitute \(x = 0\) into the function to find the value of \(y\). This gives us \(y=\frac{14}{41^{2}}(0+41)(0-41)\)

Step 3: Simplify the equation

On simplifying the equation, we get \(y= -\frac{14}{41}\)

Step 4: Write the coordinates

The coordinates of the center of the dish are thus \((0, -\frac{14}{41})\).

Key Concepts

Algebraic Modeling

Understanding algebraic modeling is crucial because it allows you to represent real-world problems with mathematical expressions and functions that can be analyzed and solved. In the context of the original exercise, the polynomial function
\(y=\frac{14}{41^{2}}(x+41)(x-41)\)
models the shape of a telescope's dish. The equation was derived from the physical dimensions and properties of the telescope dish, illustrating how algebra can be used to create a model that represents something tangible.

When creating an algebraic model, it’s important to identify what each variable represents. Here, \(x\) and \(y\) stand for the horizontal and vertical distances, respectively, from the center of the dish, which is marked by \(x = 0\). The model helps in predicting the dish's curvature or finding specific points on the dish, like its center, which is essential for focusing the telescope's lens.

Function Substitution

Function substitution is a technique used to evaluate functions for specific values of their variables. It’s a straightforward but powerful tool in algebra that facilitates the understanding of how changes in variable values affect the overall function. In the step by step solution above, function substitution is used in
Step 2: Substitute \(x = 0\)
to find the point where the cross-section of the dish meets its central axis. By substituting \(x\) with \(0\), you effectively measure the value of \(y\) at the center of the dish. Substitution is employed in various mathematical contexts, from solving equations to graphing functions, and is fundamental in understanding how to manipulate and work with functions.

Simplifying Equations

Why Simplify?

Simplifying equations is a key aspect of solving problems in algebra. The goal is to make equations easier to work with or to reveal more information about the solution or behavior of the function. In the context of the solution provided:
Step 3: Simplify the equation
simplifying the expression after function substitution highlights the y-value of the telescope dish's center. By simplifying an equation, we can reduce computation errors, make patterns more visible, and ultimately, arrive at more accessible, interpretable solutions. It's important to remember that simplifying equations can involve factoring, combining like terms, reducing fractions, and applying arithmetic operations efficiently.