Problem 54 - Algebra Form and Function

Question

A window is in the shape of a square with a semicircle on top. If the side of the square is \(l\) feet long then the area of the glass sheet in the window is $$ A=l^{2}+\frac{\pi}{2}\left(\frac{l}{2}\right)^{2} \mathrm{ft}^{2} $$ (a) Is \(A\) a power function of \(l\) ? If so, identify the coefficient \(k\) and the exponent \(p\). (b) Without computing the area, say whether the area is larger when \(l=1.5\) feet or when \(l=2.5\) feet. Explain your answer in algebraic terms.

Step-by-Step Solution

Answer

Answer: No, the given function is not a power function, as it combines a quadratic term and a term with square and constants. The area of the window is larger when l = 2.5 feet.

Step 1: Identify the power function

A power function is any function that can be represented in the form f(x) = k*x^n where k and n are constants. Looking at the given function A = l^2 + π/2*(l/2)^2, we can observe that it doesn't match the structure of a power function. The reason for it is that our function is a combination of a quadratic term (l^2) and a term that incorporates square and constants (π/2*(l/2)^2).

Step 2: Compare the function values

Without computing the values of l = 1.5 and l = 2.5, the comparison can be achieved by analyzing the function itself. From the function, A = l^2 + π/2*(l/2)^2, we can see the function is dependent on l^2. This implies that the function value (area of the window) increases as the value of l increases. Therefore, the area of the window will be larger when l = 2.5 compared to when l = 1.5 feet.

Key Concepts

Understanding Quadratic Terms

A quadratic term is a key component in many algebraic expressions. It involves the square of a variable, typically represented as \( ax^2 \). In the context of the given problem, \( l^2 \) is our quadratic term, where \( l \) is the side length of the square part of the window. This quadratic term contributes significantly to the total area of the window.

• The quadratic term grows quickly as \( l \) increases, because squaring a number amplifies its size.
• This is why an increase in \( l \) results in a much larger area; the \( l^2 \) term becomes dominant.

Recognizing a quadratic term in algebraic expressions is essential for solving many mathematical problems, as it dictates how the expression behaves as the variable changes.

Algebraic Expressions in Context

Algebraic expressions are combinations of numbers, variables, and operations that represent a particular value. In our problem, the expression \( A = l^2 + \frac{\pi}{2}\left(\frac{l}{2}\right)^{2} \) represents the total area of the window. Each part of the expression has its purpose:

• \( l^2 \) accounts for the area of the square section.
• \( \frac{\pi}{2}\left(\frac{l}{2}\right)^{2} \) represents the area of the semicircular top.

This expression is not a simple power function of \( l \), as it combines different forms: the quadratic term from the square and terms related to the geometry of the semicircle. Understanding such expressions involves recognizing each component and considering their combined effect. This step helps in assessing changes, like how the area shifts with different values of \( l \).

Calculating Area of Composite Shapes

A composite shape is a figure that is composed of a combination of simpler shapes. In this window problem, the shape combines a square and a semicircle. To find the area of such a shape, it is crucial to calculate each section's area and then sum them.

• The square's area is straightforward: \( l^2 \).
• For the semicircle, the radius is \( \frac{l}{2} \), so the area is \( \frac{\pi}{2}\left(\frac{l}{2}\right)^{2} \), which comes from halving the area of the full circle, \( \pi r^2 \).

To solve problems involving composite shapes, divide the figure into recognizable parts, calculate individual areas, and then combine these results. This method ensures a complete and accurate total area calculation.