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Chapter 1: Problem 63

Geometry Write the area \(A\) of a square as a function of its perimeter \(P .\)

Short Answer

Expert verified
The area \(A\) of a square as a function of its perimeter \(P\) is \(A = P^2/16\).

Step by step solution

01

Express the side length in terms of the perimeter

Let the length of one side be \(s\). Since a square has four equal sides, and its perimeter \(P\) is obtained by adding up all its sides, \(P = 4s\). Solving the equation \(P = 4s\) for \(s\), we can express the length of the side in terms of the perimeter: \(s = P/4\).
02

Express the area in terms of the side length

The area \(A\) of the square can be obtained by squaring the length of the side, i.e. \(A = s^2\). Substituting the value of \(s\) from Step 1, we get \(A = (P/4)^2\).
03

Simplify the expression

Squaring the right side of the equation from Step 2, we get \(A = P^2/16\). This is the area of the square expressed as a function of its perimeter.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perimeter of a Square
The perimeter of a square is the total distance around the outside of the square. Since all four sides of a square are equal, the perimeter can be calculated simply by multiplying one side's length by four. Here鈥檚 the formula you can use:
  • Perimeter ( P ): P = 4s , where {s} is the length of one side of the square.
This means if you know the perimeter of the square, you can follow simple steps to find the side length:
  • Take the perimeter value and divide it by 4 to find {s} .
Understanding how to calculate the perimeter is crucial in many areas of geometry, especially when dealing with square areas.
Function of Perimeter
A function describes how an output depends on the inputs. When talking about a square, if you know the perimeter, you can determine other properties like the side length or area. For example:
  • From P = 4s , solve for {s} to find: {s = P/4} .
Once you have the side length {s} , you can quickly find the area {A} using:
  • A = s^2 .
But, if you're interested in how area changes as a function of perimeter, substitute {s = P/4} into the formula for area:
  • A = (P/4)^2
  • Simplified, you get A = P^2/16 .
This shows how the area {A} relates to perimeter {P} directly, only using functions of {P} without needing intermediate calculations.
Geometry Formula
In geometry, formulas provide a way to solve problems using known relationships between figures' properties. For squares, we have special formulas to find lengths, perimeter, and areas that are directly connected. The key formula covered in this exercise ties the area of a square to its perimeter. It gives us a new insight:
  • The relationship is given by A = P^2/16 .
This formula tells us that:
  • As the perimeter increases, the area grows proportionally to the square of the perimeter.
  • It's essential to understand these formulas as they allow us to solve complex problems by evaluating simple changes in geometry.
Learning to express one property as a function of another helps deepen comprehension and problem solving in mathematical contexts.

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Most popular questions from this chapter

True or False? In Exercises \(71-74\) , determine whether the statement is true or false. Justify your answer. Predicting Graphical Relationships Use a graphing utility to graph \(f, g\) , and \(h\) in the same viewing window. Before looking at the graphs, try to predict how the graphs of \(g\) and \(h\) relate to the graph of \(f .\) (a) $$f(x)=x^{2}, \quad g(x)=(x-4)^{2}h(x)=(x-4)^{2}+3 h(x)=(x-4)^{2}+3$$ (b) $$f(x)=x^{2}, g(x)=(x+1)^{2} h(x)=(x+1)^{2}-2 $$ (c) $$f(x)=x^{2}, \quad g(x)=(x+4)^{2} h(x)=(x+4)^{2}+2$$

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Right Triangle Explain how you could use slope to show that the points \(A(-1,5), B(3,7),\) and \(C(5,3)\) are the vertices of a right triangle.
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