Problem 80
Question
The area of a square is given by \(A(s)=s^{2}\) where \(s\) is the length of a side in inches. Compute the expression for \(A(2 s)\) and explain what it represents.
Step-by-Step Solution
Verified Answer
The expression for \(A(2s)\) is \(4s^{2}\). This represents the area of a square whose side length is twice that of another square, resulting in an area four times larger.
1Step 1: Substitute s with 2s in the area equation
The area of a square with side length \(2s\) is given by \(A(2s) = (2s)^{2}\)
2Step 2: Simplify the expression
Simplify the right side of the function equation. Remember that \((2s)^{2}\) equals \(2^{2} \cdot s^{2} = 4s^{2}\). Therefore, \(A(2s) = 4s^{2}\)
3Step 3: Explain the meaning of the expression
So, the expression \(A(2s) = 4s^{2}\) means that the area of a square whose side length is twice that of another square is four times the area of the smaller square. This is because squaring the factor by which the side length is increased also squares the area.
Key Concepts
Algebraic ExpressionsSimplifying ExpressionsGeometry Concepts
Algebraic Expressions
Understanding algebraic expressions is foundational in solving geometric problems. An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \( s \)), and operations (such as addition and multiplication). In the context of the exercise, \( A(s) = s^2 \) is an algebraic expression where \( s \) represents the side length of a square, and the expression itself represents the area of that square.
When we encounter \( A(2s) \) in the problem, it is telling us to evaluate the area expression \( A(s) \) by replacing \( s \) with \( 2s \) - which means we are looking at a square that has a side length that is twice as long. As expressions represent quantities, it’s crucial to understand how changing the variables will affect the whole expression, which, in this case, impacts the area calculation.
When we encounter \( A(2s) \) in the problem, it is telling us to evaluate the area expression \( A(s) \) by replacing \( s \) with \( 2s \) - which means we are looking at a square that has a side length that is twice as long. As expressions represent quantities, it’s crucial to understand how changing the variables will affect the whole expression, which, in this case, impacts the area calculation.
Simplifying Expressions
Simplifying expressions is a vital skill that reduces an expression to its simplest form, making it easier to understand and work with. Simplification often involves applying the rules of exponents and carrying out the operations.
In our square area example, the expression \( A(2s) = (2s)^2 \) is simplified by squaring both the numeric factor \(2\) and the variable \(s\). It’s important to recognize that \( (2s)^2 \) is not the same as \( 2s^2 \), because the exponent applies to both the coefficient \(2\) and the variable \(s\). Thus, the expression becomes \( 4s^2 \) after applying the rule that \( (ab)^2 = a^2b^2 \), where \(a\) and \(b\) are any numbers or variables. This simplification step is crucial in understanding the relationship between the side length of the square and its area.
In our square area example, the expression \( A(2s) = (2s)^2 \) is simplified by squaring both the numeric factor \(2\) and the variable \(s\). It’s important to recognize that \( (2s)^2 \) is not the same as \( 2s^2 \), because the exponent applies to both the coefficient \(2\) and the variable \(s\). Thus, the expression becomes \( 4s^2 \) after applying the rule that \( (ab)^2 = a^2b^2 \), where \(a\) and \(b\) are any numbers or variables. This simplification step is crucial in understanding the relationship between the side length of the square and its area.
Geometry Concepts
Geometry concepts often involve understanding and applying formulas to compute areas, volumes, and other properties of shapes. In this case, we’re focused on the geometry of squares. The key geometric concept demonstrated here is that the area of a square (represented by the algebraic expression \( A(s) = s^2 \)) is dependent on the square of its side length.
This geometric relationship implies that if the side length of the square is doubled (\(s\) is replaced by \(2s\)), the area is not merely doubled but quadrupled. This is because the area of a square is proportional to the square of its side length (not directly to the side length). As a consequence, any change in the side length has a squared effect on the area, which is a fundamental concept in geometry that applies to all squares regardless of their size.
This geometric relationship implies that if the side length of the square is doubled (\(s\) is replaced by \(2s\)), the area is not merely doubled but quadrupled. This is because the area of a square is proportional to the square of its side length (not directly to the side length). As a consequence, any change in the side length has a squared effect on the area, which is a fundamental concept in geometry that applies to all squares regardless of their size.
Other exercises in this chapter
Problem 80
The height of a ball after being dropped from the roof of a 200 -foot-tall building is given by \(h(t)=-16 t^{2}+200,\) where \(t\) is the time in seconds since
View solution Problem 80
A chartered bus company has the following price structure. A single bus ticket costs \(\$ 30 .\) For each additional ticket sold to a group of travelers, the pr
View solution Problem 80
In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=|x| ; g(x)=\f
View solution Problem 80
Use the following definition. A complex number \(a+b i\) is often denoted by the letter \(z .\) Its conjugate, \(a-b i\) is denoted by \(\bar{z}\). Show that th
View solution