Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

If \(S(r)=4 \pi r^{2}\) and \(D(t)=2 t+5\), find \((S \circ D)(t)\).

Step-by-Step Solution

Answer

\((S \circ D)(t) = 16\pi t^2 + 80\pi t + 100\pi\).

Step 1: Understand Function Composition

Function composition involves plugging one function into another. In this case, we need to replace the variable in function \( S(r) \) with the entire function \( D(t) \). This means we will be substituting \( r \) in \( S(r) \) with \( D(t) \).

Step 2: Substitute \( D(t) \) into \( S(r) \)

To find \((S \circ D)(t)\), we take \(D(t) = 2t + 5\) and substitute it into \(S(r) = 4 \pi r^2\). Thus, \(S(D(t)) = 4 \pi (D(t))^2 = 4 \pi (2t + 5)^2\).

Step 3: Expand the Expression

Now, let's expand \((2t + 5)^2\). We use the formula \((a+b)^2 = a^2 + 2ab + b^2\), where \(a = 2t \) and \(b = 5\). Thus, \((2t+5)^2 = (2t)^2 + 2 \cdot 2t \cdot 5 + 5^2\).

Step 4: Simplify the Expansion

Calculate each part of the expansion: \((2t)^2 = 4t^2\), \(2 \cdot 2t \cdot 5 = 20t\), and \(5^2 = 25\). Combine these to get \(4t^2 + 20t + 25\).

Step 5: Multiply by \(4\pi\)

Substitute the expanded expression back into \(S(D(t))\): \(S(D(t)) = 4 \pi (4t^2 + 20t + 25)\). Distribute \(4\pi\) through the expression to get \(16\pi t^2 + 80\pi t + 100\pi\).

Step 6: Write the Final Expression

The composition \((S \circ D)(t)\) is \(16\pi t^2 + 80\pi t + 100\pi\). This is the simplified form obtained by substituting \(D(t)\) into \(S(r)\).

Key Concepts

Area of a Sphere

The area of a sphere is an essential concept in geometry, especially if you want to understand different properties of spheres. The formula for the surface area of a sphere is given by \( S(r) = 4 \pi r^2 \), where \( r \) is the radius of the sphere.
This formula tells us the complete area covering the sphere, like wrapping a tennis ball with your hand.

• The symbol \( \pi \) represents the constant pi, approximately 3.14159.
• The \( r^2 \) indicates that the area is a function of the square of the radius.

Understanding this can help in various areas, such as physics and real-world applications where you need to measure round objects. Spheres can be found everywhere, from planets to balls. When working with the area, remember it increases with the square of the radius, meaning even small increases in radius can lead to larger increases in area.

Polynomial Expansion

When dealing with functions that involve terms like \((x + y)^n\), polynomial expansion becomes useful. A polynomial expansion turns a product of factors into a sum of terms by applying algebraic identities, such as the binomial theorem.For example, the expression \((2t + 5)^2\) involves expanding a binomial. To expand:\- Apply the formula, \((a+b)^2 = a^2 + 2ab + b^2\), where \( a = 2t \) and \( b = 5 \).- Calculate each part separately to ensure no mistakes.The steps lead to \((2t+5)^2 = 4t^2 + 20t + 25\). This process is known as polynomial expansion and allows you to break complex expressions into simpler components when solving them. Polynomial expansion is vital in algebra, calculus, and even physics scenarios where predicting outcomes with precision is necessary.

Substitution in Functions

Substitution in functions is a fundamental tool in algebra and calculus. It involves replacing a variable in a function with another expression or another function. This is common in function composition, where you substitute one function into another.In our example, substituting \( D(t) = 2t + 5 \) into \( S(r) = 4 \pi r^2 \) means you replace the variable \( r \) in \( S(r) \) with \( 2t + 5 \), making it \( S(D(t)) = 4 \pi (2t + 5)^2 \).

• Always pay attention to what expression is replacing the variable.
• More complex expressions require careful handling of terms, especially negative signs or fractions.

Substitution is a step towards transforming and simplifying expression evaluations.
It can empower deeper understanding and reach solutions that are not immediately visible with original expressions.

Algebraic Simplification

Algebraic simplification is the process of making an expression easier to understand and work with. During simplification, you often expand terms, combine like terms, and reduce expressions to a more manageable form.In our problem, after expanding the polynomial \( (2t+5)^2 \) to get \( 4t^2 + 20t + 25 \), you multiply by \( 4\pi \). \The expression becomes \( 16\pi t^2 + 80\pi t + 100\pi \) after distributing the multiplication across each term.

• Always perform operations in the correct order: expand, simplify inside terms first, then multiply or add as required.
• Combining like terms helps to reduce complexity and highlight the expression's main components.

A simplified form allows better insight and shows the main contributing factors to the expression. This practice not only aids in solution finding but also boosts clarity in more complex problem-solving scenarios. Remember, simplification doesn’t change the expression’s value, it only makes it easier to work with.