Problem 40 - Precalculus with Limits

Question

In Exercises 37-52, evaluate the function at each specified value of the independent variable and simplify. \(S(r) = 4\pi r^2\) (a) \(S(2)\) (b) \(S(\frac{1}{2})\) (c) \(S(3r)\)

Step-by-Step Solution

Answer

The solutions are: (a) \(16\pi\), (b) \(\pi\), (c) \(36\pi r^2 \)

Step 1: Evaluate \(S(2)\)

Substitute \(r = 2\) into the equation: \(S(2) = 4\pi (2)^2 = 4\pi * 4 = 16\pi\)

Step 2: Evaluate \(S(\frac{1}{2})\)

Substitute \(r = \frac{1}{2}\) into the equation: \(S(\frac{1}{2}) = 4\pi (\frac{1}{2})^2 = 4\pi * \frac{1}{4} = \pi\)

Step 3: Evaluate \(S(3r)\)

Substitute \(r = 3r\) into the equation: \(S(3r) = 4\pi (3r)^2 = 4\pi * 9r^2 = 36\pi r^2 \)

Key Concepts

Surface Area of a Sphere

The surface area of a sphere is a concept used frequently in geometry. A sphere is a three-dimensional, perfectly round shape, and its surface area can be calculated with the formula \(S(r) = 4\pi r^2\). Here, \(\pi\) is a constant approximately equal to 3.14159, and \(r\) is the radius of the sphere.

• The formula stems from integrating the circumference of a circle, which is \(2\pi r\), over the differentials of the sphere’s radius.
• This gives us the total surface covering a sphere, helping us understand its spatial dimensions.

Understanding how this formula works is vital for many real-life applications, like designing basketballs, planetary science, or calculating paint needed to cover a spherical object.
Recognizing the formula and knowing how to plug in the radius will allow you to quickly find the outer 'skin' measurement of any sphere, whether tiny like a marble or immense like a planet.

Algebraic Manipulation

Algebraic manipulation is a fundamental skill used to modify mathematical equations or expressions for simplification. When working with formulas like the surface area of a sphere, understanding algebraic manipulation makes the computation easier and more intuitive.

• Start by substituting known values into expressions.
• Simplify the expression by performing operations like squaring the radius, as seen in the formula \((r^2)\), or multiplying coefficients together.
• Ensure all terms are simplified to their most basic form, such as reducing fractions and combining like terms.

These skills are not just for math problems but are useful in solving everyday problems that involve calculation. By rearranging and simplifying, complex problems become understandable. For instance, the function \(S(r) = 4\pi r^2\) might initially seem a bit complex, but by substituting values precisely and performing operations, it becomes straightforward.

Substitution Method

The substitution method is an essential technique often used in algebra to solve equations and evaluate expressions. It's particularly handy when working with functions, like when determining the surface area of a sphere for different values \(r\).
This method involves replacing a variable with a particular number or another expression to make the equation easier to solve.

• First, identify the variable to substitute, as seen in the problems: substituting \(r = 2\), \(r = \frac{1}{2}\), and \(r = 3r\).
• Carefully replace the variable in the equation with the new value or expression.
• Perform the necessary mathematical operations to simplify the result.

By substituting directly and simplifying, we efficiently evaluate the function for any given input. This method streamlines solving equations that might otherwise be cumbersome to tackle, providing a systematic path to the correct answer.