Problem 23 - Algebra for College Students

Question

Find the value of the indicated variable. Round approximate answers to three decimal places. Find \(h\) if \(r=3, V=12 \pi,\) and \(r^{2}=\frac{V}{\pi h}\)

Step-by-Step Solution

Answer

The value of \( h \) is approximately 1.333.

Step 1: Identify and Write Down the Given Values

Given values are:\( r = 3 \ V = 12 \pi \)

Step 2: Write Down the Formula

The equation given is: \( r^2 = \frac{V}{\pi h} \)

Step 3: Substitute the Given Values into the Formula

Substitute \(r\) and \(V\) with the given values:\[ 3^2 = \frac{12 \pi}{\pi h} \]

Step 4: Simplify the Equation

First, simplify \( \pi \) in the denominator and numerator:\[ 9 = \frac{12}{h} \]

Step 5: Solve for \(h\)

Rearrange the equation to solve for \(h\): \[ h = \frac{12}{9} \]\[ h = \frac{4}{3} \approx 1.333 \]

Key Concepts

solving equations

Solving equations is the process of finding the value of a variable that makes an equation true. In the given exercise, we needed to find the value of the variable \(h\) using the given equation \(r^{2} = \frac{V}{\text{π} h}\). To solve for \(h\), we performed several steps:
We started by plugging in the provided values for \(r\) and \(V\). Then we simplified the equation to isolate \(h\) on one side. Finally, we performed basic arithmetic to solve for \(h\).
Remember, the goal is to manipulate the equation until the variable you are solving for is by itself on one side of the equation.

algebraic simplification

Algebraic simplification involves reducing an equation or expression to its simplest form. In this problem, we needed to simplify the equation \(3^{2} = \frac{12π}{π h}\) after substituting the given values.
Here's how we simplified it: First, we noticed the \(\text{π}\) terms in the numerator and denominator and canceled them out, resulting in \(3^{2} = \frac{12}{h}\).
Simplifying further, \(3^{2}\) is equal to \(9\), so we get \(9 = \frac{12}{h}\). This simplified form made it easier to isolate the variable \(h\) and find its value.
Simplifying equations can help you see the relationships between variables more clearly and make it easier to solve them.

substitution method

The substitution method involves replacing variables with their given values or expressions to solve an equation. In the exercise, we substituted \(r = 3\) and \(V = 12π\) into the original formula \(r^{2} = \frac{V}{\text{π} h}\).
This substitution changed the abstract equation into a more concrete form: \(3^{2} = \frac{12\text{π}}{\text{π} h}\).
By substituting these values, you turn an equation with unknowns into a simpler equation with known values that is easier to solve.
The substitution method is particularly useful in problems where multiple variables are involved, and you need to simplify the problem step-by-step.

volume formulas

Volume formulas are mathematical expressions that calculate the space inside a 3-dimensional object. Here, the provided volume \(V\) was given as \(12\text{π}\), which might relate to the formula for the volume of a cylinder or similar shape.
Although this problem didn't explicitly state the shape of the object, knowing volume formulas helps us understand where the equation \(r^{2} = \frac{V}{\text{π} h}\) might come from.
Be familiar with common volume formulas like those for cylinders, cones, and spheres. For example,

• Volume of a cylinder: \(V = \text{π} r^{2} h\)
• Volume of a sphere: \(V = \frac{4}{3} \text{π} r^{3}\)
• Volume of a cone: \(V = \frac{1}{3} \text{π} r^{2} h\)

Understanding these formulas can help you recognize the type of problem and accurately solve for the desired variable.