Problem 103 - Precalculus Essentials

Question

Solve for \(h: \pi r^{2} h=22 .\) Then rewrite \(2 \pi r^{2}+2 \pi r h\) in terms of \(r\).

Step-by-Step Solution

Answer

Solving for \(h\) yields \(h = \frac{22}{\pi r^{2}}\). Substituting the expression for \(h\) into the second equation and simplifying results in \(2 \pi r^{2} + 44/r\).

Step 1: Solve for \(h\)

To find \(h\), the formula \(\pi r^{2} h = 22\) needs to be rearranged. First, the sides of the equation should be divided by \(\pi r^{2}\) on both sides, which results in the equation \(h = \frac{22}{\pi r^{2}}\).

Step 2: Rewrite \(2 \pi r^{2} + 2 \pi r h\) in terms of \(r\)

Now the expression for \(h\) found previously can be substituted into the second equation. Replacing \(h\) with \(\frac{22}{\pi r^{2}}\), \(2 \pi r^{2} + 2 \pi r h\) becomes \(2 \pi r^{2} + 2 \pi r \cdot \frac{22}{\pi r^{2}}\).

Step 3: Simplify the equation

We proceed to simplify the equation. The \(\pi r^{2}\) cancels out the \(\pi r^{2}\) in the denominator, leaving \[ 2 \pi r^{2} + 2 \cdot 22/r = 2 \pi r^{2} + 44/r \].

Key Concepts

Solving Equations

When faced with an equation like \(\pi r^{2} h = 22\), you need to isolate the variable you want to solve for, in this case, \(h\). Start by understanding the structure of the equation and identify that \(h\) is being multiplied by the product \(\pi r^{2}\). To isolate \(h\), you divide both sides of the equation by \(\pi r^{2}\).
This operation effectively "cancels out" the \(\pi r^{2}\) on the side with \(h\), allowing you to solve for \(h\):

• Original equation: \(\pi r^{2} h = 22\)
• Divide both sides by \(\pi r^{2}\) to isolate \(h\)
• The result is \(h = \frac{22}{\pi r^{2}}\)

Solving equations often involves such strategic moves, centered around undoing operations by performing the inverse operation. Taking the time to rearrange equations correctly is fundamental to solving them.

Substitution Method

After finding \(h = \frac{22}{\pi r^{2}}\), substitution comes into play. This method involves replacing a variable in an equation with its equivalent expression from another equation you have solved.
In this problem, you substitute the value we found for \(h\) into the expression \(2 \pi r^{2} + 2 \pi r h\). The variable \(h\) is replaced with its expression \(\frac{22}{\pi r^{2}}\).

• Initial expression: \(2 \pi r^{2} + 2 \pi r h\)
• Substitute \(h = \frac{22}{\pi r^{2}}\) into the expression
• New expression: \(2 \pi r^{2} + 2 \pi r \cdot \frac{22}{\pi r^{2}}\)

Substitution helps when you continuously work with expressions and equations, allowing you to simplify or solve complex equations by replacing variables with known values or expressions.

Simplifying Expressions

Once you've substituted \(h\) back into the equation, your next task is to simplify. Simplification is about making an equation easier to handle, often by combining like terms, cancelling out factors, or reducing fractions.
For the equation \(2 \pi r^{2} + 2 \pi r \cdot \frac{22}{\pi r^{2}}\), notice that the term \(\pi r^{2}\) in the denominator of \(\frac{22}{\pi r^{2}}\) cancels out with the \(\pi r\) in the numerator following basic fraction cancelation rules:

• The term \(2 \pi r \cdot \frac{22}{\pi r^{2}}\) becomes \(\frac{44}{r}\)
• The expression simplifies to \(2 \pi r^{2} + \frac{44}{r}\)

By simplifying equations, you make them more manageable and clearly laid out, which is crucial especially in mathematics where accuracy and efficiency is key. Always look for common factors or terms to combine for a more streamlined equation.