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Chapter 1: Problem 13

Solve. \(V=l w h,\) for \(h\) (a volume formula)

Short Answer

Expert verified
The height \( h \) can be found using the formula: \[ h = \frac{V}{l w} \]

Step by step solution

01

- Write the Given Formula

Start with the given volume formula: \[ V = l w h \] where \( V \) is the volume, \( l \) is the length, \( w \) is the width, and \( h \) is the height.
02

- Isolate the Height (\(h\))

To solve for \( h \), isolate \( h \) on one side of the equation. Start by dividing both sides of the equation by \( l w \):\[ \frac{V}{l w} = \frac{l w h}{l w} \]
03

- Simplify the Equation

On the right side of the equation, the terms \( l \) and \( w \) will cancel out, leaving:\[ \frac{V}{l w} = h \]
04

- Final Formula for \(h\)

Rewrite the equation from Step 3 to clearly show the solution for \( h \):\[ h = \frac{V}{l w} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Equations
Solving equations is a fundamental skill in algebra. It involves finding the value of a variable that makes the equation true. In this exercise, we solve the volume formula to find the height (\(h\)).

Here's how we do it:
  • Identify the variable you need to solve for (in our case, it's \(h\)).
  • Perform operations to isolate this variable. This often involves adding, subtracting, multiplying, or dividing both sides of the equation by the same number to maintain equality.

In the original solution, we isolated \(h\) by dividing both sides by \(l w\). This is a standard technique in solving equations. Remember, the goal is to get the variable by itself on one side of the equation. Once you've done this, you have your solution!
Algebraic Manipulation
Algebraic manipulation refers to the techniques used to rearrange and simplify equations. This is key in solving for a variable, as it allows us to isolate that variable effectively.

In our exercise, we used the following algebraic manipulations:
  • Division: We divided both sides of the equation by \(l w\) to isolate the variable \(h\).
  • Simplification: The terms \(l\) and \(w\) on the right side cancelled out, simplifying our equation to \(h = \frac{V}{l w}\).

These steps illustrate how algebraic manipulation helps to transform complex equations into simpler forms. Always ensure to perform the same operation on both sides of the equation to maintain balance. Practice these techniques to become proficient in solving different types of algebraic equations.
Volume Formula
The volume formula is crucial in geometry, helping us determine the space occupied by a 3-dimensional object. The formula \(V = l w h\) represents the volume of a rectangular prism, where:
  • \(V\) is the volume
  • \(l\) is the length
  • \(w\) is the width
  • \(h\) is the height

To find \(h\), we rearranged the formula by isolating \(h\). This gave us \(h = \frac{V}{l w}\). Understanding and using the volume formula is essential in many real-life applications, such as finding the space inside a box or determining the capacity of a container. Practicing these concepts will enhance your problem-solving skills in both mathematics and everyday tasks.

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Most popular questions from this chapter

The height and sides of a triangle are four consecutive integers. The height is the first integer, and the base is the third integer. The perimeter of the triangle is 42 in. Find the area of the triangle.

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ |-2-3| \cdot 4^{2}-3 $$

Find an equivalent expression by factoring. $$ 6 y-9 x-3 w $$

Write an equivalent expression using an associative law. $$ -7(a b) $$

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ -4^{2} $$
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