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Chapter 9: Problem 329

In the following exercises, use the formula \(A=\frac{1}{2} b h\) Solve for h: a when A = 375 and b = 25 b in general

Short Answer

Expert verified
a) h = 30b) h = \frac{2A}{b}

Step by step solution

01

Understand the formula

The area of a triangle is given by the formula \(A = \frac{1}{2} b h\), where \(A\) is the area, \(b\) is the base, and \(h\) is the height.
02

Rearrange the formula to solve for h

Rearrange the formula \(A = \frac{1}{2} b h\) to solve for the height (\(h\)). Multiply both sides by 2 to get \(2A = bh\). Then, divide both sides by \(b\) to isolate \(h\): \(h = \frac{2A}{b}\).
03

Substitute the given values

For part (a), substitute \(A = 375\) and \(b = 25\) into the equation \(h = \frac{2A}{b}\): \(h = \frac{2(375)}{25}\).
04

Simplify the expression

Calculate the value of the expression \(h = \frac{750}{25} = 30\). So, the height \(h\) is 30.
05

General solution for h

To find the height \(h\) for any given area \(A\) and base \(b\), use the formula derived: \(h = \frac{2A}{b}\).

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Key Concepts

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

Area of a Triangle
Understanding the area of a triangle is a fundamental concept in geometry. The formula for the area is given by \(A = \frac{1}{2} b h\), where:
  • \(A\) stands for the area
  • \(b\) is the base of the triangle
  • \(h\) represents the height of the triangle
This formula works for any triangle as long as you can determine the base and the height. These two values are essential because they allow you to calculate how much space the triangle occupies.
Formula Rearrangement
Rearranging formulas is a key skill in algebra and prealgebra. When we talk about rearranging a formula, we mean changing its structure so we can solve for a different variable. For example, starting with \(A = \frac{1}{2} b h\), if we need to solve for \(h\), we:
  • First, multiply both sides by 2 to get rid of the fraction: \(2A = bh\)
  • Then, divide both sides by \(b\) to isolate \(h\): \(h = \frac{2A}{b}\)
Now, we have a new formula \(h = \frac{2A}{b}\), which helps us find the height if we know the area and the base.
Substitution and Simplification
Substitution is when you replace a variable in an equation with a given value. Simplification is the process of reducing an equation to its simplest form. Let's go through this step-by-step with our example:
  • We are given \(A = 375\) and \(b = 25\).
  • We substitute these values into our rearranged formula for height \(h = \frac{2A}{b}\).
  • This gives us \(h = \frac{2(375)}{25}\).
  • Next, we perform the multiplication and division: \(h = \frac{750}{25} = 30\).
Therefore, the height \(h\) of the triangle is 30 units. Substitution and simplification make it easy to solve equations by following a series of logical steps.
Prealgebra
Prealgebra is the foundation for many math concepts like solving equations, working with formulas, and understanding basic geometry. In our exercise, we're dealing with:
  • Rearranging the area formula for a triangle
  • Substituting given values
  • Simplifying the results
These skills are not just important for higher-level math, but for problem-solving in general. By mastering prealgebra, students set themselves up for success in more complex topics later on. Remember, practice is key. The more you work on problems like these, the more comfortable you'll become with the concepts!

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