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Chapter 9: Problem 457

In the following exercises, use the formula \(A=\frac{1}{2} b h\) Solve for b: a when A = 416 and h = 32 b in general

Short Answer

Expert verified
For A = 416 and h = 32, b = 26. In general, b = \( \frac{2A}{h} \).

Step by step solution

01

- Understand the formula

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

- Isolate the variable b

Rearrange the formula to solve for the base b. Start by multiplying both sides by 2 to get rid of the fraction: \[ 2A = b h \] Then divide both sides by h: \[ b = \frac{2A}{h} \]
03

- Substitute the given values

For part (a): Substitute A = 416 and h = 32 into the rearranged formula: \[ b = \frac{2 \times 416}{32} \]
04

- Perform the calculations

Calculate the right side of the equation: \[ b = \frac{832}{32} = 26 \]
05

- State the solution for part (a)

The value of b when A = 416 and h = 32 is: \[ b = 26 \]
06

- General Solution

For part (b), the general formula for b is already derived in Step 2: \[ b = \frac{2A}{h} \]

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Key Concepts

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

Solving Equations
Solving equations often involves isolating a variable of interest. In this triangle problem, we used algebraic manipulation to solve for the base (b). First, we started with the given area formula of a triangle, which is \( A = \frac{1}{2} b h \). Here are the main steps to isolate b:
  • Multiply both sides by 2 to remove the fraction: \( 2A = b h \)
  • Divide both sides by h to solve for b: \( b = \frac{2A}{h} \)
By rearranging the original formula, we made b the subject of the equation. Once isolated, solving for b becomes straightforward, especially once you substitute the given values for A and h. This approach makes complex problems more manageable by breaking them down into simpler steps.
Algebra
Algebra provides powerful tools to manipulate and solve equations. In the given problem, algebraic techniques allowed us to solve for the base (b) of the triangle. Here鈥檚 a closer look at the algebraic steps used:
  • Start with the formula: \( A = \frac{1}{2} b h \)
  • Multiply both sides by 2: \( 2A = b h \)
  • Divide both sides by h: \( b = \frac{2A}{h} \)
These steps demonstrate basic algebra principles, such as the distributive property and inverse operations, which are used to isolate variables. Understanding these principles is crucial for solving various equations in mathematics. Using algebra helps us systematically approach problems by transforming equations into simpler and more solvable forms.
Geometry
Geometry deals with shapes, their properties, and measurements. In this exercise, we focused on the formula for the area of a triangle, which is \( A = \frac{1}{2} b h \). This formula tells us that the area can be found by multiplying the base (b) by the height (h) and then dividing by two. Here's the breakdown:
  • The base (b) and height (h) are the two essential dimensions of the triangle.
  • The height is perpendicular to the base.
  • Multiplying base and height gives us a product related to the area of a rectangle, but since a triangle is half of a rectangle, we divide by 2.
By rearranging this formula, we can solve for any unknown dimension if the area and one other dimension are known. Understanding this geometric relationship allows us to solve practical problems, whether in construction, design, or other fields that require precise measurements and spatial reasoning.

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