Problem 67 - College Algebra and Calculus: An Applied Approach

Question

Geometry A triangular sign has a height that is equal to its base. The area of the sign is 4 square feet. Find the base and height of the sign.

Step-by-Step Solution

Answer

The base and the height of the sign are approximately 2.83 feet.

Step 1: Understand the problem

We are given that the triangular sign has an area of 4 square feet and the base equals the height. We are asked to find the base and the height of the sign. The formula for the area of a triangle is \(\frac{1}{2} \times base \times height\). Substituting the area to 4 and since the base equals the height, we can denote them both as \(x\) and setup the equation \(\frac{1}{2}x^2=4\).

Step 2: Solve for \(x\)

First, we can get rid of the fraction by multiplying both sides of the equation \(\frac{1}{2}x^2=4\) by 2. This gives us the equation \(x^2=8\). To solve for \(x\), we need to take the square root of both sides of the equation. The square root of 8 is approximately 2.83, to three significant figures.

Step 3: Interpret the result

Our calculated value for \(x\) represents the length of the base and the height of the sign. Therefore, the sign has a base length of approximately 2.83 feet and a height of approximately 2.83 feet.

Key Concepts

Area of a Triangle

The area of a triangle is an essential concept in geometry. It helps us understand how much space the triangle occupies on a plane. To calculate the area of a triangle, you use the formula: \(\frac{1}{2} \times \text{base} \times \text{height}\). This simple formula comes from the fact that a triangle is essentially half of a rectangle.

• This formula requires two critical components: the base and the height, both of which must be perpendicular to each other.
• The area is measured in square units, such as square feet, square meters, etc.
• The height is directly perpendicular to the base at one of its endpoints.

Knowing how to apply the area formula is crucial for solving many geometry problems, such as finding how much paint is needed to cover a triangular surface.

Base and Height of Triangle

In a triangle, the base and the height are fundamental dimensions. They can vary depending on your choice of base.

• The base is any one of the sides of the triangle.
• The height is the perpendicular distance from the base to the opposite vertex.
• When the base and height are equal, it simplifies equations and calculations.

For example, consider a triangle where the base equals the height. If you denote both as \(x\), the formula for the area becomes \(\frac{1}{2}x \times x\), or \(\frac{1}{2}x^2\). This equality can make calculations straightforward and often leads to simple, solvable equations.

Solving Equations

Solving equations is a fundamental skill in mathematics that allows us to find unknown values. It often involves a bit of detective work—balancing both sides of an equation until you uncover the variable. Here’s how it works with our problem:

• We start with the area formula \(\frac{1}{2}x^2 = 4\).
• To eliminate the fraction, multiply both sides by 2 to get \(x^2 = 8\).
• Next, find the square root of both sides to solve for \(x\). Thus, \(x \approx \sqrt{8} \approx 2.83\).

Solve equations step-by-step, keeping your operations balanced, to effectively find the dimensions of a geometric figure.