Select The Correct Answer.The Height Of A Triangle Is 2 Less Than 5 Times Its Base. If The Base Of The Triangle Is X X X Feet, And The Area Of The Triangle Is 12 Square Feet, Which Equation Models This Situation?A. $5x^2 - 2x - 12 =

In geometry, the relationship between the base and height of a triangle is crucial in determining its area. The area of a triangle can be calculated using the formula: Area = (base × height) / 2. In this article, we will explore how to model a situation where the height of a triangle is 2 less than 5 times its base.

Defining the Variables

Let's assume the base of the triangle is represented by the variable x. Since the height of the triangle is 2 less than 5 times its base, we can express the height as 5x - 2.

Calculating the Area of the Triangle

The area of the triangle is given as 12 square feet. Using the formula for the area of a triangle, we can set up an equation to model this situation:

Area = (base × height) / 2 12 = (x × (5x - 2)) / 2

Simplifying the Equation

To simplify the equation, we can start by multiplying both sides by 2 to eliminate the fraction:

24 = x × (5x - 2)

Next, we can distribute the x to the terms inside the parentheses:

24 = 5x^2 - 2x

Rearranging the Equation

To put the equation in standard quadratic form, we can add 2x to both sides:

24 + 2x = 5x^2

Subtracting 24 from both sides gives us:

2x = 5x^2 - 24

Modeling the Situation

Now that we have simplified the equation, we can model the situation using the following equation:

5x^2 - 2x - 24 = 0

This equation represents the relationship between the base and height of the triangle, and it can be used to solve for the value of x.

Conclusion

In this article, we explored how to model a situation where the height of a triangle is 2 less than 5 times its base. We defined the variables, calculated the area of the triangle, simplified the equation, and rearranged it to put it in standard quadratic form. The resulting equation, 5x^2 - 2x - 24 = 0, represents the relationship between the base and height of the triangle and can be used to solve for the value of x.

Discussion

What are some real-world applications of the relationship between the base and height of a triangle? How can this concept be used to solve problems in geometry and other areas of mathematics?

Answer

In our previous article, we explored how to model a situation where the height of a triangle is 2 less than 5 times its base. We defined the variables, calculated the area of the triangle, simplified the equation, and rearranged it to put it in standard quadratic form. In this article, we will answer some frequently asked questions about modeling the relationship between the base and height of a triangle.

Q: What is the formula for the area of a triangle?

A: The formula for the area of a triangle is: Area = (base × height) / 2.

Q: How do I calculate the height of a triangle if I know the base and the area?

A: To calculate the height of a triangle, you can use the formula: height = (2 × area) / base.

Q: What is the relationship between the base and height of a triangle in this problem?

A: In this problem, the height of the triangle is 2 less than 5 times its base. This can be expressed as: height = 5x - 2.

Q: How do I simplify the equation 24 = x × (5x - 2) to put it in standard quadratic form?

A: To simplify the equation, you can start by multiplying both sides by 2 to eliminate the fraction: 48 = x × (5x - 2). Next, you can distribute the x to the terms inside the parentheses: 48 = 5x^2 - 2x. Finally, you can add 2x to both sides to put the equation in standard quadratic form: 48 + 2x = 5x^2.

Q: What is the correct equation that models the situation?

A: The correct equation that models the situation is: 5x^2 - 2x - 24 = 0.

Q: How can I use the equation 5x^2 - 2x - 24 = 0 to solve for the value of x?

A: To solve for the value of x, you can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, where a = 5, b = -2, and c = -24.

Q: What are some real-world applications of the relationship between the base and height of a triangle?

A: The relationship between the base and height of a triangle has many real-world applications, including:

• Calculating the area of a triangle in geometry and trigonometry
• Determining the height of a building or a bridge
• Calculating the volume of a pyramid or a cone
• Modeling the motion of an object under the influence of gravity

Q: How can I use the concept of the relationship between the base and height of a triangle to solve problems in other areas of mathematics?

A: The concept of the relationship between the base and height of a triangle can be used to solve problems in other areas of mathematics, including:

• Algebra: The relationship between the base and height of a triangle can be used to solve quadratic equations and systems of equations.
• Geometry: The relationship between the base and height of a triangle can be used to calculate the area and perimeter of triangles and other polygons.
• Trigonometry: The relationship between the base and height of a triangle can be used to calculate the sine, cosine, and tangent of angles in triangles.

Conclusion

In this article, we answered some frequently asked questions about modeling the relationship between the base and height of a triangle. We hope that this article has provided you with a better understanding of the concept and its applications in mathematics.