Math Problem Statement

A rectangle has a length of (5 + 2x) inches and a width of 10 inches. A triangle has a base of 30 inches and a height of (4x – 10) inches. The area in square inches of the rectangle is equal to the area in square inches of the triangle.

What is the value of x?

Solution

To find the value of xx, we will use the information given about the areas of both the rectangle and the triangle. We are told that the area of the rectangle is equal to the area of the triangle.

Step 1: Area of the rectangle

The formula for the area of a rectangle is: Area of the rectangle=length×width\text{Area of the rectangle} = \text{length} \times \text{width} The length of the rectangle is 5+2x5 + 2x inches, and the width is 10 inches. So, the area of the rectangle is: Area of the rectangle=(5+2x)×10=50+20x\text{Area of the rectangle} = (5 + 2x) \times 10 = 50 + 20x

Step 2: Area of the triangle

The formula for the area of a triangle is: Area of the triangle=12×base×height\text{Area of the triangle} = \frac{1}{2} \times \text{base} \times \text{height} The base of the triangle is 30 inches, and the height is 4x104x - 10 inches. So, the area of the triangle is: Area of the triangle=12×30×(4x10)=15×(4x10)=60x150\text{Area of the triangle} = \frac{1}{2} \times 30 \times (4x - 10) = 15 \times (4x - 10) = 60x - 150

Step 3: Set up the equation

We are told that the area of the rectangle is equal to the area of the triangle. Therefore, we can set up the equation: 50+20x=60x15050 + 20x = 60x - 150

Step 4: Solve for xx

First, subtract 20x20x from both sides to get all terms involving xx on one side: 50=40x15050 = 40x - 150 Now, add 150 to both sides: 200=40x200 = 40x Finally, divide both sides by 40: x=20040=5x = \frac{200}{40} = 5

Conclusion:

The value of xx is 55.

Would you like further details on any part of this explanation? Here are some related questions to explore:

  1. What is the final area of both the rectangle and triangle when x=5x = 5?
  2. How does the length of the rectangle change as xx increases?
  3. What happens to the height of the triangle if xx is negative?
  4. Can the value of xx ever result in a negative area for either shape?
  5. How do different values of xx affect the proportionality of the rectangle's and triangle's areas?

Tip: When solving problems with equations, it's useful to ensure units are consistent throughout.

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Math Problem Analysis

Mathematical Concepts

Algebra
Equations
Geometry
Area

Formulas

Area of a rectangle = length × width
Area of a triangle = 1/2 × base × height

Theorems

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Suitable Grade Level

Grades 8-10