Math Problem Statement

Write and solve an equation to find the value of x so that the figures have the same area. The area of a trapezoid is \( \frac{1}{2} h (b_1 + b_2) \). The two shapes should have the same area.

Solution

From the image, the problem is asking to find the value of xx so that the areas of a trapezoid and a rectangle are the same.

The formula for the area of a trapezoid is given as:

Atrapezoid=12h(b1+b2)A_{\text{trapezoid}} = \frac{1}{2} h (b_1 + b_2) where:

  • hh is the height (6 ft in the trapezoid),
  • b1b_1 and b2b_2 are the lengths of the two parallel sides (one is 12 ft, and the other is xx).

The formula for the area of a rectangle is:

Arectangle=length×widthA_{\text{rectangle}} = \text{length} \times \text{width} where:

  • The length is 12 ft,
  • The width is 6 ft.

Step-by-step solution:

  1. Area of the trapezoid: Atrapezoid=12×6×(12+x)=3(12+x)=36+3xA_{\text{trapezoid}} = \frac{1}{2} \times 6 \times (12 + x) = 3(12 + x) = 36 + 3x

  2. Area of the rectangle: Arectangle=12×6=72A_{\text{rectangle}} = 12 \times 6 = 72

  3. Equating the areas: 36+3x=7236 + 3x = 72

  4. Solving for xx: 3x=72363x = 72 - 36 3x=363x = 36 x=363=12x = \frac{36}{3} = 12

So, the value of xx is 12 feet.

Would you like more details or have any questions about this? Here are some related questions:

  1. How can you check the areas of both shapes after finding xx?
  2. What would happen if xx was smaller or larger than 12 ft?
  3. Can you apply this method to any trapezoid and rectangle?
  4. How do you solve for xx if both shapes have different dimensions?
  5. Why does the equation involve adding the base lengths of the trapezoid?

Tip: Always ensure to substitute the correct values into area formulas to avoid errors in calculations.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Geometry
Area Calculations

Formulas

Area of trapezoid: A = (1/2)h(b1 + b2)
Area of rectangle: A = length × width

Theorems

Equal Area Theorem

Suitable Grade Level

Grades 7-9