Math Problem Statement

Find the volume of a trapezoidal prism with bases 12 yards and 8 yards, height of the trapezoid 6 yards, and a depth of 6 yards.

Solution

The image shows a trapezoidal prism with labeled dimensions:

  • The top base of the trapezoid: 12 yards
  • The bottom base of the trapezoid: 8 yards
  • The height of the trapezoid (vertical distance between bases): 6 yards
  • The slant height (left side): 10 yards
  • The depth (length of the prism): 6 yards

To solve any problem related to this figure, such as finding the volume or surface area, the following steps can be used:

1. Finding the Area of the Trapezoidal Base

The area of a trapezoid is given by:

A=12×(b1+b2)×hA = \frac{1}{2} \times (b_1 + b_2) \times h

Where:

  • b1=12ydb_1 = 12 \, \text{yd} (top base)
  • b2=8ydb_2 = 8 \, \text{yd} (bottom base)
  • h=6ydh = 6 \, \text{yd} (height of the trapezoid)

Substitute values:

A=12×(12+8)×6A = \frac{1}{2} \times (12 + 8) \times 6

A=12×20×6=60square yardsA = \frac{1}{2} \times 20 \times 6 = 60 \, \text{square yards}

2. Finding the Volume of the Prism

The volume of a prism is given by:

V=Abase×depthV = A_{\text{base}} \times \text{depth}

Here:

  • Abase=60yd2A_{\text{base}} = 60 \, \text{yd}^2
  • depth=6yd\text{depth} = 6 \, \text{yd}

Substitute values:

V=60×6=360cubic yardsV = 60 \times 6 = 360 \, \text{cubic yards}

Final Answer:

The volume of the trapezoidal prism is 360 cubic yards.


Let me know if you need further clarifications or additional calculations (e.g., surface area). Here are related questions:

  1. What is the surface area of this trapezoidal prism?
  2. How do you calculate the slant height of the trapezoidal face?
  3. What would happen to the volume if the depth increased to 10 yards?
  4. How can the Pythagorean theorem be used for other parts of this prism?
  5. How do you find the lateral surface area of this prism?

Tip:

Always verify dimensions and ensure consistent units when solving geometry problems involving prisms.

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Math Problem Analysis

Mathematical Concepts

Volume of a Prism
Area of a Trapezoid

Formulas

Area of a trapezoid: A = (1/2) * (b1 + b2) * h
Volume of a prism: V = A_base * depth

Theorems

Basic Volume and Area Formulas

Suitable Grade Level

Grades 7-10