Math Problem Statement

What is the volume of this shape in cubic yards with given measurements: one side of 56 feet, another side of 85 feet, and an included angle of 26 degrees?

Solution

The given image shows a shape with the following key measurements:

  • One side of 56 ft
  • Another side of 85 ft
  • The angle between the sides, θ=26\theta = 26^\circ

We are tasked with finding the volume of this shape in cubic yards. The shape in the image resembles a triangular prism, where the triangular base is defined by the two sides (56 ft and 85 ft) and the included angle θ=26\theta = 26^\circ.

Steps to Find the Volume:

  1. Area of the triangular base: To calculate the area of the triangular base, we can use the formula for the area of a triangle when two sides and the included angle are known: Area of base=12absin(θ)\text{Area of base} = \frac{1}{2} a b \sin(\theta) where:

    • a=56fta = 56 \, \text{ft}
    • b=85ftb = 85 \, \text{ft}
    • θ=26\theta = 26^\circ
  2. Volume of the prism: Once the area of the triangular base is known,

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Math Problem Analysis

Mathematical Concepts

Geometry
Trigonometry
Volume Calculation

Formulas

Area of triangle = (1/2) * a * b * sin(θ)
Volume of prism = Area of base * height

Theorems

Trigonometric functions (sine) for triangle area calculation

Suitable Grade Level

Grades 9-12