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Chapter 8: Problem 8

Determine the surface area ( \(S A\) ) of each right prism described. Equilateral triangular base with side 2.8 yd; height 1.5 yd

Short Answer

Expert verified
The surface area is approximately 19.38 square yards.

Step by step solution

01

Calculate the area of the base

To find the area of an equilateral triangle with side length 2.8 yards, use the formula for the area of an equilateral triangle: \[ A = \frac{\sqrt{3}}{4} s^2 \] Here, \( s = 2.8 \) yards, so: \[ A = \frac{\sqrt{3}}{4} (2.8)^2 \] \[ A = \frac{\sqrt{3}}{4} (7.84) \] \[ A \approx 3.39 \text{ square yards} \]
02

Calculate the lateral surface area

The lateral surface area of a prism can be found using the perimeter of the base and the height of the prism. The perimeter of the equilateral triangle is: \[ 3 \times 2.8 = 8.4 \text{ yards} \] The height of the prism is given as 1.5 yards. The lateral surface area (LSA) is: \[ \text{LSA} = \text{Perimeter} \times \text{Height} \] \[ \text{LSA} = 8.4 \times 1.5 \] \[ \text{LSA} = 12.6 \text{ square yards} \]
03

Calculate the total surface area

The total surface area (SA) of the prism is the sum of the lateral surface area and the area of the two bases. So: \[ \text{SA} = \text{LSA} + 2 \times \text{Area of the base} \] \[ \text{SA} = 12.6 + 2 \times 3.39 \] \[ \text{SA} = 12.6 + 6.78 \] \[ \text{SA} \approx 19.38 \text{ square yards} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equilateral Triangle Area
Understanding the area of an equilateral triangle is essential for finding the surface area of prisms with equilateral triangular bases. An equilateral triangle has all three sides of equal length. The formula to find the area of an equilateral triangle is: \[ A = \frac{\sqrt{3}}{4} s^2 \] where \( s \) is the side length.
  • First, ensure you know the side length of the triangle. Here, it's 2.8 yards.
  • Substitute the side length into the formula: \( A = \frac{\sqrt{3}}{4} (2.8)^2 \)
  • Simplify the expression under the square root and multiply correctly: \( A = \frac{\sqrt{3}}{4} (7.84) \)
  • Finally, calculate the approximated area: \( A \approx 3.39 \) square yards.
Knowing this method helps in finding the base area for further surface area calculations of a prism.
Lateral Surface Area
Next, let's discuss the lateral surface area (LSA) of a prism. The LSA is the area of all the sides or faces that are perpendicular to the base. For a prism with an equilateral triangle base:

  • First, determine the perimeter of the base, which is an equilateral triangle. Here, the perimeter is calculated as \( 3 \times 2.8 = 8.4 \) yards.
  • Next, identify the height of the prism given in the problem. For our example, the height is 1.5 yards.
  • The formula to find the LSA is: \( \text{LSA} = \text{Perimeter} \times \text{Height} \)
  • Substitute the values into the formula: \( \text{LSA} = 8.4 \times 1.5 = 12.6 \) square yards.
The lateral surface area calculation is a crucial step in the overall surface area determination.
Total Surface Area Calculation
Finally, we bring everything together to find the total surface area (SA) of the prism. The total surface area includes both the lateral area and the area of the two bases:
  • First, recall the lateral surface area we previously calculated: 12.6 square yards.
  • Then, remember that the prism has two triangular bases. Each base has an area of approximately 3.39 square yards.
  • The total surface area formula is: \( \text{SA} = \text{LSA} + 2 \times \text{Area of the base} \)
  • Substitute the values into the formula: \( \text{SA} = 12.6 + 2 \times 3.39 \)
  • Simplify to find the total surface area: \( \text{SA} = 12.6 + 6.78 = 19.38 \) square yards.
Accurately determining the total surface area is important for applications in real-life situations, such as material requirements for construction.

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