Problem 36 - Intermediate Algebra

Question

Find the volume of each figure to the nearest hundredth. See Example 4. (Answers may vary slightly depending on which approximation of is used.) A pyramid whose base is a square with each side measuring \(2.57 \mathrm{cm}\) and with a height of \(12.32 \mathrm{cm}\)

Step-by-Step Solution

Answer

The volume of the pyramid is approximately 27.12 cm³.

Step 1: Identify the Volume Formula for a Pyramid

The volume of a pyramid can be calculated using the formula \( V = \frac{1}{3} B h \), where \( B \) is the area of the base of the pyramid and \( h \) is the height of the pyramid.

Step 2: Calculate the Area of the Base

Since the base of the pyramid is a square, we calculate the area by squaring the length of one of its sides. Here, each side of the base is \(2.57\, \text{cm}\). So, \( B = (2.57)^2 \).

Step 3: Compute the Area of the Base

Calculate the square of \(2.57\). So, \( B = 2.57 \times 2.57 = 6.6049 \, \text{cm}^2 \).

Step 4: Calculate the Volume of the Pyramid

Now substitute the values of \( B \) and \( h \) into the volume formula: \( V = \frac{1}{3} \times 6.6049 \times 12.32 \).

Step 5: Perform the Calculations

First calculate the product \( 6.6049 \times 12.32 = 81.360168 \, \text{cm}^3 \). Divide this result by 3 to find the volume: \( V = \frac{81.360168}{3} = 27.12 \, \text{cm}^3 \).

Step 6: Round to the Nearest Hundredth

The calculated volume is \( 27.12 \, \text{cm}^3 \), which is already rounded to the nearest hundredth.

Key Concepts

Pyramid Volume Formula

To find the volume of a pyramid, we use a special formula that is very helpful: \( V = \frac{1}{3} B h \). In this formula, \( B \) represents the area of the base of the pyramid, while \( h \) stands for the height or the vertical distance from the base to the apex. This formula makes it simple to calculate volume because it considers how a pyramid's volume is proportionately smaller than that of a prism with the same base and height. The factor of \( \frac{1}{3} \) is crucial as it reflects this difference.

Square Base Area

Since the base of the pyramid in our problem is a square, finding its area is straightforward. For any square, the area can be calculated by squaring the length of one side.
Here, we have a base with sides of length \(2.57\,\text{cm}\). This means the area \( B \) of the square is given by \( B = (2.57)^2 \). Squaring \(2.57\) means multiplying the number by itself, so \(2.57 \times 2.57 = 6.6049\, \text{cm}^2\).
This result gives us the exact area of the base of the pyramid.

Mathematical Calculations for Volume

Once we have the area of the base calculated, we can proceed to find the volume of the pyramid using our volume formula. The previously calculated base area is \(6.6049\, \text{cm}^2\) and the given height of the pyramid is \(12.32\, \text{cm}\).
We substitute these values into the volume formula: \( V = \frac{1}{3} \times 6.6049 \times 12.32 \). It's important to perform the multiplication first:

• The product \( 6.6049 \times 12.32 \) results in \(81.360168\, \text{cm}^3\).
• Then, divide this by 3, as per the formula: \( V = \frac{81.360168}{3} \).
• The volume of the pyramid is, therefore, \(27.12\, \text{cm}^3\).

This number represents the total space the pyramid occupies.

Geometry Problem-solving Steps

Breaking down a geometry problem into manageable steps can make complex calculations much easier. Here’s a quick recap of how we solved this volume issue:

• First, determine the appropriate formula to use based on the shape of the object. For a pyramid, it's \( V = \frac{1}{3} B h \).
• Next, deal with the base. Calculate its area using the relevant formula for the shape. In this case, we figured out the area of the square base.
• After that, substitute the measurements into the volume formula. This requires being careful with units and operations.
• Lastly, perform the calculations step by step, ensuring to handle multiplication and division as required by the formulas.

Through these structured steps, tackling geometry problems becomes less intimidating and more systematic, promoting a clear understanding of each component involved.