Problem 50 - Prealgebra

Question

Solve each of the following word problems. Note that not all of the problems are solved by simply multiplying the numbers in the problems. Many of the problems involve addition and subtraction as well as multiplication. This is a 3 D model of the Louvre Museum in Paris, France. The pyramid that dominates the Napoleon Courtyard has a height of 21.65 meters and a square base with sides of 35.50 meters. What is the volume of the pyramid to the nearest whole number? Hint: The volume of a pyramid can be found by the equation \(V=\left(\frac{1}{3}\right)(\text { area of the base })(\text { height })\)

Step-by-Step Solution

Answer

The volume of the Louvre pyramid is approximately 9110 cubic meters.

Step 1: Understand the Problem

We need to find the volume of the pyramid in the Louvre Museum. The pyramid has a height of 21.65 meters, and its base is a square with sides measuring 35.50 meters each.

Step 2: Calculate the Area of the Base

The base of the pyramid is a square. To find the area of the base, we use the formula for the area of a square, which is \[ \text{Area of the base} = \text{side}^2 \]Given that the side length is 35.50 meters, the area of the base is \[ (35.50)^2 = 1260.25 \text{ square meters} \].

Step 3: Apply the Volume Formula for a Pyramid

The volume of a pyramid is given by the formula \[ V = \left(\frac{1}{3}\right) \times (\text{area of the base}) \times (\text{height}) \]We have calculated the area of the base as 1260.25 square meters and are given the height as 21.65 meters. Substitute these values into the formula:\[ V = \left(\frac{1}{3}\right) \times 1260.25 \times 21.65 \].

Step 4: Calculate the Volume

Perform the multiplication:\[ V = \left(\frac{1}{3}\right) \times 1260.25 \times 21.65 = 9109.7875 \]This calculation gives us the volume in cubic meters.

Step 5: Round to the Nearest Whole Number

The final step is to round the calculated volume to the nearest whole number.\[ V \approx 9110 \text{ cubic meters} \].

Key Concepts

Volume of a Pyramid

Finding the volume of a pyramid is a fascinating process because it involves just a few simple calculations but provides a glimpse into the majestic architecture of structures like the Louvre's pyramid. The formula to calculate the volume is:

• \[ V = \left(\frac{1}{3}\right) \times (\text{area of the base}) \times (\text{height}) \]

This formula essentially tells us that the volume is one-third of the product of the base area and the height. By using this formula, you can calculate the amount of space inside the pyramid.
### Breakdown of the Formula- The fractional part \(\frac{1}{3}\) relates to how the shape of a pyramid tapers as it rises from the base to the point.- The base area (

• \(\text{area of the base}\)

) represents the foundation or the footprint of the pyramid.- The height (

• \(\text{height}\)

) measures the perpendicular distance from the base to the apex.Understanding these elements helps comprehend why the formula captures the essence of a pyramid’s space.

Area Calculation

Almost every geometric problem begins with area calculation, a fundamental aspect that supports various dimensions of mathematical problems, including volume. For the pyramid in the problem, we start by focusing on its base's area.
### Area of a SquareThe base in this word problem is a square, and the formula to find the area of a square is:

• \[\text{Area of the base} = \text{side}^2\]

With each side of the base measuring 35.50 meters, the calculation becomes simple:

• \[(35.50)^2 = 1260.25 \text{ square meters}\]

This tells you how much surface the base covers. It’s always crucial to double-check these calculations since the base area significantly influences the final volume.

Rounding Whole Numbers

Rounding whole numbers is a skill often needed to present results in a simpler, more digestible form, especially when dealing with large or complex numbers like in our calculation of the pyramid's volume. After calculating a precise value, you may round to gain an approximate but practically useful number.
### Steps to Rounding 1. Identify the digit at the place value to which you are rounding (e.g., nearest tens, hundreds). 2. Look at the digit immediately to the right.

• If it's 5 or more, increase the rounding digit by 1.
• If it's less than 5, keep the rounding digit the same.

3. Change all digits to the right of the rounding digit to zero.
### Application to the Problem In our problem, the calculated volume was 9109.7875 cubic meters. Rounding to the nearest whole number requires looking at the tenths place (7 in this case).

• Since 7 is greater than 5, we round 9109.7875 up to 9110.

The rounded result provides a cleaner, easier-to-reference volume suitable for practical usage.