Problem 67
Question
The formula for the volume of a pyramid is \(\frac{1}{3}\) times the height times the area of the base. The Rain Forest Pyramid in Mood Gardens near Galveston, Texas, is 100 feet high and 200 feet along each side of its square base. What is the volume of space inside the Rain Forest Pyramid?
Step-by-Step Solution
Verified Answer
The volume of space inside the Rain Forest Pyramid is approximately 1,333,333.33 cubic feet.
1Step 1: Calculate the Area of the Base
The base of this pyramid is a square, the area of which can be calculated by squaring the length of one side. Given that each side of the base measures 200 feet, the area of the base is therefore \(200^2 = 40000\) square feet.
2Step 2: Substitute values into the volume formula
The volume of a pyramid is given by the formula \(\frac{1}{3}\) times the height times the area of the base. Substituting the given pyramid's height of 100 feet and the calculated area of the base of 40000 square feet, the volume is thus: \(\frac{1}{3} * 100 * 40000\).
3Step 3: Compute the Volume
Carry out the multiplication to find the volume of the pyramid: \(\frac{1}{3} * 100 * 40000 = 1333333.33\) cubic feet. It is important to note that the volume is expressed in cubic units.
Key Concepts
Area of the baseGeometric formulasSubstitute valuesSquare area calculation
Area of the base
The 'area of the base' is a critical factor in calculating the volume of three-dimensional geometric shapes like pyramids. For the Rain Forest Pyramid in Moody Gardens, with a square base, finding this area is straightforward.
In general, the area of a square can be calculated using the formula: \( \text{area} = \text{side}^2 \), where 'side' refers to the length of one of the square's sides. Since the base of the Rain Forest Pyramid is a square with each side measuring 200 feet, the calculation would be \( 200^2 = 40000 \) square feet. This simple squared value is essential and used as a multiplier in further calculations for the pyramid's volume.
In general, the area of a square can be calculated using the formula: \( \text{area} = \text{side}^2 \), where 'side' refers to the length of one of the square's sides. Since the base of the Rain Forest Pyramid is a square with each side measuring 200 feet, the calculation would be \( 200^2 = 40000 \) square feet. This simple squared value is essential and used as a multiplier in further calculations for the pyramid's volume.
Geometric formulas
Understanding 'geometric formulas' is indispensable for solving problems related to the volume of various shapes. A pyramid, which tapers to a point from a flat base, has its own volume formula.
The volume of a pyramid is determined by the formula: \( V = \frac{1}{3} \times \text{height} \times \text{area of the base} \). It is vital to recognize that the fraction \( \frac{1}{3} \) reflects the pyramid's tapering shape. To visualize this, imagine that it would take three pyramids with the same base and height to fill a prism that has the same base area and height.
The volume of a pyramid is determined by the formula: \( V = \frac{1}{3} \times \text{height} \times \text{area of the base} \). It is vital to recognize that the fraction \( \frac{1}{3} \) reflects the pyramid's tapering shape. To visualize this, imagine that it would take three pyramids with the same base and height to fill a prism that has the same base area and height.
Substitute values
To progress from 'geometric formulas' to actual solutions, we must be skilled at 'substitute values'. This means replacing the variables in the formula with the measurements provided in the question.
In our case, after calculating the area of the pyramid's base, we substitute the base area of 40000 square feet and the height of 100 feet into the volume formula: \( V = \frac{1}{3} \times 100 \times 40000 \). By substituting these values carefully, avoiding mistakes, we are one step away from the answer. Accurate substitution ensures that the only step left is to perform the arithmetic.
In our case, after calculating the area of the pyramid's base, we substitute the base area of 40000 square feet and the height of 100 feet into the volume formula: \( V = \frac{1}{3} \times 100 \times 40000 \). By substituting these values carefully, avoiding mistakes, we are one step away from the answer. Accurate substitution ensures that the only step left is to perform the arithmetic.
Square area calculation
'Square area calculation' is a fundamental skill in geometry. When faced with a square base as in the Pyramid example, we use the formula \( \text{area} = \text{side}^2 \), where 'side' is the length of one of the square's sides.
For the Rain Forest Pyramid, with each side of the base measuring 200 feet, the calculation is simply \( 200 \times 200 = 40000 \) square feet. It's crucial to square the side's length, not just double it, to correctly calculate a square's area. Often, students make the mistake of multiplying the side length by 2 rather than squaring it, which leads to incorrect results.
For the Rain Forest Pyramid, with each side of the base measuring 200 feet, the calculation is simply \( 200 \times 200 = 40000 \) square feet. It's crucial to square the side's length, not just double it, to correctly calculate a square's area. Often, students make the mistake of multiplying the side length by 2 rather than squaring it, which leads to incorrect results.
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