Problem 35 - Algebra 2

Question

An amusement park owner wants to add a new wilderness water ride that includes a mountain that is shaped roughly like a square pyramid. Before building the new attraction, engineers must build and test a scale model. If the height of the scale model is 9 inches less than its length, write a polynomial function that describes the volume of the model in terms of its length. Use the formula for the volume of a pyramid, \(V=\frac{1}{3} B h\).

Step-by-Step Solution

Answer

The volume polynomial is \( V = \frac{1}{3} l^3 - 3l^2 \).

Step 1: Understand the Formula

The volume of a pyramid is given by the formula \( V = \frac{1}{3} B h \), where \( B \) is the area of the base and \( h \) is the height. Since the pyramid is square-based, the base \( B = s^2 \), where \( s \) is the side length of the square.

Step 2: Relate Variables to Length

Let the side length \( s \) of the base be equal to the length \( l \) of the pyramid. Therefore, the base area can be expressed as \( B = l^2 \). Also, according to the problem, the height \( h \) is 9 inches less than the length, so \( h = l - 9 \).

Step 3: Substitute in the Volume Formula

Substitute \( B = l^2 \) and \( h = l - 9 \) into the volume formula. Thus, the volume \( V \) is given by \( V = \frac{1}{3} l^2 (l - 9) \).

Step 4: Expand the Polynomial

Expand the expression \( \frac{1}{3} l^2 (l - 9) \) to get the polynomial function in standard form: \[ V = \frac{1}{3} (l^3 - 9l^2). \] This simplifies to \[ V = \frac{1}{3} l^3 - 3l^2. \]

Key Concepts

Volume of a Pyramid

Understanding how to calculate the volume of a pyramid is essential in various fields, from architecture to engineering. The formula to find the volume is: \[ V = \frac{1}{3} B h \] where:

• \( V \) represents the volume of the pyramid,
• \( B \) is the area of the base, and
• \( h \) is the height of the pyramid.

For a square pyramid specifically, the base is a square. This means that if the side of the square is \( s \), then the area of the base \( B = s^2 \). Thus, the volume formula for a square pyramid becomes \( V = \frac{1}{3} s^2 h \).
This formula helps in determining how much space is contained within the shape, which is crucial for projects involving physical constructions.

Square Pyramid

A square pyramid is a three-dimensional geometric figure with a square base and four triangular faces that converge to a single point called the apex. Understanding the structure of a square pyramid is essential for calculating its volume. There are characteristics unique to a square pyramid:

• The base is a perfect square, which means all four sides are equal in length.
• The height is the perpendicular distance from the center of the base to the apex.
• The lateral faces are congruent isosceles triangles.

In solving problems related to a square pyramid, knowing these properties simplifies the process of relating dimensions such as base side length and height to find other measurements.

Scale Model

A scale model is a smaller or larger physical copy of an object. These models are used in project planning to predict and test outcomes before full-scale construction. Creating a scale model of complex structures like a pyramid helps engineers test ideas feasibly and affords flexibility to change designs without the cost of full-scale execution. Some points to keep in mind about scale models:

• They preserve the proportions of the original object, making it easier to apply findings to real-life versions.
• Measurements and dimensions on the model are usually in a predefined ratio to real-life sizes.
• They offer an opportunity to evaluate visual aesthetics and structural stability without risking real-time resources.

These models are crucial in fields like architecture and engineering for both feasibility studies and demonstrations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They form the foundation for formulating mathematical problems and solutions in algebra. An algebraic expression, like the polynomial derived in the pyramid volume problem, expresses relationships between varying quantities. To create these expressions, you must:

• Identify the variables, which serve as placeholders for values that can change.
• Use constants, which are numbers that remain fixed within the context of the expression.
• Apply mathematical operations such as addition, subtraction, multiplication, and division to combine these elements.

In the case of the square pyramid, understanding these expressions allows one to represent and calculate geometric and volumetric properties as the dimensions vary. This makes them dynamic and highly useful in mathematical modeling and solving real-world problems.