Problem 89 - College Algebra

Question

The number of cubic yards of dirt, \(D\), needed to cover a garden with area \(a\) square feet is given by \(D=g(a)\) a. A garden with area \(5,000 \mathrm{ft}^{2}\) requires \(50 \mathrm{yd}^{3}\) of dirt. Express this information in terms of the function \(g\). b. Explain the meaning of the statement \(g(100)=1\).

Step-by-Step Solution

Answer

(a) \( g(5000) = 50 \). (b) \( g(100) = 1 \) means 100 sq ft of garden needs 1 yd³ of dirt.

Step 1: Understanding the function representation

We are given the function \( D = g(a) \), which relates the area of a garden in square feet \( a \) to the amount of dirt needed in cubic yards \( D \). For part (a), we need to express provided information using this function notation.

Step 2: Applying the given data to the function

From the problem, we know that a garden with an area of \( 5,000 \mathrm{ft}^2 \) requires \( 50 \mathrm{yd}^3 \) of dirt. This means when \( a = 5000 \), \( D = 50 \). Therefore, in function notation, we can write this as \( g(5000) = 50 \).

Step 3: Interpreting the statement for part b

The statement \( g(100) = 1 \) means that if the garden has an area of \( 100 \mathrm{ft}^2 \), the amount of dirt required to cover it is \( 1 \mathrm{yd}^3 \). It demonstrates how the function \( g \) relates a specific garden area to the dirt needed.

Key Concepts

Function Notation

Function notation is a special method used to describe relationships in mathematics, especially in algebra. When we say a function is expressed as \( D = g(a) \), it means we have a rule called \( g \) that tells us how to find \( D \) based on the value of \( a \). Here, \( D \) is the output or dependent variable, and \( a \) is the input or independent variable. This notation helps clarify which variable depends on the other and makes it simpler to refer to specific parts of the function's relationship.

• **Key parts**: The function name \( g \), the input \( a \), and the output \( D \).
• **Why use it**: It provides a tidy and clear way to interpret mathematical relationships.

In our exercise, \( g(5000) = 50 \) shows that when the area of the garden is 5,000 square feet, the function \( g \) tells us to use 50 cubic yards of dirt.

Interpreting Functions

Interpreting functions involves understanding what the function tells us about the relationship between variables. If we look at the expression \( g(100) = 1 \), it's more than just numbers on a page. It shows the direct relationship between the area and the needed dirt. When \( a \) is 100 square feet, \( D \) becomes 1 cubic yard. This means our function helps convert area to volume efficiently.

• **Understanding numbers**: Each input \( a \) has a specific output \( D \) determined by \( g \).
• **Real-world link**: It builds a bridge between a real-world situation (a garden) and the mathematical model (the function \( g \)).

By understanding this, we can predict or compute other results using the same function, as in our example where different garden sizes require different amounts of dirt. This interpretation skill is crucial in applying math to solve practical issues.

Cubic Measurements

Cubic measurements relate to the three-dimensional volume of an object or space. When we talk about dirt needed to cover a garden, we're referring to the volume of dirt, which is measured in cubic yards. This indicates that the dirt will fill a three-dimensional space with length, width, and height, even if we only see it spread as a layer over the area.

• **Unit insights**: Cubic yards (\text{yd}^3) is a common unit used for volume in landscaping.
• **Volume significance**: It tells us not just the area but how deep or full something should be.

In our exercise, understanding cubic measurements helps in correctly interpreting the function's output. For example, knowing that \( g(5000) = 50 \) signifies more than just a result—it involves understanding the volume necessary for a given garden area.