Problem 34
Question
A box of fixed volume \(V\) has a square base with side length \(x .\) Write a formula for the height, \(h,\) of the box in terms of \(x\) and \(V .\) Sketch a graph of \(h\) versus \(x\)
Step-by-Step Solution
Verified Answer
The height formula is \(h = \frac{V}{x^2}\), and the graph is a hyperbola.
1Step 1: Understanding the Problem
We are given a box of fixed volume \(V\) with a square base. We need to express the height \(h\) of the box in terms of the side length \(x\) of the base and the volume \(V\).
2Step 2: Volume Formula for a Box
The volume \(V\) of a box is calculated as the area of the base multiplied by the height. Since the base of the box is square with side length \(x\), the area of the base is \(x^2\). Therefore, \(V = x^2 \cdot h\).
3Step 3: Solving for Height
We need to solve the volume formula \(V = x^2 \cdot h\) to find \(h\) in terms of \(x\) and \(V\). Dividing both sides by \(x^2\), we get \(h = \frac{V}{x^2}\).
4Step 4: Sketching the Graph
The formula \(h = \frac{V}{x^2}\) suggests that \(h\) is inversely proportional to \(x^2\). This means as \(x\) increases, \(h\) should decrease, following a curve decreasing rapidly at small \(x\) values, and flattening out as \(x\) gets larger. On graph paper, plot \(h\) on the y-axis and \(x\) on the x-axis to visualize this relationship.
Key Concepts
Volume of a BoxInverse ProportionalityGraph SketchingFunction of a Variable
Volume of a Box
To understand the concept of a box's volume, imagine you need to fill it up. The volume tells us how much space is inside the box. When the box has a square base with a side length of \(x\), the formula to find the volume \(V\) is based on a simple multiplication.
The area of the base is \(x^2\) because a square's area is the side length times itself. To find the volume of the box, we multiply the base area by the height \(h\). Therefore, the formula becomes:
The area of the base is \(x^2\) because a square's area is the side length times itself. To find the volume of the box, we multiply the base area by the height \(h\). Therefore, the formula becomes:
- \(V = x^2 \cdot h\)
Inverse Proportionality
Inverse proportionality is a unique relationship between two variables where one increases as the other decreases. In this exercise, the height \(h\) of the box and the square of the base side length \(x^2\) are inversely proportional.
What this means is that for a fixed volume \(V\), if the base area increases, the height must decrease to keep the volume constant. The mathematical expression for this relationship is:
What this means is that for a fixed volume \(V\), if the base area increases, the height must decrease to keep the volume constant. The mathematical expression for this relationship is:
- \(h = \frac{V}{x^2}\)
Graph Sketching
Graph sketching is a visual way to understand how two quantities relate to each other. In this problem, we are interested in sketching the graph of height \(h\) against the side length \(x\).
Based on the inverse proportionality formula \(h = \frac{V}{x^2}\), we see that as \(x\) increases, the height \(h\) decreases. The graph forms a curve starting high when \(x\) is small, and flattens out as \(x\) increases.
To draw this, plot \(h\) on the y-axis and \(x\) on the x-axis. Begin with small values for \(x\), where \(h\) is large, and plot the points as \(x\) becomes larger, showing \(h\) lowering and leveling off. This graphical approach enables an easier and more intuitive way to understand the inverse relationship between \(h\) and \(x\).
Based on the inverse proportionality formula \(h = \frac{V}{x^2}\), we see that as \(x\) increases, the height \(h\) decreases. The graph forms a curve starting high when \(x\) is small, and flattens out as \(x\) increases.
To draw this, plot \(h\) on the y-axis and \(x\) on the x-axis. Begin with small values for \(x\), where \(h\) is large, and plot the points as \(x\) becomes larger, showing \(h\) lowering and leveling off. This graphical approach enables an easier and more intuitive way to understand the inverse relationship between \(h\) and \(x\).
Function of a Variable
A function of a variable is a rule that describes how one quantity changes as another changes. In this situation, we are looking at how the height \(h\) of the box changes as the side length \(x\) of the base changes. This is represented by the function:
This function shows that \(h\) and \(x\) have an inverse relationship. It helps in predicting how the height will adjust as you modify the box's base size, given a fixed volume. Understanding functions is crucial for exploring how changes in one variable impact another, revealing deeper insights into dynamic relationships in calculus.
- \(h(x) = \frac{V}{x^2}\)
This function shows that \(h\) and \(x\) have an inverse relationship. It helps in predicting how the height will adjust as you modify the box's base size, given a fixed volume. Understanding functions is crucial for exploring how changes in one variable impact another, revealing deeper insights into dynamic relationships in calculus.
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Problem 34
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