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Chapter 2: Problem 68

An aquarium of height 1.5 feet is to have a volume of \(6 \mathrm{ft}^{3}\). Let \(x\) denote the length of the base and \(y\) the width (see the figure). (a) Express \(y\) as a function of \(x .\) (b) Express the total number \(S\) of square feet of glass needed as a function of \(x .\) (PICTURE CANNOT COPY)

Short Answer

Expert verified
(a) \( y(x) = \frac{4}{x} \); (b) \( S(x) = 12 + \frac{12}{x} \).

Step by step solution

01

Understanding the Volume Formula

The volume of an aquarium can be found using the formula: \( V = x \cdot y \cdot h \), where \( V \) is the volume, \( x \) is the length, \( y \) is the width, and \( h \) is the height. Here, \( h = 1.5 \text{ ft} \), and \( V = 6 \text{ ft}^3 \).
02

Solve for y in Terms of x

Since \( V = 6 = x \cdot y \cdot 1.5 \), rearrange this to solve for \( y \): \[ y = \frac{6}{1.5x} \]. Simplifying, we get the function \( y(x) = \frac{4}{x} \).
03

Determine Total Surface Area S

The total surface area \( S \) of the aquarium includes the area of the base and the areas of the four sides. The base is \( x \times y \) and each pair of opposite sides are \( x \times h \) and \( y \times h \).
04

Express Each Component of Surface Area

Calculate the area of the base: \( A_{base} = x \cdot \frac{4}{x} = 4 \). Calculate the area of the two longer sides: \( A_{sides1} = 2xy \left(= 2x \times \frac{4}{x} \right) = 8 \). Calculate the area of the two shorter sides: \( A_{sides2} = 2yh = 2 \times \frac{4}{x} \times 1.5 = \frac{12}{x} \).
05

Combine Areas for Total Surface Area S

Now combine these to find the total surface area function \( S(x) \):\[ S(x) = 4 + 8 + \frac{12}{x} = 12 + \frac{12}{x} \].
06

Final Solution

To summarize:(a) The expression for \( y \) in terms of \( x \) is \( y(x) = \frac{4}{x} \).(b) The total glass surface area \( S \) needed as a function of \( x \) is \( S(x) = 12 + \frac{12}{x} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Surface Area Calculation
Understanding how to calculate the surface area of an aquarium can be invaluable in many real-world applications. When dealing with a rectangular aquarium, it's crucial to consider both the base and the side areas. The surface area helps determine the amount of material (like glass) needed for construction.

To find the total surface area, consider:
  • **The base area:** This is straightforward when you know the length and width. It's simply the product of these dimensions, here determined by the function: \( A_{base} = x \times y \).
  • **The side areas:** The aquarium also has four sides. Each pair of opposite sides is calculated by recognizing them as rectangles. Two longer sides have dimensions \( x \text{ and } h \), while the two shorter ones use \( y \text{ and } h \).
By calculating each of these areas and summing them, we derive the total surface area function: \[ S(x) = 4 + 8 + \frac{12}{x} \] Understanding this not only practices essential math skills but allows for practical assessment of materials needed for building an aquarium.
Functions and Graphs
Functions play a vital role in transforming relationships between variables into visual concepts. In our aquarium exercise, we focus on the variable relationship between the length \( x \) and width \( y \) of the aquarium's base.

The initial equation derived from the volume, \( V = x \cdot y \cdot h \), can be simplified to express \( y \) as a function of \( x \): \[ y(x) = \frac{4}{x} \]. This equation can be graphed, providing a visual way to see how changes in \( x \) affect \( y \).

With a graph:
  • The function \( y(x) = \frac{4}{x} \) generally shows an inverse relationship. As \( x \) increases, \( y \) decreases, demonstrating how the two dimensions adjust to maintain a constant volume.
  • The vertical asymptote suggests that \( y \) becomes undefined (tends toward infinity) as \( x \) approaches zero, naturally reflecting the impossibility of a length with no value.
  • Real-world interpretations become clearer, such as understanding feasible dimensions for different base measurements.
Using functions and their graphs, we can easily interpret and predict various dimensions' impacts on the overall shape and structure of the aquarium.
Mathematical Modeling
Mathematical modeling is a powerful technique used to represent real-world scenarios through mathematical expressions. In our aquarium problem, mathematical modeling includes representing the physical characteristics (like volume and surface area) with equations.

In this exercise:
  • The **Volume of the aquarium** is mathematically represented: \( V = x \cdot y \cdot h \). This model helps us relate the base dimensions (\( x \) and \( y \)) with the given height to ensure the desired volume is achieved.

  • The **Surface area formula**: \( S(x) = 12 + \frac{12}{x} \), which incorporates the base and side areas. This function is essential for understanding material requirements, correlating directly with construction costs.
  • By using these models, one can analyze how different dimensions impact the design and construction, facilitating efficient planning and material usage.
Mathematical models not only provide a formulaic representation but lead to beneficial insights when planning and optimizing real-world applications like aquarium construction.

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