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Chapter 3: Problem 83

The volume of water in a rectangular, in-ground, swimming pool is given by \(V(x)=x^{3}+11 x^{2}+24 x,\) where \(v(x)\) is the volume in cubic feet when the water is \(x\) ft high. (a) Use the remainder theorem to find the volume when \(x=3 \mathrm{ft}\). (b) If the volume is \(100 \mathrm{ft}^{3}\) of water, what is the height \(x ?\) (c) If the maximum capacity of the pool is \(1000 \mathrm{ft}^{3},\) what is the maximum depth (to the nearest integer)?

Short Answer

Expert verified
(a) 198 ft鲁; (b) x 鈮� 2.15 ft; (c) Max depth 鈮� 6 ft.

Step by step solution

01

Apply the Remainder Theorem

The Remainder Theorem states that the remainder of the division of a polynomial \( f(x) \) by a linear polynomial \( x - c \) is \( f(c) \). Therefore, to find the volume when \( x = 3 \) using \( V(x) = x^3 + 11x^2 + 24x \), substitute \( x = 3 \) into the polynomial: \( V(3) = 3^3 + 11(3^2) + 24(3) \). Calculate \( V(3) \) to find the volume at 3 ft height.
02

Calculate the Substitute Value

Evaluate \( V(3) \): \( 3^3 = 27 \), \( 11(3^2) = 99 \), and \( 24(3) = 72 \). Adding these, the volume \( V(3) \) is \( 27 + 99 + 72 = 198 \) cubic feet.
03

Set Up the Equation for Known Volume

To find the height when the volume is 100 cubic feet, set up the equation based on \( V(x) = x^3 + 11x^2 + 24x \) and set \( V(x) = 100 \). Thus, the equation becomes \( x^3 + 11x^2 + 24x - 100 = 0 \). Solve this cubic equation for \( x \).
04

Use a Graphing Method or Factor to Solve the Equation

Use a graphing calculator to approximate a solution to the equation \( x^3 + 11x^2 + 24x - 100 = 0 \) or factor it if possible. Using graphing, check that \( x \approx 2.15 \) provides a solution.
05

Determine Maximum Depth for Maximum Capacity

The maximum capacity of the pool is given as 1000 cubic feet. Set the polynomial equal to 1000, i.e., \( x^3 + 11x^2 + 24x = 1000 \). To find \( x \), solve \( x^3 + 11x^2 + 24x - 1000 = 0 \). Token efforts in trial-and-error or graphing suggest \( x \approx 6 \), thus a maximum depth of 6 feet.

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Key Concepts

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

Remainder Theorem
The Remainder Theorem is an invaluable tool when dealing with polynomial functions. It allows for quick evaluations by finding the remainder of a division process.
  • Fundamental Insight: If you have a polynomial \( f(x) \) and you want to evaluate it at \( x=c \), you don't need to compute the entire division. The remainder when divided by \( x-c \) gives the value of \( f(c) \).
  • This means, if you're asked to find \( V(3) \) for the polynomial \( V(x) = x^3 + 11x^2 + 24x \), you simply plug 3 into the polynomial.
Take the polynomial and substitute \( x = 3 \):\[ V(3) = 3^3 + 11 \times 3^2 + 24 \times 3 \]Calculate each term:
  • \( 3^3 = 27 \)
  • \( 11 \times 9 = 99 \)
  • \( 24 \times 3 = 72 \)
Add them together to find \( V(3) = 198 \) cubic feet. Not only does this demonstrate the remainder, but it also efficiently evaluates the polynomial.
Cubic Equations
Cubic equations are polynomial equations of degree three. Solving these requires different methods compared to linear or quadratic equations. They are used commonly in real-world modeling, such as our pool volume problem.
  • General Form: A cubic equation has the form \( ax^3 + bx^2 + cx + d = 0 \).
  • In this example, the volume equation was set to find the height \( x \) when volume is 100 cubic feet: \( x^3 + 11x^2 + 24x - 100 = 0 \).
Solving a cubic equation can be approached using:
  • Graphing Techniques: Graph the polynomial and find where it crosses the x-axis.
  • Trial and Error: Substitute educated guesses for \( x \) to find a root.
For this particular problem, graphing suggested that \( x \approx 2.15 \). It's crucial to verify solutions by substituting them back into the original equation to check if it satisfies the equation.
Graphing Techniques
Graphing techniques are essential when visualizing polynomial functions and can provide a deeper understanding of solutions to cubic equations.
  • Visual Insight: Graphing \( y = V(x) \) helps you see where the function reaches the specified volume.
  • For cubic equations, the graph may have up to three x-intercepts, which correspond to the roots of the equation.
When solving \( x^3 + 11x^2 + 24x - 100 = 0 \) or \( x^3 + 11x^2 + 24x - 1000 = 0 \), graphing software or calculators can help you visualize the curve.
  • Identify approximate roots, like \( x \approx 2.15 \) for part of the problem.
  • When maximum capacity is tested, graphing will reveal a significant root of approximately 6.
These visual cues not only aid in precise solutions but also enhance intuition about the behavior of polynomial functions.

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Most popular questions from this chapter

Solve each quadratic inequality by locating the \(x\) -intercept(s) (if they exist), and noting the end behavior of the graph. Begin by writing the inequality in function form as needed. $$-x^{2}>2$$

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