Problem 71 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Give approximations of linear measures to the nearest hundredth. Buoyancy of a Spherical Object It has been determined that a spherical object of radius 4 inches with specific gravity 0.25 will sink in water to a depth of \(x\) inches, where \(x\) is the least positive root of the equation $$x^{3}-12 x^{2}+64=0$$ To what depth will this object sink if \(x<10 ?\)

Step-by-Step Solution

Answer

The object will sink to a depth of 4 inches.

Step 1: Understand the Problem

We are given a cubic equation \(x^3 - 12x^2 + 64 = 0\). We need to find the smallest positive root of this equation that is less than 10. This root represents the depth to which the spherical object will sink.

Step 2: Use the Rational Root Theorem

According to the rational root theorem, the possible rational roots of the equation are factors of the constant term (64) over the leading coefficient (1). The factors of 64 are \(\pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm 32, \pm 64\). These are the possible rational roots.

Step 3: Test Possible Rational Roots

We test these roots, one by one, using substitution to check if they satisfy the equation. Use synthetic division or direct substitution to evaluate if they are the roots of the equation.

Step 4: Verify Root Using \(x = 4\)

Substituting \(x = 4\) into the equation:\(4^3 - 12 \times 4^2 + 64 = 0\)Simplify:\(64 - 192 + 64 = 0\)Thus, \(x = 4\) is indeed a root.

Step 5: Determine If It is the Least Positive Root

Since 4 satisfies the inequality \(x < 10\) and is positive, it is also the least positive root of the equation.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a helpful tool when dealing with polynomial equations, especially cubic equations like the one in this problem: \(x^3 - 12x^2 + 64 = 0\). The theorem provides a list of possible rational roots to test, saving time and effort. It states that any rational solution \(p/q\) of a polynomial equation with integer coefficients must have \(p\) as a factor of the constant term and \(q\) as a factor of the leading coefficient.

For this cubic equation, the constant term is 64, and since our leading coefficient is 1, \(q\) will always be ±1. Therefore, the possible rational roots are the factors of 64: \( \pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm 32, \pm 64\).

We test these potential roots by substituting them into the polynomial. If substituting a potential root results in a zero for the equation, then it is indeed a rational root. This step drastically narrows down the possibilities, simplifying the process of solving the cubic equation.

Synthetic Division

Synthetic division is a simplified form of polynomial division used to test potential roots derived from the Rational Root Theorem. It is quicker and less prone to errors than traditional long division for this purpose.

In synthetic division, we set up the coefficients of the polynomial—here, \(1, -12,\) and \(64\)—and perform operations using a potential root. The goal is to perform the division in such a way that a zero remainder indicates the test number is a root. If the potential root is indeed a root, synthetic division will yield a remainder of zero.

To test a number, we first write the zero remainder polynomial without the variable terms, followed by synthetic division with suspect roots until we find one that returns zero, indicating the polynomial can be divided evenly by \(x - \text{root}\). This process, when performed successfully, reveals one root without evaluating the entire polynomial traditionally.

Verification of Roots

Verification is a crucial step in confirming that a found solution is accurate. After testing potential roots from the Rational Root Theorem using synthetic division, we must check if such a root satisfies the original cubic equation. For example, when \(x = 4\) is identified as a potential root, we substitute it back into the equation:

• Calculate \(4^3 = 64\)
• Calculate \(-12 \times 4^2 = -192\)
• Add 64 to the result
• If the calculation ends up being zero, \(4\) is verified as a true root

In this example, substituting \(x = 4\) results in \(64 - 192 + 64 = 0\), confirming \(x = 4\) is a root of the equation. This verification assures us that \(x = 4\) is not only a viable root but also the least positive solution less than 10, indicating the depth to which the spherical object will sink.