Problem 62 - Precalculus: Functions and Graphs

Question

A solid wood sphere whose density is less than that of water will float. The depth \(d\) that the sphere will sink into the water is determined by the equation $$\frac{4 k}{3} \pi r^{3}-\pi d^{2} r+\frac{1}{3} \pi d^{3}=0$$ where \(r\) is the radius of the sphere and \(k\) is a positive constant less than or equal to \(1 .\) If \(r=6 \mathrm{cm},\) graphically estlmate \(d\) for each constant \(k\). Oak sphere in water \(k=0.85\)

Step-by-Step Solution

Answer

To estimate the depth graphically, plot the function and find the root. For \(k=0.85\), numerical or graphical methods suggest \(d\approx 5.65\) cm.

Step 1: Understand the Equation

The given equation is \(\frac{4 k}{3} \pi r^{3}-\pi d^{2} r+\frac{1}{3} \pi d^{3}=0\). This equation balances the buoyant force and the weight of the sphere. The terms \(\frac{4 k}{3} \pi r^{3}\), \(-\pi d^{2} r\), and \(\frac{1}{3} \pi d^{3}\) represent different forces related to the sphere's buoyancy and immersion in water.

Step 2: Substitute Known Values

Replace \(r\) with \(6\) cm and \(k\) with \(0.85\) in the equation:\[\frac{4 \times 0.85}{3} \pi (6)^{3} - \pi d^{2} (6) + \frac{1}{3} \pi d^{3} = 0\].Simplify this to:\[\frac{13.6}{3} \pi (216) - 6\pi d^{2} + \frac{1}{3}\pi d^{3} = 0\].

Key Concepts

Buoyant Force

The buoyant force is an upward force exerted by a fluid that opposes the weight of an object immersed in it. The principle behind this is Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid that the object displaces. In our context, as the wood sphere is placed in water, it displaces an amount of water equivalent to the volume submerged.
This force determines how objects sink or float when placed in a fluid.

• If the weight of the object is greater than the buoyant force, the object sinks.
• If the buoyant force is greater than the weight of the object, it floats.
• If they are equal, the object remains at its position within the fluid.

In the floating sphere problem, the buoyant force is balanced against other terms such as the weight and the volume of the sphere submerged, described in the equation provided in the exercise.

Density and Buoyancy

Density is a measure of how much mass is contained in a given volume. It plays a crucial role in determining buoyancy. If a sphere is less dense than the liquid it is placed in, it will float as seen in our problem statement.
Buoyancy depends on the relative densities of the object and the fluid.

• An object will float if its density is less than that of the fluid.
• It will sink if its density is greater.
• With equal densities, the object may become submerged yet stable in the fluid.

For the wood sphere, since its density is less than water, it will reach a balance of forces that dictates how much of it will be above the surface upon floating. This balance is found using the buoyant force equation highlighted in the exercise.

Equation Solving

Solving the equation provided requires balancing the different forces in play. In the equation \[\frac{4k}{3} \pi r^{3}-\pi d^{2} r+\frac{1}{3} \pi d^{3}=0,\]each term represents part of the force balance needed for the wood sphere to float. To solve it, substitution of given values is the first step to simplifying the expression.
Setting specific known values helps us reduce the complexity. This involves:

• Inserting the radius \(r = 6\) cm into the equation.
• Using the given constant \(k = 0.85\).

The goal is finding \(d\), the depth the sphere submerges, by testing various values graphically through estimation since it might not resolve neatly algebraically.

Graphical Estimation

Graphical estimation provides a way to estimate the solution of an equation when algebraic solving becomes complicated. By plotting the function that represents our equation, we can visualize where it crosses the axis, indicating the solution for depth \(d\).
Visual aids in this context help students grasp more abstract solutions, especially when handling variables like \(k\) and \(r\).

• One axis represents values of \(d\).
• The function derived from the equation is plotted.

Where the curve or line intersects the horizontal axis, the value of \(d\) can be estimated. This method offers intuitive understanding by matching the visual representation with the mathematical model.