Problem 66 - Intermediate Algebra

Question

Fluids. See the illustration on the right. Two glass plates in contact at the left, and separated by about 5 millimeters on the right, are dipped in beet juice, which rises by capillary action to form a hyperbola. The hyperbola is modeled by an equation of the form \(x y=k\). If the curve passes through the point \((12,2),\) what is \(k ?\)

Step-by-Step Solution

Answer

The value of \( k \) is 24.

Step 1: Understand the Formula

We have the equation of a curve that takes the form of a hyperbola: \( x y = k \). Here, \( x \) and \( y \) are coordinates of points on the curve, and \( k \) is a constant that we need to find.

Step 2: Substitute Known Values

We are given that the point \((12, 2)\) lies on the hyperbola described by the equation \(x y = k\). Substitute \( x = 12 \) and \( y = 2 \) into the equation: \( 12 \times 2 = k \).

Step 3: Solve for k

Calculate the value of \( k \) by multiplying the substituted values. Thus, \( k = 12 \times 2 = 24 \).

Key Concepts

Capillary Action

Capillary action is a fascinating phenomenon that explains why liquids can move through narrow spaces without the assistance of external forces, like gravity. This happens due to the balance between adhesive and cohesive forces. Adhesive forces are the attraction between the liquid molecules and the solid surface of the tube or space they are moving through. Cohesive forces are the attraction between the liquid's own molecules.

When the adhesive forces are stronger than the cohesive forces, the liquid gets pulled into the narrow space, causing it to rise. In the context of the exercise, beet juice rises between two glass plates due to capillary action. This forms a specific shape because of the difference in plate separation, resulting in a hyperbola.

• Capillary action is essential in understanding the movement of fluids in areas such as plant biology, where it helps transport water from roots to leaves.
• This principle has practical applications in medical diagnostics and inkjet printing, among others.

Equation Modeling

Equation modeling is an important skill used in algebra and mathematics to create equations that represent real-world situations. This involves understanding how formulas relate to physical phenomena, and in our case, how they describe shapes like hyperbolas. A hyperbola is a type of smooth curve on a graph, defined by the equation \(x y = k\), where \(k\) is a constant.

In the given problem, modeling is applied to capillary action, using the hyperbola equation to describe the precise path that the beet juice takes as it rises between the glass plates. By knowing the coordinates of a point on the hyperbola, we can solve for the constant \(k\).

• Understanding how to model with equations allows us to solve complex problems by breaking them down into manageable parts.
• This approach is widely used in engineering, physics, and economics for predicting behaviors and outcomes.

Problem-Solving in Algebra

Problem-solving in algebra involves developing strategies to find unknown values using given information. The exercise requires you to understand the concept of substitution to find the value of \(k\) in the equation of a hyperbola.

Algebraic problem-solving typically includes:

• Identifying the type of problem or equation you are dealing with.
• Substituting known values into the equation.
• Solving for the unknown variable.

In this specific problem, you substitute the given point \((12, 2)\) into the hyperbola equation \(x y = k\). This means multiplying \(x\) by \(y\), leading to the calculation of \(k\). Hence, algebraic techniques let you find unknowns even in complex scenarios.