Problem 26 - Precalculus Mathematics for Calculus

Question

Express the statement as an equation. Use the given information to find the constant of proportionality. \(H\) is jointly proportional to the squares of \(l\) and \(w .\) If \(l=2\) and \(w=\frac{1}{3},\) then \(H=36\).

Step-by-Step Solution

Answer

The constant of proportionality is 81.

Step 1: Understand Joint Proportionality

When a quantity is jointly proportional to two or more other quantities, it means the quantity is proportional to the product of those quantities. Here, the statement 'H is jointly proportional to the squares of l and w' means that \( H = k \times l^2 \times w^2 \) where \( k \) is the constant of proportionality.

Step 2: Formulate the Equation

Using the relationship from step 1, express the relationship as an equation: \( H = k \cdot l^2 \cdot w^2 \). This shows that \( H \) depends on the square of \( l \) and the square of \( w \) with some constant \( k \).

Step 3: Substitute Given Values

Substitute \( H = 36 \), \( l = 2 \), and \( w = \frac{1}{3} \) into the equation: \( 36 = k \cdot 2^2 \cdot \left( \frac{1}{3} \right)^2 \).

Step 4: Simplify the Squares

Calculate the squares: \( 2^2 = 4 \) and \( \left( \frac{1}{3} \right)^2 = \frac{1}{9} \). Substitute back into the equation to get \( 36 = k \cdot 4 \cdot \frac{1}{9} \).

Step 5: Calculate the Constant of Proportionality

Simplify the right side of the equation: \( 4 \cdot \frac{1}{9} = \frac{4}{9} \). Thus, our equation becomes \( 36 = k \cdot \frac{4}{9} \). To solve for \( k \), multiply both sides by \( \frac{9}{4} \): \( k = 36 \cdot \frac{9}{4} \).

Step 6: Final Calculation

Perform the multiplication: \( 36 \cdot \frac{9}{4} = 81 \). Thus, the constant of proportionality \( k = 81 \).

Key Concepts

Constant of Proportionality

When working with proportional relationships, a crucial element is the constant of proportionality. This constant, often represented by the letter \( k \), helps define the exact relationship between variables that are deemed to be proportional. In simpler terms, it's the number you multiply by one variable to get the other.For example, if \( y \) is directly proportional to \( x \), the equation would look like this: \( y = kx \). In joint proportionality, as seen in the exercise, the quantity is proportional to the product of multiple variables, such as \( H = k \times l^2 \times w^2 \). Here, \( k \) isn't just some random number—it's the bridge between the variables.Finding this constant involves substituting given values into the equation and simplifying to solve for \( k \). Remember, \( k \) remains constant no matter the specific values of the variables, unless the relationship itself changes.

Mathematical Expressions

Mathematical expressions are like the language of math. They convey relationships using numbers, variables, and mathematical symbols. For the concept of joint proportionality, expressions are used to succinctly describe how one quantity depends on others.In our exercise, the expression says \( H = k \cdot l^2 \cdot w^2 \). This equation packs a lot of information:

• \( H \) is the quantity that depends on other variables.
• \( k \) is the constant that scales the relationship.
• \( l^2 \) and \( w^2 \) indicate that \( H \) is dependent on the square of \( l \) and \( w \).

Formulating such expressions requires understanding the underlying mathematical relationships. When given values are inserted, like \( l = 2 \) and \( w = \frac{1}{3} \), this solidifies the expression by enabling the calculation of unknowns, like the constant \( k \). Understanding how to write and manipulate these expressions is key to solving many mathematical problems involving proportionality.

Proportional Relationships

Proportional relationships describe how quantities increase or decrease together. When one variable changes, another variable changes in response at a constant rate, defined by the constant of proportionality.Let's consider simple direct proportionality: if doubling \( x \) doubles \( y \), they're proportionally related by some constant \( k \). Joint proportionality adds a twist; it involves more than one variable influencing the outcome simultaneously.The exercise dealt with joint proportionality where \( H \) depends on the squares of \( l \) and \( w \). This means:

• As \( l \) or \( w \) increase, \( H \) changes by the square of these increments.
• The relationship doesn’t just have \( l \) added to \( w \); it multiplies the effect each variable has due to their contributions being squared.

These relationships are vital for understanding various scientific and economic phenomena where outcomes are influenced by multiple variables working together. Grasping joint proportionality enables students to tackle broader and more complex real-world problems.