Problem 22 - Algebra and Trigonometry

Question

13–22 ? Express the statement as an equation. Use the given information to find the constant of proportionality. \(M\) is jointly proportional to \(a, b,\) and \(c,\) and inversely proportional to \(d .\) If \(a\) and \(d\) have the same value, and if \(b\) and \(c\) are both \(2,\) then \(M=128 .\)

Step-by-Step Solution

Answer

The constant of proportionality is \(k = 32\). The equation is \(M = 32 \frac{abc}{d}\).

Step 1: Understand the Proportionality Statement

The statement says that \(M\) is jointly proportional to \(a, b,\) and \(c,\) meaning \(M\) increases with the product of \(a, b,\) and \(c\). Additionally, \(M\) is inversely proportional to \(d\), meaning \(M\) decreases as \(d\) increases.

Step 2: Express the Proportionality as an Equation

From the understanding in Step 1, the equation can be stated as \( M = k \frac{abc}{d} \), where \(k\) is the constant of proportionality. Here \(abc\) is the joint proportionality and \(d\) is the inverse proportionality.

Step 3: Substitute Given Values into the Equation

According to the problem, \(a = d\) and \(b = c = 2\). Substitute these values into the equation \( M = k \frac{abc}{d} \). Therefore, it becomes \( M = k \frac{a \cdot 2 \cdot 2}{a} \). Simplifying, we have \( M = 4k \).

Step 4: Substitute the Given Value of \(M\) to Find \(k\)

We know \(M = 128\). Substitute this into \(4k = 128\) and solve for \(k\). This gives \(k = \frac{128}{4} = 32\).

Step 5: Write the Final Expression for \(M\)

Now that we know \(k = 32\), substitute this back into the equation for \(M\). The final expression is \(M = 32 \frac{abc}{d}\).

Key Concepts

Constant of Proportionality

The constant of proportionality is a crucial concept in mathematics that links two variables with a constant ratio. In our exercise, we found that the constant of proportionality is the value 'k' in the equation. It defines the relationship between variables that vary jointly or inversely.

In the equation for joint proportionality, the constant 'k' scales the direct relationship between the variables. Check this explanation:

• When variables move in the same direction—both increase or decrease—'k' scales the product of the variables. In our example, this scaling factor is influenced by three variables: 'a', 'b', and 'c'.
• The proportionality constant helps us express 'M' as a multiple of these products, balancing the effect of both direct and inverse proportionalities.

Once 'k' is determined, you can predict one variable when others are known, maintaining the mathematical balance in proportional relationships.

Inverse Proportionality

Inverse proportionality is a key mathematical relationship where one variable increases while another decreases. In other words, they are in a reciprocal relationship. This is shown in equations such as: \( y \cdot x = k \).

In our problem, we used this concept by observing how 'M' changes as 'd' changes:

• Inverse proportion implies that with every increase in 'd', 'M' would decrease if 'abc' remained constant.
• The use of inverse proportionality in our solution helps derive the value of 'M' accurately as 'd' counteracts the joint effect of 'a', 'b', and 'c'.

Understanding inverse proportionality allows us to adapt situations like this into clear mathematical expressions, illustrating complex relationships as simple equations.

Proportional Relationships

Proportional relationships are fundamental in math, providing a simple way to relate two quantities. A proportional relationship can be recognized by a constant ratio or rate, which in equations, appears as: \( y = kx \).

In our exercise, proportional relationships played out as follows:

• 'M' was jointly proportional to 'a', 'b', and 'c', meaning that multiplying these values together scaled 'M'.
• Meanwhile, the relationship was adjusted by 'd', which inversely affected 'M'. This produced a unified equation tying all proportions together.

These relationships are powerful because they simplify complex variations into workable mathematical forms and allow for precise predictions where variable changes and outcomes are reliably connected.