Problem 30 - Intermediate Algebra

Question

If \(V\) varies jointly as \(B\) and \(h\), and \(V=51\) when \(B=17\) and \(h=9\), find \(V\) when \(B=19\) and \(h=12\).

Step-by-Step Solution

Answer

The new volume is 76.

Step 1: Understand the Relationship

Since we are told that volume \(V\) varies jointly as base \(B\) and height \(h\), we can express this relationship with the equation \( V = k \, B \, h \), where \( k \) is a constant of proportionality.

Step 2: Substitute Known Values to Find \(k\)

Given \( V = 51 \), \( B = 17 \), and \( h = 9 \), substitute these values into the equation: \( 51 = k \, (17) \, (9) \). Simplifying this, we find \( k = \frac{51}{153} = \frac{1}{3} \).

Step 3: Substitute Values for the New Situation

Using the constant \( k = \frac{1}{3} \), find \( V \) when \( B = 19 \) and \( h = 12 \): \( V = \frac{1}{3} \, (19) \, (12) \).

Step 4: Calculate the New Volume

Calculate \( V = \frac{1}{3} \, (19) \, (12) = \frac{1}{3} \, (228) = 76 \). Thus, the new volume is 76.

Key Concepts

Constant of Proportionality

The constant of proportionality is a crucial concept in joint variation. It serves as a link between variables that vary together. In our problem, we see that the volume \( V \) varies jointly as the base \( B \) and height \( h \). This means that \( V \) can be expressed in terms of \( B \) and \( h \) using the equation \( V = k \times B \times h \). Here, \( k \) is the constant of proportionality and determines how strongly \( V \) depends on \( B \) and \( h \).
To find \( k \), we use specific values of \( V \), \( B \), and \( h \) given in the problem. By substituting \( V = 51 \), \( B = 17 \), and \( h = 9 \) into the equation, we solve for \( k \):

• \( 51 = k \times 17 \times 9 \)
• \( k = \frac{51}{153} = \frac{1}{3} \)

With this constant in hand, we can predict the value of \( V \) for other \( B \) and \( h \) values. This ability to find and use a constant of proportionality makes solving joint variation problems systematic and straightforward.

Algebraic Equations

Algebraic equations form the backbone of solving problems involving joint variation. They allow us to relate multiple variables in a precise mathematical way. In the exercise, the relationship between \( V \), \( B \), and \( h \) is captured by the equation \( V = k \times B \times h \). This equation shows that \( V \) is directly influenced by both \( B \) and \( h \) via the constant \( k \).
Because \( V \) depends on both factors, adjusting either \( B \) or \( h \) will change \( V \). This interaction is the essence of joint variation. To solve these equations, we:

• Apply known values to find unknown constants.
• Substitute new values to calculate desired outcomes.

In our solution, we initially use the equation to find \( k \). Once \( k \) is known, it becomes simple to substitute different \( B \) and \( h \) values to determine a new \( V \). Algebraic equations thus provide a structured means to explore relationships between varying quantities.

Direct Variation

Direct variation is a specific type of relationship where one variable changes directly with another. In joint variation, we extend this idea to multiple variables. The exercise demonstrates how volume \( V \) varies directly with both base \( B \) and height \( h \).
In direct variation, as one variable increases or decreases, the other variable does the same, assuming a constant proportionality. For example, if \( B \) or \( h \) increases, \( V \) should also increase, given that all other factors remain the same.
The joint variation formula \( V = k \times B \times h \) embodies this direct relationship for all involved variables. In problems of this nature:

• Identify and understand how variables are linked.
• Use known values to deduce the constant of proportionality \( k \).
• Predict outcomes for different sets of values, maintaining consistency with the direct proportion concept.

Understanding direct variation, and by extension, joint variation, helps in thoroughly grasping how changes in one variable impact others in real-world scenarios.