Problem 24
Question
Suppose \(y\) varies jointly as \(a\) and the square root of \(b\) If \(y=42\) when \(a=3\) and \(b=49\) a) find the constant of variation. b) write the specific variation equation relating \(y, a\) and \(b\) c) find \(y\) when \(a=4\) and \(b=9\)
Step-by-Step Solution
Verified Answer
a) The constant of variation is 2.
b) The specific variation equation relating y, a, and b is \(y = 2a\sqrt{b}\).
c) When a = 4 and b = 9, the value of y is 24.
1Step 1: Understand the joint variation
In a joint variation, a variable is directly proportional to the product of two or more other variables. In this case, y varies jointly with a and the square root of b, so we can write the equation as:
\(y = k \cdot a \cdot \sqrt{b}\)
Here, k is the constant of variation.
2Step 2: Find the constant of variation (k)
We are given that y = 42, a = 3, and b = 49. We can plug these values into the joint variation equation to find the constant of variation, k:
\(42 = k \cdot 3 \cdot \sqrt{49}\)
Solve for k:
\(k = \frac{42}{3 \cdot 7}\)
\(k = 2\)
3Step 3: Write the specific variation equation
Now that we've found the constant of variation, k = 2, we can write the specific variation equation relating y, a, and b:
\(y = 2a\sqrt{b}\)
4Step 4: Find y when a = 4 and b = 9
Using the specific variation equation we found in step 3, we can find the value of y when a = 4 and b = 9:
\(y = 2(4)\sqrt{9}\)
\(y = 8 \cdot 3\)
\(y = 24\)
In conclusion,
a) The constant of variation is 2.
b) The specific variation equation relating y, a, and b is \(y = 2a\sqrt{b}\).
c) When a = 4 and b = 9, the value of y is 24.
Key Concepts
Constant of VariationDirect ProportionalityVariation Equation
Constant of Variation
In joint variation problems, the constant of variation, denoted usually as \( k \), is a crucial component. This constant helps link the dependent variable (like \( y \) in our problem) with two or more independent variables (such as \( a \) and \( b \)). Think of \( k \) as a "scaling factor" that adjusts the output of the relationship to fit specific given conditions.
For example, in our exercise, we understood that \( y \) is dependent on the product of \( a \) and \( \sqrt{b} \). When we calculated \( k \), we were using the known values of \( y \), \( a \), and \( b \) to find out how they are scaled together. We plugged these values into the equation to isolate and solve for \( k \), leading to a better understanding of their direct relationship. Understanding the constant of variation allows us to confidently predict what \( y \) would be for different values of \( a \) and \( b \).
For example, in our exercise, we understood that \( y \) is dependent on the product of \( a \) and \( \sqrt{b} \). When we calculated \( k \), we were using the known values of \( y \), \( a \), and \( b \) to find out how they are scaled together. We plugged these values into the equation to isolate and solve for \( k \), leading to a better understanding of their direct relationship. Understanding the constant of variation allows us to confidently predict what \( y \) would be for different values of \( a \) and \( b \).
- \( y = k \cdot a \cdot \sqrt{b} \)
- \( k = 2 \) was found using known values \( (a = 3, b = 49, y = 42) \)
Direct Proportionality
When we say a variable is directly proportional to another, it means that as one increases, the other also increases at a consistent rate, defined by a constant. In joint variation, we blend the idea of direct proportionality but extend it to multiple variables.
Our exercise illustrated this with \( y \), which directly varies with both \( a \) and \( \sqrt{b} \). Direct proportionality can be visualized in the variation equation as each independent variable impacting \( y \) by multiplication.
This means if one of the variables \( a \) or \( \sqrt{b} \) increases, \( y \) will increase proportionately based on the constant \( k = 2 \). The understanding here is that all parts are multiplicatively linked; change any part and the whole outcome changes in a directly proportional manner.
Our exercise illustrated this with \( y \), which directly varies with both \( a \) and \( \sqrt{b} \). Direct proportionality can be visualized in the variation equation as each independent variable impacting \( y \) by multiplication.
This means if one of the variables \( a \) or \( \sqrt{b} \) increases, \( y \) will increase proportionately based on the constant \( k = 2 \). The understanding here is that all parts are multiplicatively linked; change any part and the whole outcome changes in a directly proportional manner.
- \( y \) is directly proportional to both \( a \) and \( \sqrt{b} \)
- Any increase in \( a \) or \( \sqrt{b} \) results in an increase in \( y \)
Variation Equation
A variation equation captures the relationship described by joint variation. It expresses how a dependent variable changes with respect to one or more independent variables. Just as in direct proportionality, the equation comes with a constant of variation, \( k \), which is the key to balancing the equation.
In our problem, the variation equation is \( y = 2a\sqrt{b} \). This specific equation means that for any new values of \( a \) and \( b \), you can find \( y \) by substituting these values into your variation equation. It's the mathematical blueprint to predict the relationship outcomes under different circumstances.
For instance, when \( a = 4 \) and \( b = 9 \), by substiting into the equation, we computed \( y = 24 \), illustrating how these input values influence the outcome. The structured format of a variation equation simplifies problem-solving in complex relationships.
In our problem, the variation equation is \( y = 2a\sqrt{b} \). This specific equation means that for any new values of \( a \) and \( b \), you can find \( y \) by substituting these values into your variation equation. It's the mathematical blueprint to predict the relationship outcomes under different circumstances.
For instance, when \( a = 4 \) and \( b = 9 \), by substiting into the equation, we computed \( y = 24 \), illustrating how these input values influence the outcome. The structured format of a variation equation simplifies problem-solving in complex relationships.
- Example of use: Substituting \( a = 4 \) and \( b = 9 \) yields \( y = 24 \)
- Variable changes directly reflected by plugging values into the equation
Other exercises in this chapter
Problem 24
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