Question

If \(y\) is directly proportional to the \(\frac{3}{2}\) power of \(x\) and inversely proportional to \(w\) and \(y\) is 284 when \(x\) is 858 and \(w\) is \(361,\) evaluate the constant of proportional. ity, and write the complete equation for \(y\) in terms of \(x\) and \(w\).

Short answer

The constant of proportionality \(k\) is found using the equation \(284 = k \cdot \frac{858^{\frac{3}{2}}}{361}\), and the complete equation for \(y\) is \(y = k \cdot \frac{x^{\frac{3}{2}}}{w}\).

Step-by-step solution

Understanding the Proportionality

Directly proportional to the \(\frac{3}{2}\) power of \(x\) and inversely proportional to \(w\) means that \(y = k \cdot \frac{x^{\frac{3}{2}}}{w}\), where \(k\) is the constant of proportionality.

Substitute Given Values

Substitute \(y = 284\), \(x = 858\), and \(w = 361\) into the equation to find \(k\): \(284 = k \cdot \frac{858^{\frac{3}{2}}}{361}\).

Calculate the Power of \(x\)

Calculate \(858^{\frac{3}{2}}\): It's the square root of \(858\) raised to the third power. First compute \(\sqrt{858}\) then raise the result to the power of 3.

Solve for \(k\)

Once \(858^{\frac{3}{2}}\) is calculated, solve the equation \(284 = k \cdot \frac{858^{\frac{3}{2}}}{361}\) for \(k\) by multiplying both sides by \(361\) and then dividing by \(858^{\frac{3}{2}}\).

Write the Complete Equation for \(y\)

After finding \(k\), the complete equation for \(y\) is therefore \(y = k \cdot \frac{x^{\frac{3}{2}}}{w}\) with the calculated value of \(k\).

Key concepts

Constant of Proportionality

In mathematics, understanding the constant of proportionality is key to mastering proportional relationships. When we say that one variable is directly proportional to another, it means that as one variable increases or decreases, the other does as well, maintaining a constant ratio. This ratio is known as the constant of proportionality and is often denoted by the symbol \( k \).

Taking the given exercise as an example, we recognize that \( y \) is directly proportional to the \( \frac{3}{2} \) power of \( x \) and also inversely proportional to \( w \), leading to the formula \( y = k \cdot \frac{x^{\frac{3}{2}}}{w} \). To evaluate the constant of proportionality \( k \), we use the provided values: \( y = 284 \), \( x = 858 \), and \( w = 361 \). Substituting these into the equation, we can solve for \( k \), thus establishing a concrete relationship between \( y \), \( x \), and \( w \) where \( k \) acts as the scaling factor that quantifies the relationship.

Direct and Inverse Proportionality

Understanding Direct Proportionality

Direct proportionality occurs when an increase in one quantity leads to a proportional increase in another quantity. Mathematically, if we have two variables \( x \) and \( y \), and \( y \) is directly proportional to \( x \), we express this relationship as \( y = kx \), where \( k \) is the constant of proportionality.

Understanding Inverse Proportionality

In the realm of inverse proportionality, as one quantity increases, the other quantity decreases in such a way that their product remains constant. If \( y \) is inversely proportional to \( x \), it is represented as \( y = \frac{k}{x} \). In the original exercise, we deal with a combination of both: \( y \) is directly proportional to \( x \) raised to the power of \( \frac{3}{2} \), and inversely proportional to another variable \( w \). Combining these relationships leads to the formula for \( y \): \( y = k \cdot \frac{x^{\frac{3}{2}}}{w} \), illustrating how both types of proportionality can coexist within a single equation.

Exponents and Roots

Exponents and roots are fundamental concepts in mathematics, especially in understanding proportional relationships involving powers and roots.

Exponents are used to represent repeated multiplication of a number by itself. The expression \( x^n \) means '\( x \) multiplied by itself \( n \) times'. If the exponent is a fraction, it indicates a root; for example, \( x^{\frac{1}{n}} \) is the nth-root of \( x \). This is particularly relevant to our exercise, where \( x^{\frac{3}{2}} \) stands for the square root of \( x \) raised to the third power.

Understanding roots is just as crucial. The nth-root of a number \( x \) is a number which, when raised to the power of \( n \), gives \( x \). In the expression \( x^{\frac{3}{2}} \), we first find the square root of \( x \) and then raise this result to the third power. This operation is crucial in solving for the constant of proportionality when \( y \) is directly proportional to such a power of \( x \). Students should feel comfortable manipulating these expressions to effectively solve proportional relationship problems involving exponents and roots.