The cube root of x varies inversely as the square root of y. x = 8 when y = 3. What is the value of x when y = $\sqrt[3]{3} $ ?

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Prepp Team · Accepted Answer

Understanding Inverse Variation

The problem describes a relationship between two variables, x and y, based on inverse variation. Specifically, it states that the cube root of x varies inversely as the square root of y.

In mathematical terms, inverse variation means that the product of the related quantities is constant. If the cube root of x, denoted as $\(\sqrt[3]{x}$\), varies inversely as the square root of y, denoted as $\(\sqrt{y}$\), we can write this relationship as:

$\(\sqrt[3]{x} \propto \frac{1}{\sqrt{y}}$\)

To turn this proportionality into an equation, we introduce a constant of proportionality, let's call it k:

$\(\sqrt[3]{x} = \frac{k}{\sqrt{y}}$\)

This equation can be rearranged to find the constant k:

$\(k = \sqrt[3]{x} \cdot \sqrt{y}$\)

Finding the Constant of Proportionality

We are given an initial condition: x = 8 when y = 3. We can use these values to find the specific constant of proportionality (k) for this relationship.

Substitute x = 8 and y = 3 into the equation $\(k = \sqrt[3]{x} \cdot \sqrt{y}$\):

$\(k = \sqrt[3]{8} \cdot \sqrt{3}$\)

Calculate the cube root of 8:

$\(\sqrt[3]{8} = 2$\)

So, the equation becomes:

$\(k = 2 \cdot \sqrt{3}$\)

The constant of proportionality is $\(k = 2\sqrt{3}$\).

Calculating x for the New Value of y

Now we need to find the value of x when y = $\(\sqrt[3]{3}$\). We use the original variation equation with the constant k we just found:

$\(\sqrt[3]{x} = \frac{k}{\sqrt{y}}$\)

Substitute the value of k ($\(2\sqrt{3}$\)) and the new value of y ($\(\sqrt[3]{3}$\)) into the equation:

$\(\sqrt[3]{x} = \frac{2\sqrt{3}}{\sqrt{\sqrt[3]{3}}}$\)

Let's simplify the denominator. $\(\sqrt{\sqrt[3]{3}}$\) means the square root of the cube root of 3. We can write this using exponents:

$\(\sqrt[3]{3} = 3^{1/3}$\)

$\(\sqrt{\sqrt[3]{3}} = \sqrt{3^{1/3}} = (3^{1/3})^{1/2}$\)

Using the rule of exponents $\((a^m)^n = a^{mn}$\):

$\((3^{1/3})^{1/2} = 3^{(1/3) \cdot (1/2)} = 3^{1/6}$\)

So the equation becomes:

$\(\sqrt[3]{x} = \frac{2\sqrt{3}}{3^{1/6}}$\)

Now, let's express $\(\sqrt{3}$\) with an exponent:

$\(\sqrt{3} = 3^{1/2}$\)

The equation is now:

$\(\sqrt[3]{x} = \frac{2 \cdot 3^{1/2}}{3^{1/6}}$\)

Using the rule of exponents $\(\frac{a^m}{a^n} = a^{m-n}$\) for the terms with base 3:

$\(\frac{3^{1/2}}{3^{1/6}} = 3^{1/2 - 1/6}$\)

Find a common denominator for the exponents:

$\(1/2 - 1/6 = 3/6 - 1/6 = 2/6 = 1/3$\)

So, the equation is:

$\(\sqrt[3]{x} = 2 \cdot 3^{1/3}$\)

To find x, we need to cube both sides of the equation:

$\((\sqrt[3]{x})^3 = (2 \cdot 3^{1/3})^3$\)

Using the rule $\((ab)^n = a^n b^n$\):

$\(x = 2^3 \cdot (3^{1/3})^3$\)

Calculate the powers:

$\(2^3 = 8$\)

$\((3^{1/3})^3 = 3^{(1/3) \cdot 3} = 3^1 = 3$\)

Substitute these values back:

$\(x = 8 \cdot 3$\)

$\(x = 24$\)

Thus, the value of x when y = $\(\sqrt[3]{3}$\) is 24.

Revision Table: Inverse Variation Summary

Concept	Description	Equation
Inverse Variation	One variable increases as the other decreases proportionally.	$\(y = k/x$\) or $\(xy = k$\)
Cube Root of x	The number that, when multiplied by itself three times, equals x.	$\(\sqrt[3]{x}$\) or $\(x^{1/3}$\)
Square Root of y	The number that, when multiplied by itself, equals y.	$\(\sqrt{y}$\) or $\(y^{1/2}$\)
Constant of Proportionality (k)	The non-zero constant relating the variables in variation.	Found using known values of variables.

Additional Information: Types of Variation

Understanding different types of variation is key to solving such problems.

Direct Variation: When one variable increases, the other increases proportionally. The relationship is $\(y = kx$\) or $\(\frac{y}{x} = k$\). The graph is a straight line through the origin.
Inverse Variation: When one variable increases, the other decreases proportionally. The relationship is $\(y = \frac{k}{x}$\) or $\(xy = k$\). The graph is a hyperbola.
Joint Variation: When a variable varies directly as the product of two or more other variables. For example, if z varies jointly as x and y, the relationship is $\(z = kxy$\).

In this problem, we combined the concepts of inverse variation with roots (cube root and square root) to establish the relationship $\(\sqrt[3]{x} \cdot \sqrt{y} = k$\). Solving variation problems typically involves using a given set of values to find the constant of variation (k) and then using k with new values to find the unknown variable.

Prepp Team · Answer

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Understanding Inverse Variation

Finding the Constant of Proportionality

Calculating x for the New Value of y

Revision Table: Inverse Variation Summary

Additional Information: Types of Variation

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Concept	Description	Equation
Inverse Variation	One variable increases as the other decreases proportionally.	\(\(y = k/x\)\) or \(\(xy = k\)\)
Cube Root of x	The number that, when multiplied by itself three times, equals x.	\(\(\sqrt[3]{x}\)\) or \(\(x^{1/3}\)\)
Square Root of y	The number that, when multiplied by itself, equals y.	\(\(\sqrt{y}\)\) or \(\(y^{1/2}\)\)
Constant of Proportionality (k)	The non-zero constant relating the variables in variation.	Found using known values of variables.