Question

write a general formula to describe each variation. \(T\) varies directly with the cube root of \(x\) and the square of \(d ;\) \(T=18\) when \(x=8\) and \(d=3\)

Short answer

\(T = \sqrt[3]{x} \times d^2\)

Step-by-step solution

Understand the direct variation formula

Given that a quantity varies directly with other quantities, we use the formula: \[ T = k \times (\text{other quantities}) \] where\( k \)is a constant. In this case, we have T varies directly with the cube root of x and the square of d.

Write the direct variation formula

Since T varies directly with \ \ \( \sqrt[3]{x}\) and \(d^2\), we write the formula: \[ T = k \times \sqrt[3]{x} \times d^2 \]

Substitute the given values to find k

We are given that \( T = 18\) when \( x = 8\) and \(d = 3\). Substitute these values into the equation: \[ 18 = k \times \sqrt[3]{8} \times 3^2 \]

Calculate the cube root and square

First, calculate \( \sqrt[3]{8} \): \[ \sqrt[3]{8} = 2 \] Then calculate \(3^2 \): \[ 3^2 = 9 \]

Solve for k

Substitute these values back into the equation \[ 18 = k \times 2 \times 9 \] Simplify and solve for \( k \): \[ 18 = 18k \] \[ k = 1 \]

Write the general formula

Now that we know \( k = 1\), we can write the general formula for T: \[ T = \sqrt[3]{x} \times d^2 \]

Key concepts

Understanding Direct Variation

In mathematics, direct variation describes a relationship where one quantity directly depends on another. This means if one quantity increases, the other does as well, and vice versa. The formula for direct variation is generally written as:
\[ T = k \times (\text{other quantities}) \]
here,\( k \)is a constant known as the constant of variation. In our exercise, \( T \)varies directly with the cube root of \( x \)and the square of \( d \). Therefore, the direct variation formula becomes:
\[ T = k \times \sqrt[3]{x} \times d^2 \]
This means that the value of \( T \)is proportional to the product of the cube root of \( x \)and the square of \( d \). To find the specific relationship, we need to determine the value of \( k \), which will make the formula match given conditions.

Calculating Cube Roots

Finding the cube root of a number means finding a value that, when multiplied by itself three times, gives the original number. For example, to find the cube root of 8, we ask:

• What number multiplied by itself three times equals 8?

The answer is 2, because \( 2 \times 2 \times 2 = 8 \). Therefore, \( \sqrt[3]{8} = 2 \). Understanding cube roots is essential in problems involving direct variation where one of the variables is a cube root. For example, in our equation, if \( x = 8 \), then \( \sqrt[3]{8} = 2 \). This simplifies our calculations when applying the direct variation formula.

Working with Squares

The square of a number involves multiplying the number by itself. It answers the question:

• What do you get when you multiply a number by itself?

For example, the square of 3 is: \[ 3^2 = 3 \times 3 = 9 \].
Squaring is straightforward but crucial in problems dealing with direct variation where a variable is squared. In the given task, \( d \)is squared in the formula. So, if \( d = 3 \), then \( d^2 = 9 \). Plugging these simplified values into the variation formula allows for easy computation.
In the exercise, substituting the given values and simplifying:
\[ T = k \times \sqrt[3]{8} \times 3^2 \]which becomes \[ T = k \times 2 \times 9 \] helps us find \( k \), the constant of variation, and thus write the final direct variation formula.