Finding the Positive Number: Sum of Number and its Cube

The question asks us to find a positive number such that the sum of the number and its cube is 1740. Let the positive number be represented by \(x\).

According to the problem statement, the relationship can be written as:

\(x + x^3 = 1740\)

We need to find the value of \(x\) that satisfies this equation. Since we are given options, we can test each positive number from the options to see which one fits the equation.

Testing the Given Options

Let's check each option by substituting the value into the equation \(x + x^3\).

• Option 1: \(x = 16\)
• Calculate the sum: \(16 + 16^3\)
• We know that \(16^3 = 16 \times 16 \times 16 = 256 \times 16 = 4096\).
• Sum = \(16 + 4096 = 4112\).
• This sum (4112) is not equal to 1740. So, 16 is not the correct number.

• Option 2: \(x = 14\)
• Calculate the sum: \(14 + 14^3\)
• We know that \(14^3 = 14 \times 14 \times 14 = 196 \times 14 = 2744\).
• Sum = \(14 + 2744 = 2758\).
• This sum (2758) is not equal to 1740. So, 14 is not the correct number.

• Option 3: \(x = 8\)
• Calculate the sum: \(8 + 8^3\)
• We know that \(8^3 = 8 \times 8 \times 8 = 64 \times 8 = 512\).
• Sum = \(8 + 512 = 520\).
• This sum (520) is not equal to 1740. So, 8 is not the correct number.

• Option 4: \(x = 12\)
• Calculate the sum: \(12 + 12^3\)
• We know that \(12^3 = 12 \times 12 \times 12 = 144 \times 12 = 1728\).
• Sum = \(12 + 1728 = 1740\).
• This sum (1740) is equal to the given value in the problem. So, 12 is the correct number.

Based on testing the options, the positive number that satisfies the condition is 12.

Detailed Calculation for the Correct Number (x = 12)

Let's verify the sum for \(x=12\):

\(x + x^3 = 12 + 12^3\)

First, calculate the cube of 12:

\(12^3 = 12 \times 12 \times 12\)

\(12 \times 12 = 144\)

\(144 \times 12 = 1728\)

Now, add the number itself to its cube:

\(12 + 1728 = 1740\)

The sum is indeed 1740, confirming that 12 is the correct positive number.

Conclusion

The positive number whose sum with its cube is 1740 is 12.

Revision Table: Checking Options

Additional Information: Solving Cubic Equations

The problem \(x^3 + x - 1740 = 0\) is a cubic equation. While testing options is a good strategy when multiple-choice answers are provided, solving general cubic equations can be more complex. There are formulas like Cardano's method for solving cubic equations, but they are quite involved.

For integer or rational roots, the Rational Root Theorem can sometimes help identify potential roots. If a rational root \(p/q\) exists, then \(p\) must be a divisor of the constant term (-1740) and \(q\) must be a divisor of the leading coefficient (1). In this case, since the leading coefficient is 1, any rational root must be an integer divisor of -1740.

The divisors of 1740 include \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \dots \). Since we are looking for a positive number, we would test positive divisors. As we saw, testing 12 worked, meaning \(x=12\) is a root of the equation \(x^3 + x - 1740 = 0\).

Once one root (like \(x=12\)) is found, we can perform polynomial division to divide \(x^3 + x - 1740\) by \((x - 12)\) to get a quadratic equation. The roots of the quadratic equation would give the other two roots of the cubic equation. However, for this specific problem and given options, the testing method is the most efficient.