Understanding the Mathematical Problem
The question asks us to find a number, let's call it X, such that when this number X is added to the double of its cube, the result is 1467. We can translate this word problem into a mathematical equation.
The cube of the number X is \(X^3\).
The double of its cube is \(2 \times X^3\), which is \(2X^3\).
When the number X is added to the double of its cube, we get \(X + 2X^3\).
According to the problem, this sum is equal to 1467.
So, the equation we need to solve is: \(X + 2X^3 = 1467\).
Solving for the Value of X
We are given a set of options for the value of X. The most straightforward way to find the correct value is to test each option in the equation \(X + 2X^3 = 1467\) and see which one satisfies it.
| Value of X (Option) | Calculate \(X + 2X^3\) | Result | Matches 1467? |
|---|---|---|---|
| 7 | \(7 + 2 \times 7^3 = 7 + 2 \times 343\) | \(7 + 686 = 693\) | No |
| 9 | \(9 + 2 \times 9^3 = 9 + 2 \times 729\) | \(9 + 1458 = 1467\) | Yes |
| 6 | \(6 + 2 \times 6^3 = 6 + 2 \times 216\) | \(6 + 432 = 438\) | No |
| 8 | \(8 + 2 \times 8^3 = 8 + 2 \times 512\) | \(8 + 1024 = 1032\) | No |
Identifying the Correct Value of X
From the table above, we can see that when we substitute \(X = 9\) into the expression \(X + 2X^3\), the result is 1467. None of the other options (7, 6, or 8) produce the correct value.
Therefore, the value of X that satisfies the given condition is 9.
Revision Table: Key Steps to Finding Number X
| Step | Description | Action for this Problem |
|---|---|---|
| 1 | Understand the problem statement. | Identify the relationship between the number X, its cube, and the total value (1467). |
| 2 | Formulate the equation. | Translate the words into the equation: \(X + 2X^3 = 1467\). |
| 3 | Use the given options. | Substitute each option value for X into the equation. |
| 4 | Evaluate the expression. | Calculate \(X + 2X^3\) for each option. |
| 5 | Check the result. | Compare the calculated result with 1467 to find the value of X that matches. |
Additional Information: Solving Cubic Equations
The equation \(X + 2X^3 = 1467\) is a type of cubic equation because the highest power of X is 3. Cubic equations can be written in the general form \(aX^3 + bX^2 + cX + d = 0\).
In our case, the equation is \(2X^3 + X - 1467 = 0\). Here, \(a=2\), \(b=0\), \(c=1\), and \(d=-1467\).
Solving cubic equations analytically can be complex. However, when dealing with integer solutions, especially in multiple-choice questions, testing the given options is often the quickest and easiest method. This approach avoids complex algebraic manipulations or numerical methods.
If options were not provided, one might look for integer roots by considering the divisors of the constant term (-1467) divided by the divisors of the leading coefficient (2). However, this process is more involved than simply testing the small integer options given in this problem.
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