If three identical cubes have 3x^2+18x^2+36x+24 find a possible length of one side of one of the cubes
Hint:
If you check your question before you post, you may save yourself and your tutor a lot of time.
1. Is the expression 3x^2+18x^2+36x+24 supposed to be the total volume of the cubes, nowhere is this mentioned?
2. Should
3x^2+18x^2+36x+24
be written as
3x^3+18x^2+36x+24?
If both are true, the trick is to reduce the expression to the volume of one single cube,
x^3+6x^2+12x+8
If this is a perfect cube, i.e. the expression factors perfectly into (ax+b)^sup3;, then two conditions must be met:
1. a=1, since (x)³=x³
2. b=2, since (2)³=8
So check if (x+2)³ is your answer by expansion.
To find a possible length of one side of one of the cubes, we need to take the given expression, 3x^2 + 18x^2 + 36x + 24, and simplify it.
First, let's combine like terms by adding the coefficients of the same degree:
3x^2 + 18x^2 = 21x^2
Next, we have:
21x^2 + 36x + 24
To further simplify, let's factor out the greatest common factor (GCF). In this case, the GCF is 3:
3(7x^2 + 12x + 8)
Now, we need to factor the quadratic trinomial inside the parentheses. The factors will help us find possible lengths of a side of one of the cubes.
The quadratic trinomial 7x^2 + 12x + 8 can be factored as:
(7x + 4)(x + 2)
Now, we can find possible lengths by equating this expression to a perfect cube. A perfect cube has three identical factors. Since all three cubes are identical, we can set this expression equal to the cube of some value.
So, we have:
(7x + 4)(x + 2) = (k)^3
Where k represents the length of one side of a cube.
To find the possible lengths, we need to solve this equation for x.
Now, there are several possible values of k, but let's take one example and solve for x.
Let's assume k = 2.
(7x + 4)(x + 2) = (2)^3
Expanding the left side:
7x^2 + 18x + 12 = 8
Rearranging the equation:
7x^2 + 18x + 4 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
Using the values of a, b, and c from the quadratic equation above, we have:
x = (-(18) ± √((-18)^2 - 4(7)(4)))/(2(7))
Simplifying:
x = (-18 ± √(324 - 112))/(14)
x = (-18 ± √(212))/14
Therefore, we have two possible values of x:
x ≈ (-18 + √(212))/14 or x ≈ (-18 - √(212))/14
These values of x correspond to the possible lengths of one side of one of the cubes when k is 2.
Please note that this is just one example, and there may be other values of k that yield different lengths.
To find a possible length of one side of one of the cubes, we can set up an equation using the given expression and solve for x.
The given expression is 3x^2 + 18x^2 + 36x + 24. Since we are dealing with identical cubes, we can consider one cube and represent its side length as x.
So the volume of one cube would be x^3. Since we are dealing with three identical cubes, the total volume would be 3(x^3) = 3x^3.
According to the problem, this should be equal to the given expression, 3x^2 + 18x^2 + 36x + 24.
Setting up the equation:
3x^3 = 3x^2 + 18x^2 + 36x + 24
Combining like terms:
3x^3 - 21x^2 - 36x - 24 = 0
Now, we have a cubic equation. To find a possible length of one side of the cube, we need to solve this equation for x.
Solving a cubic equation can be complex and involve factoring, synthetic division, or other techniques. The specific solution for this equation will depend on the roots of the equation.
Alternatively, you could use numerical methods such as graphing the equation or using software or online tools to find the approximate solution.
Therefore, to obtain a possible length of one side of the cube, we need to solve the cubic equation 3x^3 - 21x^2 - 36x - 24 = 0.