"An aquarium with length x, width x-10, and height x-5 holds about 750 cubic feet of water. Find all real solutions of the equation (I wrote it) 750 = x(x-10)(x-5)."
I got the volume equation correct, right? I've tried dividing the factors of -750 by the factors of 1 (with this one theorem thingy), but I can't seem to get any real solutions. Does anyone know how to do this?
Your formula is correct. One root is x = 15. That is the only real solution. The other solutions are +/- sqrt(-50)
I got the 15 by guessing. The equation
x^3 -15x^2 +50x - 750 = 0 can be factored to give
(x-15)(x^2 + 50) = 0
Ohhh I see. Thanks so much!! :D
Yes, you have correctly set up the volume equation for the aquarium.
To find the real solutions of the equation, we can start by rearranging the equation into a standard polynomial form such that it equals zero:
0 = x(x-10)(x-5) - 750
Next, we can simplify the equation by expanding the expression on the right-hand side:
0 = (x^3 - 15x^2 + 50x) - 750
Now, combine like terms and rearrange the equation:
0 = x^3 - 15x^2 + 50x - 750
At this point, we have a cubic equation. To find the real solutions, we can use various methods including factoring, synthetic division, or the rational roots theorem.
One straightforward approach is to use the rational roots theorem. According to the theorem, any rational root of a polynomial equation will be in the form of p/q, where p is a factor of the constant term (-750) and q is a factor of the leading coefficient (1 in this case).
The factors of -750 are: ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±25, ±30, ±50, ±75, ±125, ±150, ±250, ±375, ±750.
Next, we need to check these factors by substituting each potential solution into the equation and see which values satisfy it. This process may involve some trial and error.
By trying out the potential rational roots, we find that x = 15 is a solution to the equation.
Dividing the cubic equation by this root using synthetic division, we get:
x = 15 | 1 -15 50 -750
| 15 0 750
-----------------
1 0 50 0
The result of the division is a quadratic equation: x^2 + 50 = 0. Solving this quadratic equation, we find that it has no real solutions.
Hence, the only real solution to the original cubic equation is x = 15.
Therefore, the length of the aquarium is 15 feet