9. The polynomial -x^3 -x^2 + 12x represents the volume of a rectangular aquatic tank in cubic feet. The length of the tank is (x + 4).
a. How many linear factors should you look for?
b. What are the dimensions of the tank?
c. Find the value of x that will maximize the volume of the box.
a) because of the x^3 terms, we should expect 3 linear factors, besides, volume is length x width x height
b) V = -x^3 -x^2 + 12x
= x(-x^2 - x +12)
we are told one of the factors is x+4
= x(x+4)(3-x)
so the dimensions of the tank are x by x+4 by 3-x
c) V = -x^3 -x^2 + 12x
dV/dx = -3x^2 - 2x + 12 = 0 for a max/min of V
3x^2 + 2x - 12 = 0
x = appr 1.69 or a negative.
the max volume = -(1.69)^3 - (1.69)^2 + 12(1.69) = 12.60 units^3
looks good, http://www.wolframalpha.com/input/?i=plot+y+%3D++-x%5E3+-x%5E2+%2B+12x
a. To find the number of linear factors, we need to consider the dimensions of the tank. Since the length of the tank is represented by (x + 4), we can expect a linear factor of (x + 4) for the length. Similarly, since the polynomial represents the volume of a rectangular tank, the other two dimensions, width and height, must be represented by the remaining terms.
Therefore, the number of linear factors we should look for is 3.
b. To find the dimensions of the tank, we need to factor the given polynomial. The given polynomial is -x^3 - x^2 + 12x. By factoring out the common factor of x, we get:
x(-x^2 - x + 12)
Now, let's factor the quadratic term further. We need to find two numbers whose product is 12 and whose sum is -1 (from the coefficient of x). The numbers -4 and 3 satisfy these conditions, so we can factor the quadratic as:
x(-x - 4)(x + 3)
Therefore, the dimensions of the tank are:
Length: x + 4
Width: -x - 4
Height: x + 3
c. To find the value of x that will maximize the volume of the tank, we need to look at the coefficient of the x^2 term, which in this case is -1. Since the coefficient is negative, the volume is an upside-down parabola opening downwards.
The maximum value of the volume occurs at the vertex of the parabola. In general, the x-value of the vertex can be found using the formula: x = -b / (2a), where a is the coefficient of x^2 and b is the coefficient of x.
In our case, a = -1 and b = -1. Substituting these values into the formula, we get:
x = -(-1) / (2 * -1) = 1 / 2
So, the value of x that will maximize the volume of the tank is 1/2.