I am having a great deal of difficulty with this problem.
An open box is formed by cutting squares out of a piece of cardboard that is 16 ft by 19 ft and folding up the flaps.
a. what size corner squares should be cut to yield a box that has a volume of 175 cubic feet.
So I have this equation
x(16-2x)(19-2x)= 175
multiplied both parenthesis and got
4x^2-70x+304
multiplied the entire thing by x
4x^3-70x^2+304x=175
subtract 175
4x^3-70x^2+304x-175=0
I'm not sure what to do next. I was thinking rational root theorem but I have tried most of the roots and it hasn't worked. I am not sure how to solve something to the third power.
Thanks
I graphed y=4x^3-70x^2+304x-175. Perhaps that would help. I think there are a couple of solutions.
Is there a mathematical way to solve this without using the graphing calculator?
A graphical solution is already a mathematical solution. In fact, the graphical solution gives three real roots, namely near x=1, x=6 and x=11.
On the basis that 2*11>16, 11 cannot be retained as a valid solution.
So the remaining real solutions are around 1 and 6.
If you are looking for an analytical way to solve the cubic, there is the Nicolo Fontana Tartaglia method which is described in detail in:
http://www.sosmath.com/algebra/factor/fac11/fac11.html
The solution consists of three steps:
1. Find the depressed equation
by removing the x² term of the general cubic by substitution, i.e.
in f(x)=Ax³+bx²+cx+d
substitute x=y-b/(3a) to give
f(y)=a*y^3 + (c-(b^2)/(3*a))*y + d-(b*c)/(3*a)+(2*b^3)/(27*a^2)
Note the absence of the y² term.
We will denote the depressed equation as
y³+Ay=B
2. Find s and t such that
3st=A
s²-t³=B
then a real solution of the cubic is
y=s-t from which x can be found.
3. Using long division, reduce the cubic to a quadratic and find the two remaining solutions.
Give it a try and the three (real) roots should be around 1, 6 and 11.
To solve a cubic equation like 4x^3 - 70x^2 + 304x - 175 = 0, there are a few methods you can try.
1. Rational Root Theorem: This method involves finding the rational roots (fractions in the form p/q) of the equation. Since the constant term is 175, the possible rational roots are factors of 175: ±1, ±5, ±7, ±25, ±35, ±175. You can try substituting these values into the equation and see if any of them make the equation equal to zero. If you find a root, you can use synthetic division or long division to factor out that root and reduce the equation to a quadratic.
2. Graphing: You can graph the equation y = 4x^3 - 70x^2 + 304x - 175 and look for the x-intercepts. The x-intercepts will represent the solutions to the equation. You can use graphing software or an online graphing calculator to plot the graph and determine the approximate solutions.
3. Numeric Methods: If the cubic equation does not have rational roots or is difficult to solve analytically, you can use numerical methods to approximate the roots. One common numerical method is using Newton's method or a similar iterative method to find the roots. This involves making an initial guess for the root and then using a recurrence relation to iteratively refine the guess until a sufficiently accurate solution is reached.
In your specific case, since the equation represents a physical problem, you may also want to check if the solutions make sense in the context of the problem. For example, negative values for the side length (x) or values that exceed the dimensions of the cardboard would not be valid solutions.
Try using one of these methods to solve the equation and find the proper size for the corner squares that will yield a box with a volume of 175 cubic feet.