V = x(16-2x)(19-2x) = 175 ft³
Ok, thanks. Now I'm trying to multiply out and having some difficulty. I am getting 4x^3-70x^2+304x = 175 which I'm not sure how to work with
I agree it is a little difficult to solve unless you have learned Newton's method. If not, you can solve it by the bisection method, which is to evaluate the function using trial values of x until the function evaluates to 175.
I will give you a kick-start.
Let
f(x)= x(16-2x)(19-2x)
and knowing that the solution is around 0.68, we try
f(0.6)=0.6(16-2*0.6)(19-2*0.6)=158.064
f(0.7)=179.872
f(0.65)=169.124
So the next tries should lie between 0.65 and 0.7.
Keep on trying until you get a solution to within 1 or two decimals.
Here is a graph of the function f(x).
http://i263.photobucket.com/albums/ii157/mathmate/Ash.png
To multiply out the expression (16-2x)(19-2x), you can use the distributive property. This means that you need to multiply each term in the first set of parentheses (16 - 2x) by each term in the second set of parentheses (19 - 2x). Let's break it down step by step:
1. Multiply 16 by 19: 16 * 19 = 304
2. Multiply 16 by -2x: 16 * -2x = -32x
3. Multiply -2x by 19: -2x * 19 = -38x
4. Multiply -2x by -2x: -2x * -2x = 4x^2
Now we have three terms: 304, -32x, -38x, and 4x^2. The expression becomes:
V = x(304 - 32x - 38x + 4x^2)
Next, distribute the x to each term inside the parentheses:
V = 304x - 32x^2 - 38x^2 + 4x^3
Combine like terms:
V = 4x^3 - 70x^2 + 304x
So, the expanded form of V = x(16-2x)(19-2x) is 4x^3 - 70x^2 + 304x.
Now, let's solve the equation 4x^3 - 70x^2 + 304x = 175:
We can rearrange the equation to make it equal to zero:
4x^3 - 70x^2 + 304x - 175 = 0
Unfortunately, this equation is not easily factorable, so we need to use other methods to solve it. One approach is to use numerical methods like Newton's method or the Bisection method to approximate the roots of the equation. Another approach is to use a graphing calculator or software to find the x-values where the equation intersects the x-axis (roots).
I hope this explanation helps! Let me know if you have any further questions.