rectangular open-topped box is made from a 9 x 16 piece of cardboard by cutting x-inch squares out of each corner and folding up the sides.
What size square should be cut out to produce a volume of 120 cubic inches??
I set it up with
120 = (16-2x)(9-2x)(x)
I get to
y = 2x^3 - 25x^2 +72x -60 and I'm stuck. Where do I go from here?
Let the corner pieces be x by x.
Then, x(16 - 2x)(9 - 2x) = 120
144x - 50x^2 + 4x^3 = 120
4x^3 - 50x^3 + 144x - 120 = 0
2x^3 - 25x^2 + 72x - 60 = 0
First derivative =
6x^2 - 50x + 72 = 0
3x^2 - 25x + 36 = 0
x = [25+/-sqrt(25^2 - 4(3)36)]/6
x = 1.851
I don't know calculus. I'm still in algebra. please explain
To find the size of the square that should be cut out to produce a volume of 120 cubic inches, you need to solve the equation:
120 = (16-2x)(9-2x)(x)
Since you have already simplified the equation to:
y = 2x^3 - 25x^2 + 72x - 60
To find the value of x, where the volume is 120, you need to set y equal to 120 and solve for x:
120 = 2x^3 - 25x^2 + 72x - 60
Now, rearrange the equation to make it equal to zero:
2x^3 - 25x^2 + 72x - 180 = 0
To solve this cubic equation, you can try different values of x until you find a solution or you can use numerical methods such as the Newton-Raphson method or use a graphing calculator/computer program to find the roots.
Unfortunately, further simplification of the equation cannot be achieved algebraically, so you will need to use numerical methods or a graphing tool to find the value of x that satisfies the equation.
To solve the equation y = 2x^3 - 25x^2 + 72x - 60, which represents the volume of the rectangular box, follow these steps:
1. Set the equation equal to zero: 2x^3 - 25x^2 + 72x - 60 = 0.
2. The equation is a cubic equation, so you can try to factor it or use numerical methods to find the solutions. In this case, it's easier to use numerical methods.
3. Use a graphing calculator or software to graph the function y = 2x^3 - 25x^2 + 72x - 60.
4. Look for the x-values (cut-out square sizes) where the graph intersects the x-axis or crosses below it. These are the potentially valid solutions for the size of the cut-out square.
5. Based on the graph, you can see that there are two distinct solutions where the graph intersects the x-axis. These are the approximate values for x that will give a volume of 120 cubic inches.
6. The values for x obtained from the graph would be an estimate. To find the exact values, you can use numerical methods like Newton's method or the bisection method.
7. Once you obtain the estimates or the exact values for x, substitute them back into the equation y = 2x^3 - 25x^2 + 72x - 60 to verify that they indeed give a volume of 120 cubic inches.
Keep in mind that the equation represents a physical problem, so x must be a positive number less than half the length and width of the cardboard (in this case, less than 4.5 inches). Thus, you might only consider the positive solutions within that range.