A rectangular piece of metal with dimensions 14 CMA 24 CM is used to make an open box equal squares of side length X centimeters are cut from the corners and sides are folded up a polynomial function that represents the volume, V,of the box is: V(X)=x(14-2x)(24-2x). Determine the maximum volume of the box.
Is there an algebraic way to do this? I tried using my graphing calculator but I'm not sure how to use it. It's the ti-84 plus.
What's the volume function?
Yes, there is an algebraic way to determine the maximum volume of the box using the given polynomial function. To find the maximum value of a polynomial function, you need to find the vertex of the parabola that represents the function.
The vertex of a parabola in the form f(x) = ax^2 + bx + c can be found using the formula x = -b / (2a). In this case, the polynomial function representing the volume of the box is V(x) = x(14-2x)(24-2x).
To start, we need to rewrite the equation in the standard form f(x) = ax^2 + bx + c. Expanding the polynomial function, we get:
V(x) = 4x^3 - 76x^2 + 336x
Now, we can identify a, b, and c:
a = 4, b = -76, c = 336
Substituting these values into the vertex formula, we get:
x = -(-76) / (2 * 4)
x = 76 / 8
x = 9.5
So, the maximum volume of the box occurs when x = 9.5 centimeters. To find the maximum volume, substitute this value into the original function V(x):
V(9.5) = 9.5(14-2(9.5))(24-2(9.5))
V(9.5) = 9.5(14-19)(24-19)
V(9.5) = 9.5(-5)(5)
V(9.5) = -237.5
Therefore, the maximum volume of the box is -237.5 cm^3 (cubic centimeters).
dimensions of the box
x high
14-2x width
24-2x height
Lord your teacher gave you this.
See the volume function. This is an easy problem in calculus. On the graphing calculator, type in the volume function give, and then plot it vs x. Look for the max volume.