a closed box has a fixed surface area A and a suqare base wide side X
a) find the formula for its volume, v, as a function x
b) sketch the graph of v against x
c) find the max value of V
To find the formula for the volume of the closed box as a function of x, let's break down the problem step by step:
a) Formula for the volume, V, as a function of x:
The closed box has a square base of side length x, which means all four sides of the base have length x. Since the surface area of the closed box is fixed, we need to consider the top and bottom surfaces, as well as the four sides.
The top and bottom surfaces together have an area of 2 times the area of the square base, which is 2x^2. The four sides have a combined area of 4xL, where L is the length of each side of the box.
The total surface area, A, is the sum of the areas of the top, bottom, and sides: A = 2x^2 + 4xL.
Since the closed box is rectangular, we need to express L in terms of x. Since there are two sides of length x, L is equal to (A - 2x^2) / (4x). Therefore, the formula for the volume, V, as a function of x is:
V = x^2 * L = x^2 * [(A - 2x^2) / (4x)] simplifying further, we get V = (A - 2x^2) * x / 4.
b) Sketching the graph of V against x:
To sketch the graph of V against x, we plot x on the x-axis and V on the y-axis. Remember that the surface area, A, is a fixed value. Plot a few points for different values of x and connect them to get a curve. The shape of the graph will depend on the value of A.
c) Finding the maximum value of V:
To find the maximum value of V, we can take the derivative of V with respect to x and set it equal to zero. Then, we solve for x.
Let's find the derivative of V with respect to x:
dV/dx = (A - 2x^2) / 4 - (2x(2x^2))/4 simplifying further, we get dV/dx = (A - 6x^2) / 4.
To find the maximum value, we set dV/dx equal to zero:
(A - 6x^2) / 4 = 0
Solving for x, we get:
A - 6x^2 = 0
6x^2 = A
x^2 = A / 6
x = √(A / 6) or x = -√(A / 6)
Since x represents the side length of the square base, we disregard the negative value. Therefore, the maximum value of V occurs when x = √(A / 6). Plug this value of x into the formula for V to find the maximum volume.