A closed box is to be made in the shape of a cubiod, of height h cm and with a square base that has sides of length x cm.
Its volume V is required to be 500 cm^3.
A) write an expression for the V (volume)in terms of h and x.
B)Write an expression for the surface area A in terms of h and x. use the result for A) to write A in terms of x only.
C) Use differentiation to determine the value of x which has A as a minumum. What is the coresponding value of h?
x^2h=500, so h = 500/x^2
v = x^2h
A = x^2 + 4xh = x^2 + 2000/x
dA/dx = 2x - 2000/x^2 = 2x(x^2-1000)/x^2
A) To find the volume V of the cuboid, we need to multiply the base area (x^2) by the height (h):
V = x^2 * h
B) The surface area A of the cuboid consists of six sides: two squares (top and bottom) with an area of x^2 each, and four rectangles (sides) with an area of x * h each. So the total surface area A can be expressed as:
A = 2 * x^2 + 4 * x * h
Let's simplify this expression to express A in terms of x only:
A = 2x^2 + 4xh
Factor out x from both terms:
A = x(2x + 4h)
C) To find the value of x that minimizes A, we can take the derivative of A with respect to x and set it equal to zero:
dA/dx = 2(2x + 4h) + 2x(0)
dA/dx = 4x + 8h
Setting this derivative equal to zero, we have:
4x + 8h = 0
Now, solve for x:
4x = -8h
x = -2h
Since x represents the length of a side, it cannot be negative, so we discard this solution.
We can conclude that there is no value of x that minimizes A. However, we can find the corresponding value of h by plugging the value of x in the original expression for A:
A = x(2x + 4h)
A = -2h(2(-2h) + 4h)
A = -2h(-4h + 4h) = 0
Therefore, when x = -2h, the corresponding value of A is 0.