A rectangular box open at the top has a square base. The internal side of the base is x cm long and the total internal surface area of the box is 432cm2.
Express in terms of x.
(i) the internal height h, of the box;
(ii) the internal volume V, of the box.
(b) Find:
i) the value of x for which the volume V is maximum;
(ii) the maximum internal volume of the box.
I need answer
4x^2h = 432, so h = 108/x^2
v = x^2h = 108h
dv/dx = 2xh + x^2 dh/dx = 2xh - 216/x
now find where dv/dx = 0 for max v.
To find the expressions in terms of x for the internal height of the box, h, and the internal volume, V, we need to consider the surface area and volume formulas for a rectangular box.
1) Finding the internal height (h):
The total internal surface area of the box is given as 432 cm^2. This includes the area of the base and the four sides of the box.
The base is a square, so its area is x * x = x^2 cm^2.
The area of each of the four sides is h * x cm^2 since the sides have a height of h and a length of x.
So, the total surface area is x^2 + 4hx cm^2, which is equal to 432 cm^2.
Therefore, we have the equation: x^2 + 4hx = 432.
2) Finding the internal volume (V):
The volume of a rectangular box with a base area of x^2 cm^2 and a height of h cm is given by V = x^2 * h cm^3.
Next, we need to find the value of x for which the volume V is maximum, as well as the maximum internal volume of the box.
3) Finding the value of x for maximum volume (V):
To find the value of x for maximum volume, we can differentiate the volume equation with respect to x and set it equal to zero.
Differentiating V = x^2 * h with respect to x, we get:
dV/dx = 2xh = 0.
Simplifying, we find x = 0 or h = 0.
Since the length of a side cannot be zero, x must be nonzero.
4) Finding the maximum internal volume of the box:
Substituting x = 0 or h = 0 into the volume equation will result in zero volume, so it is not the maximum.
To find the maximum volume, we need to substitute the value of h obtained from the equation x^2 + 4hx = 432 back into the volume equation.
Since we found h = 0 is not valid, we need to solve x^2 + 4hx = 432 for h and substitute it into the volume equation.
This step requires solving the quadratic equation x^2 + 4hx - 432 = 0 for h using the quadratic formula:
h = (-b ± sqrt(b^2 - 4ac)) / (2a).
By substituting the values of a, b, and c from the quadratic equation, we can calculate the values of h for different values of x.
Then, substitute the values of h and x into the volume equation V = x^2 * h to find the internal volume of the box for those values.
Finally, compare the volumes obtained and find the maximum volume.
To find the internal height (h) and volume (V) of the rectangular box, we need to set up equations using the given information.
(i) To find the internal height (h), we need to determine the height of the rectangular box. Since the box has a square base, the base area is given by
base area = x * x = x^2 cm^2
The total internal surface area of the box is given as 432 cm^2, which consists of the area of the base and the surface area of the four sides. The area of each side can be calculated as the product of height (h) and the corresponding side length.
Therefore, the total internal surface area can be expressed as:
total surface area = base area + 4 * (side area)
432 = x^2 + 4 * (x * h)
Simplifying the equation, we get:
432 = x^2 + 4xh
(ii) To find the internal volume (V) of the box, we need to multiply the base area by the height (h) of the box.
volume = base area * height
V = x^2 * h
(b) To find the value of x for which the volume V is maximum, we need to differentiate the volume equation with respect to x and set it equal to 0.
dV/dx = 2xh + x^2 * dh/dx = 0
Solving this equation for x will give us the value for which the volume is maximum.
To find the maximum internal volume of the box, substitute the value of x obtained from the previous step into the volume equation: V = x^2 * h.