A parcel delivery service has contracted you to design an open box with square base
length x cm and height y cm has a volume of 3600 cubic cm. Find the total surface area in
terms of x and y. Also determine the dimension for minimum surface area (round answers
correct to two decimal places if needed).
To find the total surface area of the open box with a square base, we need to consider the area of each face of the box.
The box has 5 faces:
1. The bottom face, which is a square with side length x cm.
2. The top face, which is also a square with side length x cm.
3. The front face, which is a rectangle with dimensions y cm (height) and x cm (base width).
4. The back face, which is also a rectangle with dimensions y cm (height) and x cm (base width).
5. The left side face, which is a rectangle with dimensions y cm (height) and x cm (base width).
Therefore, the total surface area (A) of the box is given by the equation:
A = x^2 + x^2 + yx + yx + yx
Simplifying, we have:
A = 2x^2 + 3yx
To determine the dimensions for minimum surface area, we need to find the values of x and y that minimize the surface area function A.
Given that the volume of the box is 3600 cubic cm, we have:
x^2 * y = 3600
To find the minimum surface area, we can use the method of calculus. Taking the derivative of the surface area function A with respect to x, and setting it equal to zero, we can solve for x:
dA/dx = 4x + 3y = 0
Solving for x, we have:
4x = -3y
x = (-3/4)y
Substituting this back into the volume equation, we can solve for y:
((-3/4)y)^2 * y = 3600
Simplifying, we have:
(9/16)y^3 = 3600
Solving for y, we get:
y^3 = (3600 * 16) / 9
y^3 = 6400
y = 16
Substituting the value of y back into the equation for x, we have:
x = (-3/4) * 16
x = -12
Since we're dealing with dimensions, we consider only the positive values of x and y. Therefore, the dimension for minimum surface area is x = 12 cm and y = 16 cm.
To find the total surface area in terms of x and y, we substitute these values into the equation for A:
A = 2(12)^2 + 3(12)(16)
A = 576 + 576
A = 1152 square cm
So, the total surface area of the box is 1152 square cm, and the dimensions for minimum surface area are x = 12 cm and y = 16 cm.