1. A rectangular box has a square base, four sides, but no top. It has a volume of 20 cubic feet.
Let A be the surface area of the box, and L the length of one side of the base.
(a) The volume is measured in cubic feet. What units are convenient to use for the area A
and the side length L?
(b) Is A a function of L? If no, explain why not. If yes, �nd the formula of A in terms of L
and point out its domain.
a) It is convenient to use square feet as the unit for the surface area A and the side length L since the area is two-dimensional and the length is one-dimensional.
b) Yes, A is a function of L. The formula for the surface area A in terms of L can be derived as follows:
The box has a square base, so the area of the base is L^2 square feet.
The box has four sides, each with an area equal to L times the height of the box. The height can be found by dividing the volume of the box (20 cubic feet) by the area of the base (L^2). Therefore, the height is 20/L^2 feet.
The total surface area A is the sum of the area of the base and the four sides:
A = L^2 + 4L * (20/L^2)
Simplifying the equation:
A = L^2 + 80/L
The domain of the function A is all positive real numbers for L, excluding L = 0 since it represents the length of one side of the base.
(a) The volume is measured in cubic feet, so it is convenient to use square feet for the area A and feet for the side length L. This is because when measuring the volume of a three-dimensional object, we are measuring how much space it occupies in terms of cubic units. Similarly, when measuring the surface area or the length of the sides of a two-dimensional object, we use square units and linear units, respectively.
(b) Yes, A is a function of L. A function is a relationship where for every input (independent variable), there is a unique output (dependent variable). In this case, we can express the surface area A as a function of the side length L.
To find the formula for A in terms of L, let's break down the rectangular box into its components. It has four sides, but no top, so the total surface area is the sum of the areas of the four sides.
The two sides of the box that are rectangles have the same dimensions: L (the length of one side of the base) as the width and 20/L (since volume = L * L * 20/L = 20) as the height. Therefore, the area of each rectangular side is A_rect = L * (20/L) = 20.
The other two sides of the box are squares, and since they have the same dimensions, we can calculate the area of one side and then multiply it by 2 to account for both sides. The side length of each square is L, so the area of each square side is A_square = L * L = L^2.
Therefore, the total surface area A of the rectangular box with a square base is given by:
A = A_rect + A_rect + A_square + A_square
A = 20 + 20 + L^2 + L^2
A = 40 + 2L^2
The formula for A in terms of L is A = 40 + 2L^2. The domain of this function is all positive values of L since the side length cannot be negative or zero in this context.