express the surface area S of arectangular box with no top as a function of the dimensions x,y and z.
If x,y form the base, and z is the height, then
Area of base = xy
Lateral area = perimeter*height
= 2(x+y)z
Total surface area of open box
=xy+2xz+2yz
To express the surface area (S) of a rectangular box with no top as a function of the dimensions (x, y, and z), we need to consider the six sides of the box.
The surface area of a rectangular box can be calculated by adding the areas of all six sides:
S = 2(xy + xz + yz)
So, the function that represents the surface area S in terms of the dimensions x, y, and z is:
S(x, y, z) = 2(xy + xz + yz)
To find the surface area (S) of a rectangular box without a top, we need to consider the dimensions of its sides. Let's label the sides of the box as follows:
- Length (L) corresponds to dimension x.
- Width (W) corresponds to dimension y.
- Height (H) corresponds to dimension z.
Now, let's break down the surface area into its individual components:
1. The bottom area consists of the length (L) multiplied by the width (W). Therefore, the area of the bottom is LW.
2. The front and back areas are equal and determined by the height (H) and width (W). Each of these areas is the product of HW, so the total is 2HW.
3. The left and right side areas are equal and determined by the length (L) and height (H). Each of these areas is the product of LH, so the total is 2LH.
Adding up all these individual areas, we get the formula for the surface area (S) of the rectangular box:
S = LW + 2HW + 2LH
In summary, the surface area of a rectangular box with dimensions x, y, and z is given by the formula S = LW + 2HW + 2LH.