The total surface area of a rectangular solid is the sum of the areas of the six faces. If each dimension of a given rectangular solid is doubled, what effect does this have on the total surface area?
if the dimensions are scaled by a factor of f, then the area grows by a factor of f^2
This is easy to see, since
area = length * width
if each is doubled, then the
newarea = (2*length)*(2*width) = 2^2 * (length*width) = 2^2 * area
unknown
Thanks much!
To determine the effect of doubling each dimension on the total surface area of a given rectangular solid, let's break down the problem:
1. Understanding the basic formula: The total surface area of a rectangular solid is calculated by summing the areas of all six faces. The formula is given as:
Total Surface Area = 2*(lw + lh + wh)
where l, w, and h represent the length, width, and height of the rectangular solid, respectively.
2. Doubling each dimension: When each dimension of a rectangular solid is doubled, it means that the length (l), width (w), and height (h) will all become twice their initial values.
Therefore, the new dimensions of the rectangular solid would be: 2l, 2w, and 2h.
3. Calculating the new total surface area: Substituting the new dimensions into the formula, we have:
New Total Surface Area = 2*(2l * 2w + 2l * 2h + 2w * 2h)
= 2*(4lw + 4lh + 4wh)
= 8*(lw + lh + wh)
So, the new total surface area is eight times the initial total surface area.
In conclusion, if each dimension of a given rectangular solid is doubled, the effect on the total surface area is that it becomes eight times larger.