Matthew is constructing a rectangular prism with a volume of 729 cubic inches. The prism will have the least possible surface area.
a) describe the prism and what will be its dimensions
b) what will be its surface area
a cube has minimum area
729 = 9^3
delete the one on top of this one
V = s^3
729 = s^3
s = 9 inches (A 9” x 9” x 9” cube)
SA = 6s^2
= 6(9)^2
= 486 square inches
is my answer correct?
is my answer correct?
yes
To find the dimensions of a rectangular prism with the least possible surface area and a given volume of 729 cubic inches, we need to determine the factors of 729 and choose the factors that will result in the smallest surface area.
a) Dimensions of the prism:
To find the dimensions of the prism, we need to look for factors of 729. Prime factorizing 729, we get:
729 = 3 * 3 * 3 * 3 * 3 * 3
The factors of 729 are:
1, 3, 9, 27, 81, 243, and 729.
To minimize the surface area, we need to choose two factors that are closest to each other. In this case, 9 and 81 are the closest factors.
Therefore, the dimensions of the prism will be 9 inches, 9 inches, and 81 inches.
b) Surface area of the prism:
The surface area of a rectangular prism can be calculated using the formula:
Surface Area = 2(length * width + width * height + height * length)
In this case, the dimensions are 9 inches, 9 inches, and 81 inches.
Plugging the values into the formula, we get:
Surface Area = 2(9 * 9 + 9 * 81 + 81 * 9)
Surface Area = 2(81 + 729 + 729)
Surface Area = 2(1539)
Surface Area = 3078 square inches
So, the surface area of the prism with the least possible surface area and a volume of 729 cubic inches is 3078 square inches.