a block prism has a volume of 36 cubic units, what is the least and greatest surface area it could have

assuming integer dimensions,

36 = 2*2*3*3

4x3x3: area=66
1x1x36: area=146

To determine the least and greatest surface area of a block prism, we need to consider that the volume remains constant at 36 cubic units.

The surface area of a block prism can be calculated using the formula: SA = 2lw + 2lh + 2wh, where l, w, and h represent the dimensions, namely the length, width, and height of the block prism, respectively.

Since the volume is given as 36 cubic units, we can express this in terms of the dimensions using the formula: V = lwh = 36.

Now, we need to consider the different possible combinations of l, w, and h that satisfy the volume equation.

Let's list down the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

We can use these factors to find the possible dimensions (l, w, h) that would result in a volume of 36 cubic units.

Using these factors, we can calculate the surface area for each possible dimension set. The set with the smallest surface area would represent the "least" surface area, while the set with the largest surface area would represent the "greatest" surface area.

Here are the possible sets of dimensions (l, w, h) and their corresponding surface areas:

Set 1: (1, 1, 36)
Surface Area = 2(1)(1) + 2(1)(36) + 2(1)(36) = 2 + 72 + 72 = 146 square units

Set 2: (2, 2, 9)
Surface Area = 2(2)(2) + 2(2)(9) + 2(9)(2) = 8 + 36 + 36 = 80 square units

Set 3: (3, 3, 4)
Surface Area = 2(3)(3) + 2(3)(4) + 2(4)(3) = 18 + 24 + 24 = 66 square units

Set 4: (4, 4, 3)
Surface Area = 2(4)(4) + 2(4)(3) + 2(3)(4) = 32 + 24 + 24 = 80 square units

Set 5: (6, 6, 1)
Surface Area = 2(6)(6) + 2(6)(1) + 2(1)(6) = 72 + 12 + 12 = 96 square units

Set 6: (9, 9, 4)
Surface Area = 2(9)(9) + 2(9)(4) + 2(4)(9) = 162 + 72 + 72 = 306 square units

Set 7: (12, 12, 1)
Surface Area = 2(12)(12) + 2(12)(1) + 2(1)(12) = 288 + 24 + 24 = 336 square units

Set 8: (18, 18, 1)
Surface Area = 2(18)(18) + 2(18)(1) + 2(1)(18) = 648 + 36 + 36 = 756 square units

Set 9: (36, 36, 1)
Surface Area = 2(36)(36) + 2(36)(1) + 2(1)(36) = 2592 + 72 + 72 = 2736 square units

From the above calculations, we can conclude that the least surface area for a block prism with a volume of 36 cubic units is 66 square units (Set 3), while the greatest surface area is 2736 square units (Set 9).