A rectangular prism has a volume of 8 cubic yards. Assume the dimensions are whole numbers. What dimensions yield a prism with the greatest surface area? the leaset surface area?
Try a cube of 2 yards on each side and a rectangular solid of 1x1x8.
Calculate the surface area of each for your answer.
1x1x8
To find the dimensions that yield a rectangular prism with the greatest surface area and the least surface area, we can use the concept of optimization by taking the derivative of the surface area equation.
Let's assume the dimensions of the rectangular prism are length (L), width (W), and height (H).
1. To find the dimensions that yield a prism with the greatest surface area:
The surface area of a rectangular prism is given by the formula:
Surface Area = 2(LW + LH + WH)
Given the volume is 8 cubic yards, we have:
LWH = 8
To solve this problem, we need to find the dimensions that maximize the surface area while satisfying the volume constraint.
Step 1: Solve for one variable in terms of the other.
From the volume equation, we can solve for H:
H = 8 / (LW)
Step 2: Substitute the value of H into the surface area equation.
Surface Area = 2(LW + L(8 / LW) + W(8 / LW))
Surface Area = 2(LW + 8L / W + 8W / L)
Surface Area = 2[(L^2W + 8LW + 8W^2) / LW]
Step 3: Simplify the equation.
Surface Area = (2L^2W + 16LW + 16W^2) / LW
Surface Area = 2L + 16 / W + 16W / L
Step 4: Take the derivative of the surface area equation with respect to L and set it equal to zero.
d(Surface Area) / dL = 2 - 16 / W^2 + 16W^2 / L^2
Setting d(Surface Area) / dL = 0:
2 - 16 / W^2 + 16W^2 / L^2 = 0
Step 5: Solve for L.
16W^2 / L^2 - 16 / W^2 = -2
16W^4 - 16L^2 = -2L^2W^2
16W^4 + 2L^2W^2 - 16L^2 = 0
This equation is a quadratic equation in terms of W^2. We can solve it using the quadratic formula or by factoring.
Once we find the value of W, we can substitute it back into the volume equation to find the corresponding values for L and H.
By solving these equations, we can find the dimensions that yield a prism with the greatest surface area.
2. To find the dimensions that yield a prism with the least surface area:
Since the surface area equation is symmetric in L and W (length and width), the dimensions that yield the greatest surface area will also yield the least surface area.
Therefore, the dimensions that yield the greatest surface area will also yield the least surface area.
Hence, the dimensions that maximize the surface area will give us the least surface area as well.
Thus, both the greatest and least surface areas can be obtained from the same set of dimensions.
Note: The above steps provide an outline for finding the dimensions that yield the maximum (and minimum) surface area. The exact numerical solution will depend on solving the equations obtained.
To find the dimensions that yield a rectangular prism with the greatest surface area and the least surface area, we need to understand the relationship between volume and surface area for a rectangular prism.
The volume of a rectangular prism is given by the formula:
Volume = length × width × height
The surface area of a rectangular prism is given by the formula:
Surface Area = 2(length × width + width × height + height × length)
Let's start with finding the dimensions that yield the prism with the greatest surface area.
To find the dimensions that maximize the surface area, we need to find the dimensions that satisfy the volume constraint (8 cubic yards) and maximize the surface area formula.
One way to do this is to make a table to explore different possibilities.
Let's consider factors of 8:
1 × 1 × 8 = 8 (surface area = 58)
1 × 2 × 4 = 8 (surface area = 36)
1 × 4 × 2 = 8 (surface area = 36)
1 × 8 × 1 = 8 (surface area = 58)
2 × 1 × 4 = 8 (surface area = 36)
2 × 2 × 2 = 8 (surface area = 32)
4 × 1 × 2 = 8 (surface area = 36)
4 × 2 × 1 = 8 (surface area = 36)
8 × 1 × 1 = 8 (surface area = 58)
Looking at the table, we can conclude that the dimensions that yield a prism with the greatest surface area are 1 yard, 1 yard, and 8 yards, or 8 yards, 1 yard, and 1 yard. Both configurations yield the same surface area of 58 square yards.
Now let's find the dimensions that yield the prism with the least surface area.
Again, we need to find the dimensions that satisfy the volume constraint (8 cubic yards) and minimize the surface area formula.
Using the same table as before, we can see that the dimensions that minimize the surface area are 2 yards, 2 yards, and 2 yards. This configuration yields a surface area of 32 square yards, which is the smallest possible surface area for a prism with a volume of 8 cubic yards.
To summarize:
Dimensions that yield a prism with the greatest surface area: 1 yard, 1 yard, and 8 yards (or 8 yards, 1 yard, and 1 yard).
Dimensions that yield a prism with the least surface area: 2 yards, 2 yards, and 2 yards.