Volume of prism 1296 cube cm height 16/√3cm find whole surface area.
lwh = V
lw(16/√3) = 1296
lw = 81/√3
SA = 2lw + 2lh + 2hw
the only value I know is lw, so you have missing information.
If it is a square base....
the each side of the base is 27√3
and you can then find the SA
To find the whole surface area of a prism, we need to calculate the lateral surface area (the area of the sides, excluding the top and bottom faces) and then add the area of the two bases.
First, let's calculate the volume of the prism. The volume of a prism is given by the formula:
Volume = Base Area × Height
Given that the volume is 1296 cubic cm and the height is 16/√3 cm, we can rearrange the formula to solve for the base area:
Base Area = Volume ÷ Height
Substituting the given values:
Base Area = 1296 cubic cm ÷ (16/√3) cm
To simplify this, we need to rationalize the denominator by multiplying it by its conjugate:
Base Area = 1296 cubic cm ÷ (16/√3) cm × (√3)/(√3)
Simplifying further:
Base Area = (1296 × √3) cubic cm ÷ 16 cm
Base Area = 81√3 cubic cm
Now, let's calculate the lateral surface area. Since a prism has two identical bases, the lateral surface area can be found by multiplying the base perimeter by the height:
Lateral Surface Area = 2 × Base Perimeter × Height
To find the base perimeter, we need to use the base area we just calculated:
Base Perimeter = √(Base Area)
Base Perimeter = √(81√3 cubic cm)
To simplify this, we can rewrite the square root as a power of 1/2:
Base Perimeter = (81√3 cubic cm)^(1/2)
Taking the square root of 81, we get:
Base Perimeter = 9√3 cm
Now we can calculate the lateral surface area:
Lateral Surface Area = 2 × 9√3 cm × (16/√3) cm
Lateral Surface Area = 288 cm^2
Finally, the whole surface area is the sum of the lateral surface area and the area of the two bases:
Whole Surface Area = Lateral Surface Area + 2 × Base Area
Whole Surface Area = 288 cm^2 + 2 × 81√3 cm^2
Whole Surface Area = 288 cm^2 + 162√3 cm^2
So, the whole surface area of the prism is 288 cm^2 + 162√3 cm^2.