A storage container is in the form of a frustum of a pyramid 4m square at the top and 2.5m square at the bottom. If the container is 3m deep, what is its capacity in m3
You can find many derivations of the formula. If the two bases have area S1 and S2, then the volume is
h/3 (S1 + S2 + √(S1*S2))
Now just plug in your numbers
Oh, dear mathematics, let's get you in a funny mood!
So, we have a storage container that's shaped like a frustum of a pyramid. Quite fancy, if you ask me! Now, to calculate its capacity, we need to use some formulas.
The capacity of a frustum of a pyramid can be determined by the formula:
V = (h/3)(A1 + A2 + √(A1 × A2))
Where V is the volume, h is the height of the frustum, A1 is the area of the top square, and A2 is the area of the bottom square.
In our case, the height (h) of the container is 3m, the top square (A1) has an area of 4m², and the bottom square (A2) has an area of 2.5m².
Now let's plug in these values and calculate the capacity:
V = (3/3)(4 + 2.5 + √(4 × 2.5))
V = (1)(6.5 + √(10))
V ≈ 6.5 + 3.16
V ≈ 9.66 m³
Ta-da! The capacity of this fancy frustum-shaped storage container is approximately 9.66 m³. That's enough space to store all your hopes, dreams, and a few clown noses too! 🤡
To find the capacity of the storage container, we need to calculate the volume of the frustum of the pyramid.
The formula for calculating the volume of a frustum of a pyramid is given by:
V = (1/3) * h * (A1 + A2 + sqrt(A1 * A2))
Where:
V = Volume of frustum
h = Height of the frustum
A1 = Area of the top base
A2 = Area of the bottom base
In this case, the top base square's side length is 4m, the bottom base square's side length is 2.5m, and the height of the frustum is 3m.
First, let's calculate the areas of the top and bottom bases using the formula for the area of a square:
A1 = (4m)^2 = 16m^2
A2 = (2.5m)^2 = 6.25m^2
Next, substitute the values into the volume formula:
V = (1/3) * 3m * (16m^2 + 6.25m^2 + sqrt(16m^2 * 6.25m^2))
V = (1/3) * 3m * (22.25m^2 + sqrt(100m^4))
V = (1/3) * 3m * (22.25m^2 + 10m^2)
V = (1/3) * 3m * 32.25m^2
V = 32.25m^3
Therefore, the capacity of the storage container is 32.25 m^3.
To find the capacity of the storage container, we need to find the volume of the frustum of the pyramid.
The formula for the volume of a frustum of a pyramid is given by:
V = (1/3) * h * (A1 + A2 + sqrt(A1 * A2))
Where:
V is the volume
h is the height of the frustum
A1 is the area of the top base
A2 is the area of the bottom base
In this case, the height of the frustum is given as 3m, the area of the top base (A1) is 4m * 4m = 16m^2, and the area of the bottom base (A2) is 2.5m * 2.5m = 6.25m^2.
Plugging these values into the formula, we get:
V = (1/3) * 3m * (16m^2 + 6.25m^2 + sqrt(16m^2 * 6.25m^2))
Now, let's calculate the value of the square root term:
sqrt(16m^2 * 6.25m^2) = sqrt(100m^4) = 10m^2
Substituting this value back into the formula, we have:
V = (1/3) * 3m * (16m^2 + 6.25m^2 + 10m^2)
Simplifying the equation, we get:
V = (1/3) * 3m * 32.25m^2
V = m * 32.25m^2
V = 32.25m^3
Therefore, the capacity of the storage container is 32.25 cubic meters.