A pedestal is constructed in the form of frustum of a pyramid, the sides of the square ends of the frustum being 360cm and 160 cm and slant height 260cm. Find volume , lateral surface area including area of the top and the cost of construction @ rs.50 /cubic metre and plastering it @ rs.10 /square metre
To find the volume of the frustum of the pyramid, we can use the formula:
V = (1/3) * h * (A1 + A2 + √(A1 * A2))
Where:
V = Volume of the frustum of the pyramid
h = Height of the frustum
A1 = Area of the top square base
A2 = Area of the bottom square base
Given that the sides of the square ends of the frustum are 360 cm and 160 cm, and the slant height is 260 cm, we can calculate the height of the frustum using the Pythagorean theorem:
h = √(slant height^2 - (side1 - side2)^2)
Let's plug in the values and calculate the height first:
h = √(260^2 - (360 - 160)^2)
= √(260^2 - 200^2)
= √(67600 - 40000)
= √27600
≈ 166.063 cm
Now, let's calculate the areas of the top and bottom square bases:
A1 = side1^2 = 360^2 = 129,600 cm^2
A2 = side2^2 = 160^2 = 25,600 cm^2
Now, we can calculate the volume using the given formula:
V = (1/3) * h * (A1 + A2 + √(A1 * A2))
= (1/3) * 166.063 * (129,600 + 25,600 + √(129,600 * 25,600))
= (1/3) * 166.063 * (129,600 + 25,600 + √(3,323,520,000))
= (1/3) * 166.063 * (155,200 + √(3,323,520,000))
= (1/3) * 166.063 * (155,200 + 57,664.315)
V ≈ 13,456,805.29 cm^3
To find the lateral surface area of the frustum, we can calculate the sum of the areas of the four trapezoidal faces:
Lateral surface area = (1/2) * (side1 + side2) * slant height + A1 + A2
Lateral surface area = (1/2) * (360 + 160) * 260 + 129,600 + 25,600
= 520 * 260 + 129,600 + 25,600
= 135,200 + 129,600 + 25,600
= 290,400 cm²
To find the total surface area, including the area of the top, we need to add the areas of the top and bottom square bases:
Total surface area = Lateral surface area + 2 * (A1 + A2)
= 290,400 + 2 * (129,600 + 25,600)
= 290,400 + 2 * 155,200
= 290,400 + 310,400
= 600,800 cm²
Lastly, let's calculate the cost of construction and plastering:
Volume of frustum = 13,456,805.29 cm³
Cost of construction = Volume * Cost per cubic meter
= 13,456,805.29 * 50
= 672,840,264.5 Rs
Surface area = 600,800 cm²
Cost of plastering = Surface area * Cost per square meter
= 600,800 * 10
= 6,008,000 Rs
Therefore, the cost of construction is 672,840,264.5 Rs and the cost of plastering is 6,008,000 Rs.
To find the volume of the frustum of a pyramid, you can use the following formula:
V = (1/3) * h (A₁ + A₂ + √(A₁ * A₂))
Where:
V is the volume of the frustum,
h is the height of the frustum,
A₁ and A₂ are the areas of the two square ends of the frustum.
In this case, the dimensions of the frustum are given as follows:
Sides of the square ends: 360 cm and 160 cm
Slant height: 260 cm
First, let's calculate the height (h) of the frustum using the Pythagorean theorem:
h = √(slant height² - ((side₁ + side₂)/2)²)
Plugging in the values:
h = √(260² - ((360 + 160)/2)²)
h = √(67600 - (520/2)²)
h = √(67600 - 260²)
h = √(67600 - 67600)
h = √0
h = 0 cm
Since the height is 0 cm, the frustum degenerates into a flat square. The volume of a flat square is 0, so the volume of this frustum is also 0 cm³.
Now let's calculate the lateral surface area (including the top) and the cost of construction.
The lateral surface area of the frustum can be calculated as the sum of the areas of the two square ends and the lateral surface area of the trapezoidal part connecting them.
Lateral Surface Area = A₁ + A₂ + Area of Trapezoidal Part
The areas of the square ends are:
A₁ = side₁² = 360² = 129600 cm²
A₂ = side₂² = 160² = 25600 cm²
The area of the trapezoidal part can be calculated using the formula:
Area of Trapezoid = ((side₁ + side₂) / 2) * slant height
Plugging in the values:
Area of Trapezoidal Part = ((360 + 160) / 2) * 260
Area of Trapezoidal Part = 520 * 260 = 135200 cm²
Lateral Surface Area = A₁ + A₂ + Area of Trapezoidal Part
Lateral Surface Area = 129600 + 25600 + 135200 = 290400 cm²
To calculate the cost of construction, we need to convert the volume from cubic centimeters to cubic meters:
Volume = 0 cm³ = 0 m³
Cost of Construction = Volume * Cost per Cubic Meter
Cost of Construction = 0 * 50 rs/m³ = 0 rs
To plaster the frustum, we need to calculate the total surface area (including the top) and then multiply it by the cost per square meter.
Total Surface Area = Lateral Surface Area + 2 * Area of Top
Area of Top = side₁² = 360² = 129600 cm²
Total Surface Area = 290400 + 2 * 129600 = 549600 cm²
Cost of Plastering = Total Surface Area * Cost per Square Meter
Cost of Plastering = 549600 * 10 rs/m² = 5496000 rs
Therefore, the volume of the frustum is 0 cubic cm, the lateral surface area (including the top) is 290400 cm², the cost of construction is 0 rs, and the cost of plastering is 5496000 rs.
If you draw a careful diagram, you will see that the altitude is 240cm between the bases.
Using similar triangles, you can see that if the pyramid were not chopped off, its height would be 320/3.
So, the missing part is a pyramid of base side 80 and height 400/3.
Now just work with the missing part and the retained part.
Then you can figure the cost.
Or, you can find useful formulas here
http://mathworld.wolfram.com/PyramidalFrustum.html