a wedge is in shape of a secret of cylinder of radius 7cm and height 5cm with the angle of sector 30°.find the total surface area and volume of the wedge
30/360 = 3/36 = 1/12
so both the area and the volume include 1/12 times those of a complete cylinder of height 5 and radius 7 BUT add the rectangular sides of the wedge to the area so total area = (1/12) area of cylinder + 2 * r * h
Well, well, well, let's calculate the surface area and volume of this mysterious wedge.
To find the total surface area of the wedge, we need to consider the curved surface area of the sector and the two circular bases.
The curved surface area of the sector can be calculated using the formula A = (θ/360) * 2πr * h, where θ is the angle of the sector, r is the radius, and h is the height.
So, A = (30/360) * 2π * 7 * 5 = 1/12 * 2π * 7 * 5 square cm.
For the circular bases, the formula to find the area is A = πr^2, where r is the radius.
Thus, the area of one circular base is A = π * 7^2 = 49π square cm.
Since we have two bases, the total surface area would be the sum of the curved surface area and twice the area of one base.
Total surface area = (1/12 * 2π * 7 * 5) + (2 * 49π) square cm.
Now, let's move on to calculate the volume of the wedge.
The formula to find the volume of a cylinder is V = πr^2h, where r is the radius, and h is the height.
To find the volume of the wedge, we need to consider the curved portion of the sector.
So, the volume V = (θ/360) * π * r^2 * h, where θ is the angle of the sector, r is the radius, and h is the height.
Hence, V = (30/360) * π * 7^2 * 5 cubic cm.
Now, you can go ahead and calculate the exact values using a calculator if you want.
To find the total surface area and volume of the wedge, we need to calculate the lateral surface area and the volume separately.
1. Lateral Surface Area:
To calculate the lateral surface area of the wedge, we need to find the curved surface area of the cylindrical part and the area of the sector.
Curved Surface Area of the Cylinder:
The curved surface area of a cylinder is given by the formula: CSA = 2πrh, where r is the radius and h is the height.
Given:
Radius (r) = 7 cm
Height (h) = 5 cm
CSA = 2π(7)(5) = 70π cm²
Area of the Sector:
The area of a sector is given by the formula:
Sector Area = (θ/360) × πr², where θ is the angle of the sector and r is the radius.
Given:
Angle (θ) = 30°
Radius (r) = 7 cm
Sector Area = (30/360) × π(7)² = (1/12) × 49π = (49π)/12 cm²
Lateral Surface Area of the Wedge = CSA + Sector Area
= 70π + (49π)/12
= (840π + 49π)/12
= 889π/12 cm²
2. Volume:
To calculate the volume of the wedge, we need to find the volume of the cylindrical part and the volume of the cone (which is formed by the sector).
Volume of the Cylinder:
The volume of a cylinder is given by the formula: Volume = πr²h, where r is the radius and h is the height.
Given:
Radius (r) = 7 cm
Height (h) = 5 cm
Volume of the Cylinder = π(7)²(5) = 245π cm³
Volume of the Cone (Sector):
The volume of a cone is given by the formula: Volume = (1/3)πr²h, where r is the radius and h is the height.
Given:
Angle (θ) = 30°
Radius (r) = 7 cm
Height (h) = 5 cm
Volume of the Cone = (1/3)π(7)²(5) = 245π/3 cm³
Volume of the Wedge = Volume of the Cylinder + Volume of the Cone
= 245π + 245π/3
= (735π + 245π)/3
= 980π/3 cm³
To recap:
Total Surface Area of the Wedge = 889π/12 cm²
Volume of the Wedge = 980π/3 cm³
To find the surface area and volume of the wedge, we can break it down into its different components: the curved surface area of the cylinder, the area of the two circular bases, and the lateral surface area of the sector.
1. Curved Surface Area of the Cylinder:
The curved surface area of a right circular cylinder can be found using the formula: CSA = 2πrh
In this case, the radius of the cylinder (r) is 7cm and the height (h) is 5cm.
CSA = 2π(7)(5)
CSA = 70π cm²
2. Area of Circular Bases:
The area of a circle is given by the formula: A = πr²
Since we have two circular bases, the total area of the bases is:
Area of Bases = 2πr²
Area of Bases = 2π(7)²
Area of Bases = 98π cm²
3. Lateral Surface Area of the Sector:
The lateral surface area of a sector is given by the formula: LSA = (θ/360°) × πr²
In this case, the angle of the sector (θ) is 30° and the radius (r) is 7cm.
LSA = (30/360) × π(7)²
LSA = (1/12) × 49π
LSA = (49/12)π cm²
Total Surface Area = CSA + Area of Bases + LSA
Total Surface Area = 70π + 98π + (49/12)π
Total Surface Area = (840 + 1176 + 49/12)π
Total Surface Area = (2844/12 + 49/12)π
Total Surface Area = (2893/12)π cm²
4. Volume of the Wedge:
The volume of a right circular cone can be found using the formula: V = (1/3)πr²h
In this case, the radius of the cone (r) is 7cm and the height (h) is 5cm.
Volume = (1/3)π(7)²(5)
Volume = (1/3)(49)(5)π
Volume = (245/3)π cm³
Therefore, the total surface area of the wedge is (2893/12)π cm² and the volume is (245/3)π cm³.