A cylindrical birthday cake of radius 23cm and height 12cm is to be shared equally among 18 people at a party,the piece of cake was shaped as AOB to give 20 degrees,hence prove that.Also calculate the area of cylindrical cake.
1cake=360
When cake is divided into 18 part equally then
18part of cake=360/18=20degree
Therefore AngleAOB =20degree
Total surface area of cylinder= 2Pie r*h+2pie r^2
=2*22/7*23*12+2*22/7*23^2
=1734.85+3325.14
=5059.99
To prove that the piece of cake is shaped as AOB with a 20 degree angle, we can use the properties of a circle.
Step 1: Draw a diagram of the cylindrical cake.
O
_______
/ /
/ /
A-------B
Here, O represents the center of the circular base, AB is the diameter of the circular base, and AOB represents the piece of cake.
Step 2: Calculate the angle AOB.
Since the entire circle has 360 degrees and the cake is shared equally among 18 people, each person will receive 360/18 = 20 degrees.
Therefore, the angle AOB is 20 degrees.
To calculate the area of the cylindrical cake, we will use the formula for the lateral surface area of a cylinder.
Step 3: Find the area of the circular base.
The area of a circle is given by the formula: A = πr^2, where r is the radius.
Given that the radius of the cake is 23 cm, the area of the circular base is:
A = π(23 cm)^2
= π(529 cm^2)
≈ 1661.9 cm^2 (rounded to one decimal place)
Step 4: Find the area of the curved surface of the cake.
The curved surface area of a cylinder is given by the formula: A = 2πrh, where r is the radius and h is the height.
Given that the radius of the cake is 23 cm and the height is 12 cm, the curved surface area of the cake is:
A = 2π(23 cm)(12 cm)
= 2π(276 cm^2)
≈ 1735.6 cm^2 (rounded to one decimal place)
Therefore, the area of the cylindrical cake is approximately 1735.6 cm^2.
To prove that the piece of cake, shaped as AOB, gives an angle of 20 degrees, we can use the properties of arcs and angles in a circle.
In a circle, the angle formed by an arc is equal to the central angle that subtends that arc. So, in this case, angle AOB (θ) is equal to the angle formed by the arc that represents the piece of cake, which is 20 degrees.
To calculate the area of the cylindrical cake, we need to first find the lateral surface area of the cylinder, and then add the area of the two circular bases.
1. Lateral Surface Area:
The lateral surface area (A) of a cylinder is given by the formula:
A = 2πrh
Where r is the radius of the circular base (23 cm) and h is the height of the cylinder (12 cm).
A = 2π(23 cm)(12 cm)
A = 552π cm²
2. Circular Base Area:
The area of a circle is given by the formula:
A = πr²
For the cylindrical cake, there are two circular bases, so the total area of the bases is:
2A = 2π(23 cm)²
2A = 1058π cm²
3. Total Area:
To calculate the total area of the cylindrical cake, we add the lateral surface area and the area of the bases:
Total Area = Lateral Surface Area + 2 × Base Area
Total Area = 552π cm² + 1058π cm²
Total Area = 1610π cm²
Therefore, the area of the cylindrical cake is 1610π square centimeters.
So, based on the given information, the piece of cake shaped as AOB gives an angle of 20 degrees, and the area of the cylindrical cake is 1610π square centimeters.