a cylindrical vessel without lid is made from a sheet of metal find the area of metal required if the length of the vessel is 35cm and its diameter is 28cm.
a = πr^2 + 2πrh
so, plug in r=14 and h=35 and do the math.
3696 cm^2
To find the surface area of the cylindrical vessel, we need to find the area of the curved surface and the area of the two circular bases.
1. Area of the curved surface:
The curved surface of the cylinder is in the shape of a rectangle when it is rolled out flat. The length of this rectangle would be the height of the cylinder (which is the same as the length of the vessel) and the width would be the circumference of the cylinder. The circumference can be calculated using the formula:
circumference = π * diameter
Given that the diameter is 28 cm, the circumference will be:
circumference = π * 28 cm
Now, the area of the curved surface is given by:
curved surface area = length * circumference
Given that the length is 35 cm, the equation becomes:
curved surface area = 35 cm * (π * 28 cm)
2. Area of the circular bases:
The area of each circular base can be calculated using the formula:
base area = π * radius^2
where radius is equal to half of the diameter.
Given that the diameter is 28 cm, the radius will be:
radius = 28 cm / 2
Now, the area of each base is given by:
base area = π * (14 cm)^2
To find the total area of the two bases, we need to multiply this by 2.
3. Total surface area:
The total surface area of the cylindrical vessel is the sum of the curved surface area and the total area of the two bases.
total surface area = curved surface area + (2 * base area)
Now, you can substitute the values into the equations and calculate the area.
To find the area of the metal required to make a cylindrical vessel without a lid, you need to calculate the surface area of the cylinder.
The surface area of a cylinder consists of three parts: the area of the curved surface (lateral area), the area of the circular top, and the area of the circular bottom (which is not included in this case because there is no lid).
1. Calculate the circumference of the circular base:
Circumference = π × Diameter
Circumference = π × 28 cm
2. Calculate the lateral area of the cylinder:
Lateral Area = Circumference × Height
Since the height is not given in the question, it is assumed that the length of the vessel refers to the height.
Lateral Area = Circumference × Length
Lateral Area = π × 28 cm × 35 cm
3. Calculate the area of the circular top (which is excluded in this case, but we will calculate it for reference):
Area of Circular Top = (π × Radius^2)
Area of Circular Top = (π × (28 cm / 2)^2)
4. Calculate the total surface area of the cylinder:
Total Surface Area = Lateral Area + Area of Circular Top
Total Surface Area = (π × 28 cm × 35 cm) + (π × (28 cm / 2)^2)
To get the final answer, you can use a calculator to evaluate the expression and find the area of the metal required.