Find the amount of sheet metal needed to form a conical funnel of base radius 30 cm and vertical height 50 cm, allowing for a 0.5 cm overlap at the join.
The lateral surface area is
a = 2πrs = 2πr√(r^2+h^2)
= 2π*30*√3400
= 600π√34
Assuming that the join is 0.5 cm at the base, but decreases approaching the point of the cone, the 0.5cm overlap at the base means that you have a portion of the surface area added. That amount is
0.5/(2π*30) = 1/(120π)
of the surface area. So, add an additional
1/(120π) * 600π√34 = 5√34 cm^2
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To find the amount of sheet metal needed to form a conical funnel, we will first calculate the slant height of the cone using the given base radius and vertical height.
Step 1: Calculate the slant height (l) of the cone using the Pythagorean theorem:
l = √(r^2 + h^2)
= √(30^2 + 50^2)
≈ √(900 + 2500)
≈ √3400
≈ 58.31 cm
Step 2: Calculate the circumference of the base of the cone using the formula:
C = 2πr
= 2 * 3.14 * 30
≈ 188.4 cm
Step 3: Add the overlap to the circumference:
Total circumference = C + overlap
= 188.4 + 0.5
≈ 188.9 cm
Step 4: Calculate the length of the sheet metal needed by multiplying the total circumference by the slant height of the cone:
Length of sheet metal = Total circumference * slant height
= 188.9 * 58.31
≈ 11,015.94 cm
Therefore, approximately 11,015.94 cm of sheet metal is needed to form the conical funnel with a base radius of 30 cm and a vertical height of 50 cm, allowing for a 0.5 cm overlap at the join.
To find the amount of sheet metal needed to form a conical funnel, we can calculate the lateral surface area of the cone, which will give us the surface area of the sheet metal required.
The lateral surface area of a cone can be calculated using the formula:
A = πr√(r^2 + h^2)
Where:
A is the lateral surface area,
r is the base radius of the cone, and
h is the vertical height of the cone.
In this case, the base radius (r) is given as 30 cm and the vertical height (h) is given as 50 cm.
Let's plug these values into the formula:
A = π * 30 cm * √((30 cm)^2 + (50 cm)^2)
Now let's solve the equation step by step:
A = 3.14 * 30 cm * √(900 cm^2 + 2500 cm^2)
= 3.14 * 30 cm * √(3400 cm^2)
= 3.14 * 30 cm * 58.31 cm
≈ 5405.7 cm^2
So the lateral surface area of the cone, and therefore the amount of sheet metal needed, is approximately 5405.7 cm².
Now, considering the 0.5 cm overlap at the join, we need to subtract the area of the overlap from the total surface area.
The overlap is essentially a strip surrounding the base of the cone, and its area can be calculated using the formula for the circumference of a circle:
C = 2πr
Where:
C is the circumference, and
r is the base radius.
Let's calculate the circumference:
C = 2π * 30 cm
≈ 188.5 cm
Since the overlap extends 0.5 cm from the base, the strip's width will be 0.5 cm. Therefore, the overlap area can be calculated as:
Overlap area = C * overlap width
= 188.5 cm * 0.5 cm
= 94.25 cm²
Finally, we subtract the overlap area from the total surface area to find the actual amount of sheet metal needed:
Total sheet metal needed = Total surface area - overlap area
= 5405.7 cm² - 94.25 cm²
≈ 5311.45 cm²
Therefore, approximately 5311.45 cm² of sheet metal is needed to form the conical funnel with a base radius of 30 cm, vertical height of 50 cm, and a 0.5 cm overlap at the join.