The flask consist of a cylindrical part and a frustum of a cone. The diameter of the base is 10 cm the diameter is 2 cm while that of neck is 2 cm. The vertical height of the flask is 12 cm.
(a) the slant height of the frustum part;
(b) the slant height of the smaller cone that was cut off to make the frustum part.
(c) the external surface area of the flask. (Take pi = 3.142)
Height = 12 - 2
= 10
Cylindrical part = 12 - 2
= 2
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To find the slant height of the frustum part, we can use the Pythagorean theorem.
(a) Slant height of the frustum part:
The slant height of the frustum can be found using the formula:
l1 = sqrt(h^2 + r1^2) + sqrt(h^2 + r2^2), where h is the height and r1, r2 are the radii of the frustum's base and top.
In this case, the height (h) is 10 cm, the larger radius (r1) is half of the diameter of the base, which is 5 cm, and the smaller radius (r2) is half of the diameter of the neck, which is 1 cm.
So, plugging these values into the formula, we get:
l1 = sqrt(10^2 + 5^2) + sqrt(10^2 + 1^2)
l1 = sqrt(100 + 25) + sqrt(100 + 1)
l1 = sqrt(125) + sqrt(101)
l1 ≈ 11.180 + 10.050
l1 ≈ 21.230 cm
Therefore, the slant height of the frustum part is approximately 21.230 cm.
Now, to find the slant height of the smaller cone that was cut off, we can use a similar formula.
(b) Slant height of the smaller cone that was cut off:
The slant height of the small cone can be found using the formula:
l2 = sqrt(h^2 + r2^2), where h is the height and r2 is the radius of the small cone (same as the radius of the frustum's neck).
In this case, we already know that the height (h) is 10 cm and the radius (r2) is 1 cm.
So, plugging these values into the formula, we get:
l2 = sqrt(10^2 + 1^2)
l2 = sqrt(100 + 1)
l2 ≈ sqrt(101)
l2 ≈ 10.050 cm
Therefore, the slant height of the smaller cone that was cut off is approximately 10.050 cm.
Lastly, to find the external surface area of the flask, we need to calculate the areas of the cylindrical part and the frustum part separately.
(c) External surface area of the flask:
The external surface area of the cylindrical part is given by the formula:
A_cylinder = 2πrh, where r is the radius of the cylindrical part and h is the height of the cylindrical part.
In this case, the radius (r) is half the diameter of the cylindrical part, which is 2.5 cm, and the height (h) is 2 cm.
So, plugging these values into the formula, we get:
A_cylinder = 2π(2.5 cm)(2 cm)
A_cylinder = 10π cm²
The external surface area of the frustum part can be calculated using the formula for the lateral surface area of a frustum:
A_frustum = π(r1 + r2)l1, where r1 and r2 are the radii of the frustum's base and top, and l1 is the slant height of the frustum.
In this case, we already know that r1 is 5 cm, r2 is 1 cm, and l1 is approximately 21.230 cm.
So, plugging these values into the formula, we get:
A_frustum = π(5 cm + 1 cm)(21.230 cm)
A_frustum = 6π(21.230 cm)
A_frustum ≈ 127.382π cm²
Now we can calculate the total external surface area of the flask by adding the areas of the cylindrical part and the frustum part:
A_total = A_cylinder + A_frustum
A_total ≈ 10π cm² + 127.382π cm²
A_total ≈ 137.382π cm²
Since we are given π = 3.142, we can substitute this value to find the approximate external surface area in square centimeters:
A_total ≈ 137.382(3.142) cm²
A_total ≈ 431.653 cm²
Therefore, the external surface area of the flask is approximately 431.653 cm².