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)
total height = 12
height = 10
Cylindrical part height = 12 - 2
cylindrical radius = 1
over and over
you ever gonna provide any input? I already suggested that you consider the top part of the cone that was cut off. Subtract that from the total cone, and you have the data for the frustrum.
If the frustrum's height is 2, and its base has diameter 10, and the top has diameter 2, then using similar triangles, you can see that the height of the missing part has height 1/2. So, the slant height of the frustrum is 4/5 of the total slant height of (5/2)√5.
To find the slant height of the frustum part, we can use the Pythagorean theorem.
(a) Slant height of the frustum part:
We know that the diameter of the base of the frustum part is 10 cm, which means the radius is 5 cm. The diameter of the top (neck) of the frustum is 2 cm, so the radius of the top is 1 cm. The height of the frustum is the same as the total height of the flask, which is 12 cm.
Using the Pythagorean theorem, we can find the slant height (l) of the frustum part:
l^2 = (5 - 1)^2 + 12^2
l^2 = 16 + 144
l^2 = 160
l = sqrt(160)
l ≈ 12.65 cm
Therefore, the slant height of the frustum part is approximately 12.65 cm.
(b) Slant height of the smaller cone that was cut off:
To find the slant height of the smaller cone that was cut off to make the frustum part, we need to subtract the slant height of the frustum from the total height of the flask.
Slant height of the smaller cone = total height of the flask - slant height of the frustum part
Slant height of the smaller cone = 12 cm - 12.65 cm
Slant height of the smaller cone ≈ -0.65 cm
Note: A negative value for the slant height of the smaller cone suggests an error in the given dimensions. Please recheck the given values.
(c) External surface area of the flask:
The external surface area of the flask consists of the curved surface area of the cylindrical part and the frustum part.
Curved surface area of the cylindrical part = 2πrh
Curved surface area of the cylindrical part = 2 * 3.142 * 1 * (12 - 2) [radius = 1 cm, height = 12 - 2 cm]
Curved surface area of the cylindrical part ≈ 75.34 cm^2
To find the curved surface area of the frustum part, we first need to find the slant height of the top of the frustum. The top of the frustum is a smaller cone.
Slant height of the top of the frustum = slant height of the frustum part − slant height of the smaller cone
Slant height of the top of the frustum ≈ 12.65 cm - (-0.65 cm)
Slant height of the top of the frustum ≈ 13.3 cm
Curved surface area of the frustum part = π(R_top + R_base) * slant height of the top of the frustum
Curved surface area of the frustum part = 3.142 * (1 + 2) * 13.3 [R_top = 1 cm, R_base = 2 cm, slant height ≈ 13.3 cm]
Curved surface area of the frustum part ≈ 119.43 cm^2
External surface area of the flask = Curved surface area of the cylindrical part + Curved surface area of the frustum part
External surface area of the flask ≈ 75.34 cm^2 + 119.43 cm^2
External surface area of the flask ≈ 194.77 cm^2
Therefore, the external surface area of the flask is approximately 194.77 cm^2.