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)
Consider the cut-off part of the cone. If the height of the frustrum is 4x, then since the radius shrinks from 5 to 1, using similar triangles, the missing top part has height x.
thus, the volume of the missing part is 1π/3 * 1^2 x = π/3 x
The volume of the frustrum is thus π/3 (5^2*5x - 1^2 x) = 8πx
(a) s1^2 = (4x)^2+(5-1)^2
(b) s2^2 = x^2+1^2
(c) a = π*5^2*(12-x) + 2π*5*s1 - 2π*1*s2
Height = 12 - 2
= 10
Cylindrical part = 12 - 2
= 2
To find the answers to the questions, we need to use some formulas and calculations. Let's break it down step by step.
(a) To find the slant height of the frustum part:
First, we need to find the slant height of the larger end of the frustum. We can use the Pythagorean theorem to do this.
The radius of the large end of the frustum (R) is half of the diameter, so R = 10 cm / 2 = 5 cm.
The height of the frustum (h) is the vertical height of the flask, which is given as 12 cm.
By using the Pythagorean theorem, we have:
Slant height of the frustum = sqrt(R^2 + h^2)
= sqrt((5 cm)^2 + (12 cm)^2)
(b) To find the slant height of the smaller cone that was cut off:
We can find the radius of the small end of the frustum (r) by using the diameter, which is 2 cm.
The height of the small cone (h') is the height of the frustum minus the height of the cylindrical part, which is 12 cm - (10 cm - 2 cm) = 4 cm.
By using the Pythagorean theorem, we have:
Slant height of the smaller cone = sqrt(r^2 + h'^2)
= sqrt((1 cm)^2 + (4 cm)^2)
(c) To find the external surface area of the flask:
The external surface area of the flask consists of the curved surface area of the frustum and the curved surface area of the cylindrical part.
Curved surface area of the frustum = π(R + r) * slant height of the frustum
= 3.142 * (5 cm + 1 cm) * slant height of the frustum
Curved surface area of the cylindrical part = 2 * π * R * height of the cylindrical part
= 2 * 3.142 * 5 cm * 10 cm
Finally, add the curved surface areas of the frustum and the cylindrical part together to get the external surface area of the flask.