The flask consist of a cylindrical part and a frustum of a cone. The diameter of the base is 10 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)
not enough information
neck could be long or short
cone height plus neck = 12 cm ... is not enough info
To find the answers to these questions, we will use the formulas for the slant height of a frustum of a cone, the slant height of a smaller cone cut off, and the external surface area of a flask.
(a) To find the slant height of the frustum part, we can use the formula:
l = √(h^2 + (r1 - r2)^2)
where:
l is the slant height,
h is the vertical height,
r1 is the radius of the larger base of the frustum, and
r2 is the radius of the smaller base of the frustum.
In this case, the radius of the larger base (r1) is half the diameter, so r1 = 10 cm / 2 = 5 cm. The radius of the smaller base (r2) is also half the diameter, so r2 = 2 cm / 2 = 1 cm. The vertical height (h) is given as 12 cm. Substituting these values into the formula, we get:
l = √(12^2 + (5 - 1)^2)
l = √(144 + 16)
l = √160
l ≈ 12.65 cm
Therefore, the slant height of the frustum part is approximately 12.65 cm.
(b) To find the slant height of the smaller cone cut off, we can use the same formula:
l = √(h^2 + r^2)
where:
l is the slant height,
h is the vertical height, and
r is the radius.
In this case, the radius of the smaller cone (r) is half the diameter, so r = 2 cm / 2 = 1 cm. The vertical height (h) is given as 12 cm. Substituting these values into the formula, we get:
l = √(12^2 + 1^2)
l = √(144 + 1)
l = √145
l ≈ 12.04 cm
Therefore, the slant height of the smaller cone cut off is approximately 12.04 cm.
(c) To find the external surface area of the flask, we need to find the curved surface areas of the cylindrical part and the frustum part separately, and then add them together.
The curved surface area of the cylindrical part can be found using the formula:
A1 = 2πrh
where:
A1 is the curved surface area,
r is the radius (half the diameter, so r = 10 cm / 2 = 5 cm), and
h is the height (12 cm).
Substituting these values into the formula, we get:
A1 = 2 * 3.142 * 5 * 12
A1 = 376.8 cm^2
The curved surface area of the frustum part can be found using the formula:
A2 = π(r1 + r2)l
where:
A2 is the curved surface area,
r1 is the radius of the larger base (5 cm), and
r2 is the radius of the smaller base (1 cm),
l is the slant height (approximately 12.65 cm from part a).
Substituting these values into the formula, we get:
A2 = 3.142(5 + 1) * 12.65
A2 = 3.142 * 6 * 12.65
A2 ≈ 239.09 cm^2
Therefore, the external surface area of the flask is:
A = A1 + A2
A ≈ 376.8 + 239.09
A ≈ 615.89 cm^2
Therefore, the external surface area of the flask is approximately 615.89 cm^2.