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)
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
You did not provide any information on how tall the cylinder or frustrum is. All we have is 12 for the total height.
To find the answers to these questions, we'll need to use some basic geometric formulas. Let's break it down:
(a) The slant height of the frustum part:
The slant height of a frustum of a cone can be found using the formula:
l = sqrt(h^2 + (r1 - r2)^2)
Where:
- l is the slant height
- h is the height of the frustum (12 cm in this case)
- r1 is the radius of the larger base (half the diameter, so 10/2 = 5 cm)
- r2 is the radius of the smaller base (half the diameter, so 2/2 = 1 cm)
Plugging in the values:
l = sqrt(12^2 + (5 - 1)^2)
l = sqrt(144 + 16)
l = sqrt(160)
l ≈ 12.65 cm
So, the slant height of the frustum part is approximately 12.65 cm.
(b) The slant height of the smaller cone that was cut off:
The slant height of the smaller cone can be found using the Pythagorean theorem. Since we know the height of the smaller cone is the difference in heights between the entire flask and the frustum part (12 cm - the frustum height), we can use the same formula as above with the appropriate values:
l2 = sqrt((12 - h)^2 + r2^2)
Plugging in the values:
l2 = sqrt((12 - 12)^2 + 1^2)
l2 = sqrt(0^2 + 1^2)
l2 = sqrt(1)
l2 = 1 cm
So, the slant height of the smaller cone that was cut off is 1 cm.
(c) The external surface area of the flask:
To find the external surface area, we need to calculate the areas of the cylindrical part and the frustum part separately and then add them together.
The lateral surface area of the cylinder can be calculated using the formula:
A1 = 2πrh
Where:
- A1 is the lateral surface area of the cylinder
- π is a mathematical constant (approximately 3.142)
- r is the radius of the cylinder (half of the diameter, so 10/2 = 5 cm)
- h is the height of the cylinder (12 cm)
Plugging in the values:
A1 = 2 * 3.142 * 5 * 12
A1 ≈ 376.992 cm²
To calculate the lateral surface area of the frustum, we use the formula:
A2 = π(r1 + r2) * l
Where:
- A2 is the lateral surface area of the frustum
- π is the mathematical constant (approximately 3.142)
- r1 is the radius of the larger base (half the diameter, so 10/2 = 5 cm)
- r2 is the radius of the smaller base (half the diameter, so 2/2 = 1 cm)
- l is the slant height of the frustum (approximately 12.65 cm)
Plugging in the values:
A2 = 3.142 * (5 + 1) * 12.65
A2 ≈ 252.095 cm²
Now, we can find the total external surface area by adding the areas of the cylindrical and frustum parts:
Total surface area = A1 + A2
Total surface area ≈ 376.992 + 252.095
Total surface area ≈ 629.087 cm²
So, the external surface area of the flask is approximately 629.087 cm².