A bucket holds 28.49 liters of milk .the radii of the top and bottom are 28cm and 21 cm respectively.find the total surface area of the bucket.
Sketch a cone with the vertex downwards , and top radius of 28 cm
create a truncated cone by sketching a cone inside this one with a top radius of 21
The top part of your diagram is your bucket.
The first part is to find the height of the bucket.
let the height of the bottom part of the cone, (the part we will cut off) be x
let the height of the bucket part be h
by ratios:
28/(x+h) = 21/x
21x + 21h = 28x
21h = 7x
x = 3h
So the whole cone has a height of x+h or 4h
and the bucket has a height of h
Volume of whole cone = (1/3)π (28^2)(4h)
volume of bottom part = (1/3)π (21^2)(h)
volume of bucket = difference between the above two expression, and factoring
= (1/3)π(h) (4(28^2) - 21^2)
= 2822.1974h
but the volume is given as 28.49 L = 28490 cm^3
so 2822.1974h = 28490
h = 10.09 cm
Now we need the lateral surface area + area of the bottom of the bucke
the last part would simply be 2π(21) or 42 π
the first part is harder:
look at
http://www.vitutor.com/geometry/solid/truncated_cone.html
and continue from there
There is also a lateral surface area of a truncated cone Calculator, (how about that ?)
enter 28 or top radius
enter 21 for bottom radius
enter 10.09 for height, (don't choose slant height)
to get 5738.87 cm^2
good luck
forgot to give you the link to the calculator
http://www.onlineconversion.com/object_surfacearea_trunc_cone.htm
extend the cone of the bucket, so it is of height H. The height h of the bucket is H/4 since 21/28 = 3/4.
subtract the volume of the 21cm cone from a 28-cm cone:
v = 1/3 pi 28^2 H - 1/3 pi 21^2 h
pi/3 (4*28^2-21^2)h = 28490 cm^3
h = 10.095 cm
H = 40.380 cm (height of 28-cm cone)
H-h = 30.285 (height of 21-cm cone)
the area of the bucket is the area of the 28-cm cone minus the 21-cm cone:
a = pi 28 √(28^2+40.380^2) - pi 21 √(21^2+30.285^2) = 1891 cm^2
hmmm. guess I blew it somewhere.
hey I confused
Steve , we both got the same answer of 10.1 for the height of the bucket, so that seems good.
I didn't actually try any calculation for the surface area, simply used the "online calculator", assuming it was correct.
formula for volume frustum of a cone is: 1/3 π h (R² + Rr + r²)
1 liter = 1000 cubic cm
28.49 liters = 28490 cubic cm
1/3 π h (R² + Rr + r²) = 28490
Substitute values to get h and then find l
by s = √((r1 - r2)2 + h2) then substitute l in tsa formula
TSA formula π(R+r)l+πR²+πr²
h = 15 cm
To find the total surface area of the bucket, we need to consider the curved surface area of the cylinder and, in addition, the areas of the two circular bases.
To start, let's calculate the curved surface area of the cylinder. The formula for the curved surface area of a cylinder is given by:
Curved Surface Area = 2πrh
Where:
π is the mathematical constant pi (approximately 3.14)
r is the radius of the base of the cylinder
h is the height of the cylinder (which is the same as the difference in radii, since the bucket is in the shape of a frustum)
Given that the radius of the bottom base is 21 cm and the radius of the top base is 28 cm, we can find the height (h) of the frustum by subtracting the two radii:
h = 28 cm - 21 cm
h = 7 cm
Now we can substitute the values into the formula to calculate the curved surface area:
Curved Surface Area = 2πrh
Curved Surface Area = 2 * 3.14 * 7 cm * 28 cm
Next, let's calculate the areas of the circular bases. The formula for the area of a circle is given by:
Area = πr^2
We'll calculate the area of the top and bottom bases separately using the respective radii:
Area of top base = π(28 cm)^2
Area of bottom base = π(21 cm)^2
Finally, we can find the total surface area by summing up the curved surface area and the areas of the circular bases:
Total Surface Area = Curved Surface Area + Area of top base + Area of bottom base
By plugging in the values and evaluating the expression, you'll find the total surface area of the bucket.