Q.1) The volume of a right circular cone is 37.68 dm3 .If the the base is 3 dm, find (1) the height (2) the curved surface area of the cone .( Take π=3.14)
V of cone = (1/3) π r^2 h
37.68 = (1/2) π (1.5)^2 h , I assume the base has a diameter of 3 dm
h = 2(37.68)/(2.25π) = .....
(you are obviously going to use your calculator, so why not use the built-in value of π in your calculator for more accuracy. Btw, don't round off h at this point, keep the value of h in your calculator's memory)
Surface area of cone = πrs, where s is the slant height
so we need s
s^2 = r^2 + h^2
= 1.5^2 + h^2 from above,
so you can find s.
sub into the surface area formula
The base is 3dm? I assume that is the diameter.
Volume=1/3 Base Area*height
height= 3*volume/base area=3*37.68/(PI(3/2)^2)
Surface area=PI*r*sqrt(r^2+h^2)
To find the height and curved surface area of the cone, we can use the formula for the volume and curved surface area of a cone.
Formula for the volume of a cone:
V = (1/3) * π * r^2 * h
Formula for the curved surface area of a cone:
CSA = π * r * l
Where:
V = Volume of the cone
π = Pi, approximately 3.14
r = Radius of the base of the cone
h = Height of the cone
CSA = Curved surface area of the cone
l = Slant height of the cone
Given:
Volume (V) = 37.68 dm3
Base radius (r) = 3 dm
π = 3.14
Let's solve for the height (h) first:
Using the volume formula:
V = (1/3) * π * r^2 * h
Plugging in the values:
37.68 = (1/3) * 3.14 * 3^2 * h
Simplifying:
37.68 = 3.14 * 9 * h
Dividing both sides by (3.14 * 9):
h = 37.68 / (3.14 * 9)
Calculating the height:
h ≈ 1.4 dm (rounded to one decimal place)
Now let's find the curved surface area (CSA) of the cone:
Using the curved surface area formula:
CSA = π * r * l
Since we don't know the slant height (l) directly, we can calculate it using the height (h), base radius (r), and Pythagoras' theorem.
Using Pythagoras' theorem:
l^2 = r^2 + h^2
Plugging in the values:
l^2 = 3^2 + 1.4^2
Simplifying:
l^2 = 9 + 1.96
l^2 = 10.96
Taking the square root of both sides:
l ≈ √10.96
Calculating the slant height:
l ≈ 3.31 dm (rounded to two decimal places)
Now we can find the curved surface area by substituting the values in the formula:
CSA = π * r * l
CSA = 3.14 * 3 * 3.31
Calculating the curved surface area:
CSA ≈ 98.37 dm2 (rounded to two decimal places)
(1) The height of the cone is approximately 1.4 dm.
(2) The curved surface area of the cone is approximately 98.37 dm2.