a quadrant of circle of radius 7 is folded to make a cone . find curved surface area and volume of cone

Question

surface area and volume

Answer 1

The formula for the volume of a cone is V = π r² h/3

 

In this case we need to find the radius of the circle at the top of the cone as well as the height of the cone.

 

We know that the radius of the original circle from which the cone was taken was 7.  Therefore the circumference of the original circle was C=2πr or 14π

 

Since the cone was taken from one quadrant of the circle, or 1/4 of the circle, the circumference of the circle at the top of the cone is (14/4)π = (7/2)π

 

Using the circumference formula again, we can determine the radius of this circle

(7/2)π = 2πr

so r =  7/4

 

To determine the height of the cone, we use the radius we just calculated and the fact that the original radius was 7.  

 

The original radius becomes the slant of the cone.

 

What we have is a right triangle with a base of 7/4 (the radius of the circle of the cone) and a hypotenuse of 7 (slant).  We can solve for the height of the cone using a² + b² = c², where a = 7/4 and c = 7

 

(7/4)² + b² = 7²

and b² = 49 - 49/16 = 735/16

and b = √(735/16) = (7/4)√15  Height

 

Using the original formula  V = π r² h/3

V = π (7/4)² [(7/4)√(15)] / 3

V = π (7/4)³ √15 / 3

V = (343/192) √15 π

V = 6.92π or 21.73

 

Hope this helps.

Answer 2

Hi Ajay, the surface area of a cone is: π * r * s + π * r². The volume of a cone is: 1 / 3 * π * r² * h. If we start with a circle of radius 7 and fold one quarter of it to make a cone we will have a cone with the following dimensions:

 

The radius of the circle, 7 units, becomes the side or slant of the cone.

 

The circumference of the circle of radius 7 units is: C = π * d = Π * 2 * r ≈ 44 units.

 

One-quarter of the circumference of the circle will be the circumference of the cone, or 11 units.

 

Use the formula for the circumference of a circle again to find the diameter and radius of the cone.

 

C = π * d

 

Substitute for C and solve for d:

 

11 = π * d

 

d = 11 / π = 3.5 units

 

Radius of cone:

 

r = d / 2 = 1.75 units

 

 Height of cone:

 

Use the Pythagorean Theorem to find the height of the cone.

 

a² + b² = c²

 

Let a represent one leg of the right triangle. Let b represent the other leg of the right triangle, and c will represent the hypotenuse.

 

a = 1.75 units

 

b = unknown or height of the cone

 

c = 7 units or slant length of the cone

 

Substitute:

 

1.75² + b² = 7²

 

Solve for b:

 

b = √(7² - 1.75²)

 

b = 6.78 units 

 

We now have all the information to answer the question:

 

Radius of cone: 1.75 units

 

Height of cone: 6.78 units

 

Slant length of the cone: 7 units

 

Surface area of the cone:

 

S = π * r * s + π * r²

 

Substituting:

 

S = π * 1.75 * 7 + π * 1.75²

 

S = 48.11 square units

 

Volume of cone:

 

V = 1 / 3 * π * r² * h

 

Substituting:

 

V = 1 /3 * π * 1.75² * h

 

V = 21.74 cubic units

 

Questions?

Answer 3

 

There are some very key words in this problem statement:  “quadrant of a circle … folded.”  That means that the lateral surface area of the cone is (1/4)(area of circle).  So,

    A_L = (1/4)(∏7²)

    A_L = (49/4)∏ 

 

If, by “curved surface area,” you mean only the lateral surface area, then A=(49/4)∏.

 

Now, it makes sense to keep ∏ as long as possible, but when you must use an approximation, PLZ realize that this problem has what I call “magic numbers” (they make the computations easy so that you can concentrate on the lesson).  PLZ use the approximation of ∏≈22/7.  [also, PLZ keep fractions until they must be changed]

      A_L=(49/4)(22/7) = 77/2   

 

Otherwise (that is, if the little circle base is part of the area),

       Circumference of original circle = ∏d = 2∏r = 14∏

       Circumference of cone’s circle = (1/4)14∏ = (7/2)∏ = 2∏r

       Radius of cone’s circle = r = 7/4

 

       Area of cone’s circle = A_C= ∏r2 = ∏ (7/4)(7/4) = (49/16)∏ = (49/16)(22/7) = 77/8

 

      A = A_L + A_C

      A = 77/2 +  77/8 =  385/8 = 48.125 sq-units

 

Volume of cone:     (1/3)∏r²h

  From the Pythagorean Theorem, h = √(72 – (7/4)2) = (7/4)√(15)

  V = (1/3)(22/7)(7/4)(7/4)( (7/4)√(15) )

  V ≈ 21.75 cu-units