Question

Cones A and B both have volume 48pie cubic units but have different dimensions.Cone A has radius 6 units and height 4 units.Select all the possibilities for the radius and height for cone B

Answer 1

r = 3 units, h = 16 units

r = 4 units, h = 9 units

r = 2 units, h = 36 units

r = 1 unit, h = 144 units

r = 12 units, h = 1 unit

Explanation:

The options to be selected are not included in this question, but here is the solution to the question.

The formula for the volume of a cone is given by :

V = 1/3 (πr²h)

From the question, V = 48π cubic units

Cone A has a radius, r = 6 units, h = 4 units

Since volume of cone A = volume of cone B

We need to get all the combinations of r and h that will give a volume of 48π for B

If r = 3 units,

48π = 1/3 ( π 3² h)

h = 16 units

If r = 4 units,

48π = 1/3 ( π 4² h)

h = 9 units

If r = 2 units.

48π = 1/3 ( π 2² h)

h = 36 units

If r = 1 units.

48π = 1/3 ( π 1² h)

h = 144 units

If r = 12 units.

48π = 1/3 ( π 12² h)

h = 1 unit

Answer 2

the possible combinations of radius and height for B are

(i) radius = 4 units and height = 9 units

(ii) radius = 3 units and height = 16 units

(i) radius = 2 units and height = 36 units

(iv) radius = 12 units and height = 1 unit

(v) radius = 1 unit and height = 12 units

Explanation:

Since the volume of a cone V = πr²h/3 and the volume of both cones is 48π cubic units.

For cone A, height, h = 4 units and radius, r = 6 units, Its volume is

V = πr²h/3 = π6² × 4/3 = 48π

6² × 4 = 48 × 3 = 144

Since for cone B also, V = πr²h/3 and r²h = 144. We look for all the factors of 144 which will contain a perfect square. The prime factorization of 144 = 2⁴ × 3². This combination could be 4² × 9,16 × 3² , 36 × 2², 1 × 12² and 12 × 1². So the possible combinations of radius and height for B are

(i) radius = 4 units and height = 9 units

(ii) radius = 3 units and height = 16 units

(iii) radius = 2 units and height = 36 units

(iv) radius = 12 units and height = 1 unit

(v) radius = 1 unit and height = 12 units