Noah and Lin are making paper cones to hold popcorn to hand out at parent math night. They want the cones to hold 9 cubic inches of popcorn. What are two different possible values for height h and radius r for the cones?

height (h)=

radius (r)=

Answer:

So, The possible values of h and r are

1. r = 3 , h = 3

2. r = 1, h = 27

3. r = 2, h = 6.75

Step-by-step explanation:

P.S - The exact ques is -

As we know that Volume of cone , V = [tex]\frac{1}{3}\pi r^{2}h[/tex]

As given,

V = [tex]9\pi[/tex]

⇒[tex]\frac{1}{3}\pi r^{2}h = 9\pi \\ \frac{1}{3} r^{2}h = 9 \\r^{2}h = 27[/tex]

So,

The possible values of h and r are

1. r = 3 , h = 3

2. r = 1, h = 27

3. r = 2, h = 6.75

In Case I-

When r = 3, h = 3

(3)² ×3 = 9×3 = 27

Satisfied

Case II-

r = 1, h = 27

(1)²×27 = 1×27 = 27

Satisfied

Case III-

r = 2, h = 6.75

(2)²×6.75 = 4×6.75 = 27

Satisfied

If the radius of the base of the cone is 2 inches then the height of the cone should be 2.1485 inches.

The volume of the cone is the volume occupied by the cone. It is given by the formula,

[tex]\text{Volume of the cone}= \dfrac{1}{3} \pi r^2h[/tex]

where r is the radius of the base, and h is the height of the cone.

Now, as it is given that cones need to hold 9 cubic inches of popcorn. therefore, the volume of the cone can be written as,

[tex]\text{Volume of the cone}= \dfrac{1}{3} \pi r^2h\\\\9 \times 3 = \pi r^2h\\\\\dfrac{9 \times 3}{\pi}=r^2 h\\\\8.5943 = r^2 h[/tex]

Thus, the product of the square of the radius and the height of the cone should be such that the result is 8.5943.

Now, let the radius of the cone be 2 inches, therefore, the height of the cone is,

[tex]8.5943 = r^2 h\\\\8.5943 = (2^2) h\\\\h = 2.1485\rm\ in[/tex]

Hence, if the radius of the base of the cone is 2 inches then the height of the cone should be 2.1485 inches.

Learn more about the volume of the cone:

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