A scoop of ice cream has a 3-inch diameter. How tall should the ice cream cone of the same diameter be in order to contain all of the ice cream inside the cone?

12 inches

Step-by-step explanation:

Suppose the ice cream scoop has the shape of a sphere of a radius (r) inches.

Thus, the top of the cone will also have a radius (r) inches.

Let consider h to be the height of the cone in inches.

So, the objective is to make the volume of the scoop of the ice cream to be the same as the volume of the cone.

To do that; we need to equate both volumes together and make h the subject of the formula:

The volume of a sphere = 4/3 πr³ and the volume of a cone = 1/3πr²h

4/3 πr³  = 1/3πr²h

Multiply both sides by 3

4πr³  = πr²h

4r = h

h = 4(3)

h = 12 inches

To contain all the ice cream inside the cone, the height of the ice cream cone needs to be 12 inches tall

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step-by-step explanation:

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240