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Core Connections Integrated II, 2015

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Core Connections Integrated II, 2015 View details

Chapter Closure

Exercise 92 Page 636

The volume of a cone is 13 π r^2 h, where r is the radius of the cone's base and h is the cone's height. The volume of a cylinder is π r^2h, where r is the radius of the cylinder's base and h is the cylinder's height.

2 inches

We have an ice cream cone that is filled with ice cream. On top of it there is a cylindrical scoop that has the same volume as the ice cream within the cone.

We can ignore the thickness of the cone, so the ice cream that is inside the cone fills the entirety of the cone. Let's recall the formula for the volume of a cone. V = 1/3 π r^2 h We know that the cone's height is 6 inches and the radius of the cone's base is 1.5 inches. Let's substitute 1.5 for r and 6 for h in the above formula and simplify.

V = 1/3 π r^2 h

r= 1.5, h= 6

V= 1/3 π ( 1.5)^2 ( 6)

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Simplify right-hand side

Calculate power

V = 1/3 π (2.25)(6)

Multiply

V = 1/3 (13.5) π

MoveRightFacToNumOne

1/b* a = a/b

V = 13.5/3 π

Calculate quotient

V = 4.5π

The volume of the ice cream within the cone is 4.5π cubic inches. Now, recall the formula for the volume of a cylinder that has a height of h and a base with a radius r. V = π r^2 h In our case, the base's radius is 1.5 inches. We also know that the volume of the cylindrical scoop is the same as the volume of ice cream within the cone — 4.5π cubic inches. Let's substitute 1.5 for r and 4.5π for V to get an equation for h. Then, let's solve this equation.

V = π r^2 h

r= 1.5, V= 4.5π

4.5π = π ( 1.5)^2 h

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Solve for h

Calculate power

4.5π = π (2.25) h

Multiply

4.5π = 2.25 π h

.LHS /2.25π.=.RHS /2.25π.

4.5π/2.25π = h

Calculate quotient

2 = h

Rearrange equation

h = 2

The height of the scoop of ice cream on top is 2 inches.

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