Respuesta :

Answer:

Part A) The volume of the ice cream scoop is [tex]36\pi\ in^{3}[/tex]

Part B) The melted ice cream won't fill the cup

Part C) The melted ice cream exceeds the volume  of Anna's cup

Part D) The height of the smallest cylindrical cup is [tex]h=8\ in[/tex]

Step-by-step explanation:

Part A) Find the volume of the ice cream scoop

we know that

The volume of the sphere (ice cream scoop) is equal to

[tex]V=\frac{4}{3}\pi r^{3}[/tex]

we have

[tex]r=3\ in[/tex]

substitute

[tex]V=\frac{4}{3}\pi (3)^{3}=36\pi\ in^{3}[/tex]

Part B) Find the volume of Anna's cylindrical cup

we know that

The volume of a cylinder is equal to

[tex]V=\pi r^{2}h[/tex]

we have

[tex]r=3\ in[/tex]

[tex]h=6\ in[/tex]

substitute

[tex]V=\pi (3)^{2}(6)=54\pi\ in^{3}[/tex]

[tex]54\pi\ in^{3}> 36\pi\ in^{3}[/tex]

The volume of Anna's cup (cylinder) is greater than the volume of melted ice cream scoop

therefore

The melted ice cream won't fill the cup.

Part C) Will two scoop of melted ice cream fit in Anna's cup?

Multiply the volume of ice cream scoop by 2 and compare with the volume of Anna's cup

so

[tex](2)36\pi\ in^{3}=72\pi\ in^{3}[/tex]

[tex]72\pi\ in^{3}> 54\pi\ in^{3}[/tex]

The volume of Anna's cup (cylinder) is less than the volume of two melted ice cream scoop

therefore

The melted ice cream exceeds the volume  of Anna's cup

Part D) Find the smallest cylindrical cup that will hold two scoops of melted ice cream

we know that

The volume of a cylinder is equal to

[tex]V=\pi r^{2}h[/tex]

we have

[tex]V=72\pi\ in^{3}[/tex] ------> volume of two scoops of melted ice cream

[tex]r=3\ in[/tex]

substitute in the formula and solve for h

[tex]72\pi=\pi (3)^{2}h[/tex]

simplify

[tex]72=(9)h[/tex]

[tex]h=8\ in[/tex]