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]