To solve this, we are going to use the volume formula for a rectangular prism: [tex]V=lwh[/tex]
where
[tex]V [/tex] is the volume of the prism
[tex]l[/tex] is the length of the prism
[tex]w[/tex] is the width of the prism
[tex]h[/tex] is the height of the prism
We are also using the volume of a sphere formula: [tex]V= \frac{4}{3} \pi r^{3}[/tex]
where
[tex]V[/tex] is the volume of the sphere
[tex]r[/tex] is the radius of the sphere
Part A: Since she wants a layer of marbles in the bottom of the tank about 5 centimeters deep, the height of this little rectangular prism will be 5 centimeters, so [tex]h=5[/tex]. Since the other measures of our little rectangular prism are the same as the aquarium, [tex]l=50[/tex] and [tex]w=25[/tex]. Lets replace those values in our volume formula:
[tex]V=lwh[/tex]
[tex]V=(50)(25)(5)[/tex]
[tex]V=6250 cm^{3}[/tex]
Now that we know that she wants to cover a volume of 6250 [tex]cm^{3}[/tex] with marbles, lets find the volume of each marble.
We know that the diameter of each marble is 1 cm. Since the radius is of a sphere is half its diameter, [tex]r= \frac{1}{2} =0.5[/tex]. Lets replace that value in our volume formula:
[tex]V= \frac{4}{3} \pi r^{3}[/tex]
[tex]V= \frac{4}{3} \pi (0.5)^{3}[/tex]
[tex]V=0.52 cm^{3}[/tex]
Now, let [tex]x[/tex] the number of marbles she will need to cover a volume of 6250 [tex]cm^{3}[/tex]:
[tex]0.52x=6250[/tex]
[tex]x= \frac{6250}{0.52} [/tex]
[tex]x=12019.23[/tex]
Since we know that each bag of marbles has 500 marbles, we are going to divide the number of marbles she will need by the number of marbles in each bag:
[tex] \frac{12019.23}{500} =24.03[/tex]
We can conclude that to fill a layer of 5 cm with marbles, she will need 24 bags of marbles.
Part B: We know that she already have 5 cm of her aquarium filled with marbles, and that she also added water until it’s level is 3 centimeters below the top of the tank, so to find the height of this new rectangular prism, we are going to subtract 5 cm plus 3 cm from the original height of her aquarium:
[tex]h=30-(5+8)[/tex]
[tex]h=30-13[/tex]
[tex]h=17cm[/tex]
Since the other measures of our little rectangular prism are the same as the aquarium, [tex]l=50[/tex] and [tex]w=25[/tex]. Lets replace those values in our volume formula:
[tex]V=lwh[/tex]
[tex]V=(50)(25)(17)[/tex]
[tex]V=21250cm^{3}[/tex]
Remember that [tex]1litre=1000cm^{3}[/tex]. So:
[tex](21250cm^3)( \frac{1litre}{1000cm^{3})} =21.25liters[/tex]
We can conclude that there are 21.25 liters of water in the tank.
