Respuesta :
Step 1
Find the Volume of the two balls
we know that
the Volume of a sphere is equal to
[tex]V=\frac{4}{3} \pi r^{3}[/tex]
In this problem we have
[tex]r=1.5\ in[/tex]
Substitute
[tex]V=\frac{4}{3} \pi 1.5^{3}=4.5\pi\ in^{3}[/tex]
so
the volume of the two balls is equal
[tex]V=2*4.5\pi=9 \pi\ in^{3}[/tex]
Step 2
Find the volume of a cylinder
we know that
The Volume of a cylinder is equal to
[tex]V=\pi r^{2}h[/tex]
In this problem we have
[tex]r=1.5\ in[/tex]
[tex]h=4*r=4*1.5=6\ in[/tex]
Substitute
[tex]V=\pi (1.5^{2})(6)=13.5 \pi\ in^{3}[/tex]
Step 3
Find the amount of wasted space in the cylinder
Subtract the volume of the two balls from the volume of the cylinder
[tex]13.5\pi\ in^{3}-9\pi\ in^{3}=4.5\pi\ in^{3}=14.14\ in^{3}[/tex]
Step 4
Find the volume of the square prism
The volume of the prism is equal to
[tex]V=Bh[/tex]
where
B is the area of the base of the prism
h is the height of the prism
In this problem we have
[tex]B=(2*1.5)^{2}=9\ in^{2}[/tex] -----> the base is equal to the diameter
[tex]h=4*r=4*1.5=6\ in[/tex]
Substitute
[tex]V=9*6=54\ in^{3}[/tex]
Step 5
Find the amount of wasted space in the prism
Subtract the volume of the two balls from the volume of the prism
[tex]54\ in^{3}-9\pi\ in^{3}=25.73\ in^{3}[/tex]
Step 6
Compare the amount of wasted space
the amount of wasted space in the cylinder is [tex]14.14\ in^{3}[/tex]
the amount of wasted space in the prism is [tex]25.73\ in^{3}[/tex]
Find the difference
[tex]25.73\ in^{3}-14.14\ in^{3}=11.59\ in^{3}=11.6\ in^{3}[/tex]
The package that has the least amount of wasted space is the cylinder
therefore
the answer is the option
the cylinder because it has approximately 11.6 in.3 less wasted space than the prism
You can calculate volume of both prism and cylinder and subtract the volume of the sphere ball so as to find the wasted space.
Option C:the cylinder because it has approximately 11.6 in^3 less wasted space than the prism. is correct.
How to find the volume of the ball?
Since we know the radius of the ball is 1.5 inch and since ball is a spherical object mostly, thus we can use volume of sphere to find volume of the considered ball:
[tex]V_{ball} = \dfrac{4}{3} \pi r^3 = \dfrac{4}{3} \pi (1.5)^3 \approx 14.137 \: \rm inch^3[/tex]
Thus, two balls will occupy 28.274 cubic inch
How to find the volume of square prism and cylinder?
Assuming that square prism and cylinder both become as short as possible, we will have:
Side of base square prism will be of diameter of the sphere = 3 inch
Height of square prism = twice of diameter = 4 times radius = 6 inch
Thus volume will be cube of side length = [tex]V_{prism} = 3\times 3 \times 6 = 54\: \rm inch^3[/tex]
Height of cylinder would be the twice the diameter of the sphere the base of cylinder would have radius same as that of radius of the sphere.
Thus, volume of cylinder would be:
[tex]V_{cylinder} = \pi r^2 h =\pi (1.5)^2 \times 6 \approx 42.412 \: \rm inch^3[/tex]
How to find the wasted space's amount?
We have the wasted volume for both container as:
[tex]W_{prism} = V_{prism} - V_{ball} = 54 - 2 \times 14.137 = 25.726 \: \rm inch^3\\ W_{cylinder} = V_{cylinder} - V_{ball} = 42.412 - 2 \times 14.137 = 14.138 \: \rm inch^3[/tex]
The difference between wasted space is:
[tex]W_{prism} - W_{cylinder} = 25.726-14.138 \approx 11.6 \: \rm inch^3[/tex]
Thus, the company should choose cylinder.
And thus, Option C:the cylinder because it has approximately 11.6 in^3 less wasted space than the prism. is correct.
Learn more about square prism and cylinder here:
https://brainly.com/question/9067687
