Mathematics, 01.07.2019 03:00 carlos200411114

Asports company wants to package a ball with a 1.5-inch radius in sets of two. they have two options: a cylinder or a square prism. the company wants to use the package that has the least amount of wasted space. the company should choose the prism because it has approximately 11.6 in.3 less wasted space than the cylinder. the prism because it has approximately 14.1 in.3 less wasted space than the cylinder. the cylinder because it has approximately 11.6 in.3 less wasted space than the prism. the cylinder because it has approximately 14.1 in.3 less wasted space than the prism

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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:

$V_{ball} = \dfrac{4}{3} \pi r^3 = \dfrac{4}{3} \pi (1.5)^3 \approx 14.137 \: \rm inch^3$

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 = $V_{prism} = 3\times 3 \times 6 = 54\: \rm inch^3$

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:

$V_{cylinder} = \pi r^2 h =\pi (1.5)^2 \times 6 \approx 42.412 \: \rm inch^3$

How to find the wasted space's amount?

We have the wasted volume for both container as:

$W_{prism} = V_{prism} - V_{ball} = 54 - 2 \times 14.137 = 25.726 \: \rm inch^3\\
W_{cylinder} = V_{cylinder} - V_{ball} = 42.412 - 2 \times 14.137 = 14.138 \: \rm inch^3$

The difference between wasted space is:
$W_{prism} - W_{cylinder} = 25.726-14.138 \approx 11.6 \: \rm inch^3$

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:
link

Answer from: issapolly

Step 1

Find the Volume of the two balls

we know that

the Volume of a sphere is equal to

$V=\frac{4}{3} \pi r^{3}$

In this problem we have

$r=1.5\ in$

Substitute

$V=\frac{4}{3} \pi 1.5^{3}=4.5\pi\ in^{3}$

the volume of the two balls is equal

$V=2*4.5\pi=9 \pi\ in^{3}$

Step 2

Find the volume of a cylinder

we know that

The Volume of a cylinder is equal to

$V=\pi r^{2}h$

In this problem we have

$r=1.5\ in$

$h=4*r=4*1.5=6\ in$

Substitute

$V=\pi (1.5^{2})(6)=13.5 \pi\ in^{3}$

Step 3

Find the amount of wasted space in the cylinder

Subtract the volume of the two balls from the volume of the cylinder

$13.5\pi\ in^{3}-9\pi\ in^{3}=4.5\pi\ in^{3}=14.14\ in^{3}$

Step 4

Find the volume of the square prism

The volume of the prism is equal to

V=Bh

where

B is the area of the base of the prism

h is the height of the prism

In this problem we have

$B=(2*1.5)^{2}=9\ in^{2}$ -----> the base is equal to the diameter

$h=4*r=4*1.5=6\ in$

Substitute

$V=9*6=54\ in^{3}$

Step 5

Find the amount of wasted space in the prism

Subtract the volume of the two balls from the volume of the prism

$54\ in^{3}-9\pi\ in^{3}=25.73\ in^{3}$

Step 6

Compare the amount of wasted space

the amount of wasted space in the cylinder is $14.14\ in^{3}$

the amount of wasted space in the prism is $25.73\ in^{3}$

Find the difference

$25.73\ in^{3}-14.14\ in^{3}=11.59\ in^{3}=11.6\ in^{3}$

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

Answer from: Quest

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step-by-step explanation:

Answer from: Quest

answer: approximately 206 bones

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