Unlock Real-World Applications of Three-Dimensional Modeling, Geometric Shapes, and Volume Calculations

	Volume of the Cylinder	Volume of the Cone
Formula	$V_{1} = π r^{2} h$	$V_{2} = \frac{1}{3} π r^{2} h$

Volume of the Cylinder

Volume of the Cone

Formula

V_{1} = π r^{2} h

V_{2} = \frac{1}{3} π r^{2} h

	Volume of the Cylinder	Volume of the Cone
Formula	$V_{1} = π r^{2} h$	$V_{2} = \frac{1}{3} π r^{2} h$
Substitute Values	$V_{1} = π (5)^{2} 5$	$V_{2} = \frac{1}{3} π (5)^{2} 5$
Calculate	$V_{1} = 125 π$	$V_{2} = \frac{1 2 5 π}{3}$

Volume of the Cylinder

Volume of the Cone

Formula

V_{1} = π r^{2} h

V_{2} = \frac{1}{3} π r^{2} h

Substitute Values

V_{1} = π (5)^{2} 5

V_{2} = \frac{1}{3} π (5)^{2} 5

Calculate

V_{1} = 125 π

V_{2} = \frac{1 2 5 π}{3}

Cavalieri Principle
Two solids with the same height and the same cross-sectional area at every altitude have the same volume.

	Aquarium $A$	Aquarium $B$
Height $(ft)$	$5$
Cross-Sectional Area $(ft^{2})$	$10 π h - π h^{2}$

Aquarium

A

Aquarium

B

Height

(ft)

5

Cross-Sectional Area

(ft^{2})

10 π h - π h^{2}

Type of Glass
Cocktail Glass	Collins Glass
$8 \cdot 6$	$5 \cdot 10$
$$ 48$	$$ 50$

Type of Glass

Cocktail Glass

Collins Glass

8 \cdot 6

5 \cdot 10

$ 48

$ 50

A cylinder-shaped cistern has an inner diameter of $10$ meters and currently holds $1200$ cubic meters of oil. The oil fills up two-thirds of the cistern. What is the difference between the height the oil reaches and the height of the cistern? Round the difference to one decimal place.

Examining the diagram, we see that the oil is in itself a cylinder with the same inner diameter as the cistern. Since we have been given the volume of the oil, we can determine how high it reaches by using the formula for the volume of a cylinder and solving for the height h. V=π r^2 h

Height of the Oil

Let's substitute the volume, 1200 cubic centimeters and radius, 102=5 meters, into the formula to solve for the height the oil reaches.

We will keep the height of the oil in exact form to maintain precision accuracy in our calculations.

Height of the Cistern

We know that the oil reaches two-thirds of the cylinder's height. If we call the cylinder's height H, we can write the following equation. 2/3H=h Let's substitute the value of h and solve for H.

The height of the cistern is 72π meters.

Difference Between Heights

To determine the height from the top of the oil to the top of the cistern, we can subtract the height where the oil reaches from the height of the cistern.

The difference between the two variables is about 7.6 meters.

A cube of copper has side lengths of

20

centimeters. How many smaller cubes can be made from the original cube if each of the smaller cubes will have a side length of

2

centimeters?

Each side length is 20 centimeters. Divide that number by 2, which is the target length of the new smaller cubes. This tells us that we can fit ten 2-centimeter segments. Therefore, along each dimension, we would be able to fit 10 small cubes.

If we multiply the number of small cubes that fit along three of the original cube's sides, we get the total number of small cubes that can be created from the original cube. (10)(10)(10)=1000

A silo for wheat consists of a cylinder and a hemisphere.

The total height of the silo is

30

meters, and it has a radius of

5

meters. How many cubic meters of space are available to hold wheat? Round the amount to the nearest cubic meter.

From the given information, we know that the radius of the cylinder is 5 meters. This means that the cylinder's height is 25 meters.

To calculate the volume of this composite solid, we have to determine the volume of the cylinder and of the hemisphere then add the results.

Volume of the Hemisphere

To calculate the volume of the hemisphere V_(HS), we divide the formula for finding the volume of a sphere by 2. V_(HS) = 43π r^3/2 Let's substitute the hemisphere's radius into the formula and evaluate.

The volume of the hemisphere is 250π3 cubic meters.

Volume of the Cylinder

To calculate the volume of the cylinder V_C, we multiply the base area, which is a circle, by its height.

The cylindrical part of the silo has a volume of 625π cubic meters.

Volume of Silo

Now that the volume of each part is known, we can determine the silo's total volume by adding its parts.

The volume of the silo is about 2225 cubic meters. That is the amount of available space to hold wheat or any other substance.

Which box gives the most cereal in relation to its cost?

To compare the prices, we have to calculate the amount of cubic inches of cereal we get per dollar for each of the boxes. In other words, we want to divide the volume of each box by their respective price. Volume of Cereal/Price We will start by finding the volume of each box.

Volume of the Larger Box

Let's calculate the volume of the large box. This is a rectangular prism. To find its volume, we multiply, its width, length and height. From the diagram, we can see w= 3, l = 8, and h= 10. Let's multiply them.

The volume of the large box is 240 cubic inches.

Volume of the Smaller Box

We will calculate the volume of the small box in the same way. From the diagram we can see that the dimensions are w= 2, l = 6, and h= 8.

The volume of the small box is 96 cubic inches.

Amount of Cereal in Cubic Inches per Dollar

Now we can find how many cubic inches of cereal we get per dollar by dividing the volume of each box by the respective price. Large box:& 240/6=40in.$^3$ per dollar [1em] Small box:& 96/3=32in.$^3$ per dollar With the large box, we get 40 cubic inches of cereal per dollar as opposed to 32 cubic inches of cereal per dollar when buying the smaller box. Therefore, buying the larger box is the better deal in terms of cost and amount of cereal.

A jeweler is offering to sale Zain either seven golden marbles with a radius of $4$ millimeters each, or one golden marble with a radius of $8$ millimeters? Which offer should Zain choose if they want the most amount of gold?

7 small golden marbles and one giant golden marble

To compare the offers, we will calculate the combined volume of the seven smaller marbles and then the volume of the giant marble. It appears that the offer with the bigger volume of gold is better. Let's check if this is the case. Since these are spheres, we will use the formula for the volume of a sphere. V=4/3π r^3

Total Volume of Seven Smaller Marbles

Before calculating the total volume of the smaller marbles, we will calculate the volume of one of them. Then, we will multiply it by 7 to account for all of the marbles.

Finally, we multiply the volume of one marble by 7 to get the total volume of the seven marbles. 7V=7(256π/3)=1792π/3

Volume of the Giant Marble

Again, we will use the same formula to calculate the volume of the giant marble. To do so, we will substitute r=8 into the formula.

Comparison

Since we have found both volumes, we can compare the offers. Small marbles:& V=1792π/3 [0.75em] Giant marble:& V=2048π/3 Since the volume of the giant marble has a greater numerator, Zain should choose the giant marble to obtain the most amount of gold.

Modeling in Three Dimensions

Catch-Up and Review

Comparing Aquarium Volumes

Comparing Aquarium Volumes II

Hint

Solution

Aquarium $A$

Aquarium $B$

Finding the Length of a Toilet Paper Roll

Hint

Solution

Cylindrical Soda Can

Hint

Solution

Choosing Type of Glass

Hint

Solution

Estimating How Many Grains of Sand a Hand Can Hold

Answer

Hint

Solution

Finding the Volume of a Grain

Finding the Mass of a Grain

Finding the Number of Grains