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Calculating Volumes of Boxes, Cubes, and Cylinders, Exercises of Analytical Geometry and Calculus

Formulas and examples for calculating the volumes of boxes or cubes and cylinders. Understanding volume is essential for various practical applications, such as determining how much concrete or liquid is needed to fill a container. Step-by-step solutions for calculating the volumes of different geometric figures.

Typology: Exercises

2021/2022

Uploaded on 09/12/2022

marylen
marylen 🇺🇸

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VOLUME OF A BOX, CUBE, OR CYLINDER
The concept of volume holds great practical importance. If you dig a hole that you need filled
with concrete, you need to know its volume to find out how much concrete to buy. Volume is
typically measured in cubic units. The easiest way to think about volume is to try to imagine
how much liquid you would need to pour into a geometric figure to fill it.
The first type of solid for which we will calculate volume is a box or cube (also sometimes
called a rectangular solid or a rectangular prism). In general, the formula used to find the
volume of a rectangular solid is given by
V l w!! !h
where l = the length
w = the width, and
h = the height of the figure
h
w
l
Find the volume of each figure.
Example 1 Solution:
13 6 8V
!
! !
624V
!
cubic feet (ft3)
Example 2 Solution:
"
#
"
#
"
#
5.2 4.9 6.3V
!
160.524V
!
cm3
13
f
eet
8
f
eet
6 feet
5.2 cm
6.3 cm
4.9 cm
pf2

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VOLUME OF A BOX, CUBE, OR CYLINDER

The concept of volume holds great practical importance. If you dig a hole that you need filled with concrete, you need to know its volume to find out how much concrete to buy. Volume is typically measured in cubic units. The easiest way to think about volume is to try to imagine how much liquid you would need to pour into a geometric figure to fill it.

The first type of solid for which we will calculate volume is a box or cube (also sometimes called a rectangular solid or a rectangular prism ). In general, the formula used to find the volume of a rectangular solid is given by

V! l w!! h where^ l^ = the^ length w = the width , and h = the height of the figure

h

w l

Find the volume of each figure.

Example 1 Solution:

V! 13 6 8!!

V! 624 cubic feet (ft^3 )

Example 2 Solution:

V! " 5.2 #" 4.9 #" 6.3#

V! 160.524 cm^3

13 feet

8 feet

6 feet

5.2 cm

6.3 cm

4.9 cm

The second solid for which we will calculate volume is a cylinder (sometimes referred to as a right circular solid ). In general, the formula used to find the volume of a cylinder is given by

V! $ r h^2 where^ r^ = the^ radius,^ and h = the height of the figure

h

r

Find the volume of each figure.

Example 3 Solution:

V! $ " 9 2 #" 8

V! 648 $ or 2035.752 m^3

(use your calculator’s! key)

22 m

9 m

Example 4 Solution:

Find the volume of a cylinder whose diameter is 32 inch and

whose height is 78 inch.

2r = d , so if diameter = 32 , then radius = 1 32 2!!^34. 3 2 7 4 8

V $

)/-.^ * 0 /^0

V! $ %)^ '+ ( %,&* ) &*

V! $ %)^ &*

V! $ or 1.546 in^3