Dimensional Analysis Dimensional analysis - Arizona State University

Dimensional Analysis

We will start our study of geometry with dimensional analysis. Dimensional analysis is the fancy name for how to convert between various types of units; feet to miles, kilometers per hour to meters per second, square feet to square inches, cubic meters to cubic centimeters; etc. Dimensional analysis is the process of using a standard conversion to create a fraction, including units in that fraction and canceling units in the same way that variables are cancelled.

We do need to know some relationships between units:

1 mile = 5280 feet, 3 feet = 1 yard, 1000 meters = 1 kilometer, 100 centimeters = 1 meter

If you write out the full process and make sure that the units cancel leaving only what you want, you should be successful in converting units. The remainder of this section is examples.

Example 1. Convert 6 feet to yards.

Solution:

6 feet 1 yard 6 yard

6 feet =

?

=

= 2 yards

1 3 feet

3

Example 2. Convert 765 inches per second to miles per hour:

Solution:

765 inches foot 1 mile

765 miles

765 in/sec =

?

?

=

second 12 inches 5280 feet 12 ? 5280 second

17 miles 60 second 60 minute 17 ? 60 ? 60 miles

=

?

?

=

1408 second 1 minute 1 hour

1408 hour

7650

3925

=

mi/hr =

mi/hr 43.4659 mi/hr

176

88

Example 3. Convert 40 square feet to square inches.

Solution:

40 ft2 = 40 ft ? ft ? 12 in ? 12 in = 5760 in2

1

1 ft 1 ft

Example 4. Convert 2 cubic yards to cubic feet.

Solution:

2 yd3 =

2 yd ?

yd ?

yd 3 ft 3 ft 3 ft ???

= 2 ? 3 ? 3 ? 3 ft3 = 54 ft3

1

yd yd yd

Example 5. Convert 75 centimeters per second to meters per hour:

Solution:

76 cm 1 meter 60 second 60 minute

75 cm/sec =

?

?

?

second 100 cm 1 minute 1 hour

76 ? 60 ? 60 miles

=

= 2736 m/hr

100 hour

1

2

Geometry

For more practice and to ensure that the process is clear, we will also do some with unfamiliar (made up) units.

Example 6. Suppose we are given that 13 horks is equivalent to one plop, 7 plops are equivalent to one wooze, 5 hons is equal to one slop and 11 slops are equal to one murk. Convert:

(1) 80 horks to woozes (2) 9 square plops to square horks (3) 2 cubic woozes to cubic plops (4) 12 horks per hon to plops per murk (5) 18 cubic horks per murk to cubic plops per slop

Solution:

(1)

80 hork plop wooze 80

80

80 horks =

?

?

=

woozes = woozes 0.8791 woozes

1 13 hork 7 plop 13 ? 7

91

(2)

9 plops2 = 9 plop ? 13 hork ? 13 hork = 1521 horks2

1

plop plop

(3) 2 woozes3 = 2woozes ? 7 plop ? 7 plop ? 7 plop = 686 plops3 1 wooze wooze wooze

(4)

12 horks plop 5 hon 11 slop 660

12 horks/hon =

?

?

?

= plops/murk 50.7692 plops/murk

hon 13 hork slop murk 13

(5)

18

horks3/murk

=

18

horks3 ?

plop

?

plop

?

plop

5 hon 11 slop 660

??

= plops/mork 50.7692 plops/

hon 13 hork 13 hork 13 hork slop murk 13

!"M#$A"T%1&4'2()- G#e%o"m*(etry

3

+"&,$"%"&(-./(0&"-(

+-1"(2(#3(45(

Perimeter and Area

!

W atethaereongoe-ind" " -gi(m0##t!!#e1$o)%n(+4#c4s(!!'oi4&')o'n'$n(t64)!ai!&+n$l8!**u$d!'7*e!i2#+s#*!ot7$!ua''%9n)r*##!c's%3e,-t#%u-(a0!!+dr:#*oy.,)!u-+*o(.n2'f!#)d'!$g//4!te(!&&-ho#,(em+'%*0#fi$e5#!)gt*7ru%#y.r!!$e1w%('a*i,nt2)h!d-*1!t*t'w2h3#-oe!(/-0td(&w#i,)mo%+#-(e'd!$n)i)$ms-4i!!o/e*(n2n&#a,s!%il*o#1fin+'5ga!3!ul rsepsa8.#cW%e(0ce#o*wv#e%irl!ledloboyk the figure. !

The perim;e2t#e!r8#%o(f0#a*#s%h!'a/p!$e!+2is$8d#e!(fi+!n-#e/d()#a-s!$t+h!*e2#d!-is(+t*a$n)7c#e!$a%r'o,)u-n!*d2#t!h+2e$8s#h5!a!!'/!$!J(%74#!

A = bh

!

J = L %

!

b = base of the rectangle

! !

&>((99((&%-;/",(:h.*:($#=3?("%&t;(h"%(;e5=EE====+6>B=====. >?5=>>==T69E>?>+!=!?him+ =e>s =u?n?Fit9s!!.:*T! his will produce a

final reading of square units (or units squared). Thus the area of the figure is 112 square

units. This fits well with the definition of area which is the number of square units that will

cover a closed figure.

Our next formula will be for the area of a parallelogram. A parallelogram is a quadrilateral with opposite sides parallel.

Area of a Parallelogram A = bh

b = base of the parallelogram h = the height of the parallelogram

$

7(8(%9"(7-*"(#3(%9"(:-&-;;";#1&-$(

$

9(8(%9"(9",19%(#3(%9"(:-&-;;";#1&-$(

$

6

A'-$2.,,$/'!.(#$!"+!$!".)$.)$!"#$)+3#$+G)$e!o"m#$e*t'ry&3-,+$*'&$!"#$+&#+$'*$+$

You will is just a

nspo&+!te"#$ic%c(#i+e!$a&5F"Ai6n(/8d1.,$t%$?-h@,e$$"a+r,e"a$&'o$%f-t,h$'e()fi*+g,u$$re below

74(

68(

565(

$

Solution: For$ thBi&s.*t%r(&a/p@$ezoid, the bases are shown as the top and the bottom of the figure.

The lengths of these sAid&e+$s%-a(0r$e%+4"58,a9&n(d1C$1%-2,1$7u"0n,i0t$s".+,I$t0-d&o!e/s$"0n$%o-t,$m%&a8t$"t/e1r$%w-,h$i7c&h%%o&6f $these we say is

b1 and which is b2. Th&e'$%h-e,i$g'(h)t*+o,3f$$=th-,e$.t,r/a)%p-e0z$&o'i$d%-i,s0,2$00(1u,n0$i"t+s,.$?WD$"h/e1n$EFwE$e*/p(%l0u3$g$G%a$ll this into the

formula, we get A = 211(&b,10+$/&b%2$6)h"%%=,+$12!(-1( ................
................

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