When a varies directly as b, we often say, "a is proportional to b." In that case, the relationship between a and b takes this algebraic form:
a = kb.
k is called the constant of proportionality.
The circumference of a circle, for example, varies directly as the diameter. The constant of proportionality is called π.
C = πD.
Example
a.) For a given spring, F has the value 35 when the spring has stretched 8 inches. What is the constant of proportionality for that spring?
b) What is the value of F when the spring has stretched 11 inches?
a) The distance d that an automobile travels varies directly as the time ta) that it travels. After 2 hours, the car has traveled 115 miles. Writea) the equation that relates d and t.
b) The units on the right must be same as those of d on the left, that is, a) distance. What are the units, then, of 57.5?
c) How far has the car traveled after 7 hours?
3.) If the side of a square doubles, how will the perimeter change?
4) a varies as the square of b. When b = 7, a = 4. What is the value of a when b = 35?
a varies as the square of b. When b = 20, a = 32. What is the value of a when b = 15?
5) The area A of a circle varies directly as the area of the circumscribed square. That is, as the area of the square changes, the area
of the circle changes proportionally.
a) Show that this implies that the area A of the circle varies as the square of the radius r.
b) If the radius of a circle changes from 6 cm to 12 cm, how will the area change?
c) What is the constant of proportionality that relates the area A to r²?
Subscribe to:
Post Comments (Atom)
a possible answer for example one is
ReplyDeletethe constant of proportionality is k(inches and for part b 35/8=4.375 therefor 11x4.375=48.125
no3
ReplyDeleteconstant is the length of side of square
p= k4(s)
for eg. if side of square is 2
then p = (1) 4(2)
p = 8
this is because the constant in this case did no change.
if side doubles then p=2(4)(2)
p= 16
the first example the constant would be the distance stretched to the distance of the spring at rest and part
ReplyDelete(b) since f=35 at 8
so 8k=35
k=35/8
therefore 11k=35/8 * 11
= 48.125
the second question
ReplyDeleted α t
d=kt.
(b)
d=kt
since
d= 57.5
57.5=kt
when t=120
120k=57.5
k=0.48
(c)
for 7 hours
d=kt
when k=.48
t= 7h (7*60)=420
d=420*.48
=201.6 miles
for the third question
ReplyDeleteif the side is x the perimeter will be 4x
and if the side doubles 2x, the perimeter will now be 8x.
so therefore if the side doubles so to does the perimeter.
for number 4
ReplyDeletea α b²
when a=4
b=7
a=kb²
4=k(7)²
4=k49
k=49/4=12.25
therfore when b=35
a=kb²
a=(12.25)(35)²
a=15006.25
for example (a and b):taking the stretch to be "S" the equation is: FαS= F=kS and if F is 35 and S is 8 then the constant proportion is 35/8=4.375.
ReplyDeletepart (b)
F= constant * stretch given= 4.375*11= 48.125
1.
ReplyDeleteF=k 35
35= k 8
35= k 11
k= 35/11
k= 3.18
2.
d=kt
115 miles= k 2hrs
115km= k 0.002km
115=k 0.002 (7)
k= 0.014
k= 115/0.014= 8214.3
3.side = y
= 4y
double = 2y
4.a=k b^2
a= k 15^2
a= 3.87
a = k b^2
a= k 352
a= k 5.91
1.a.F is propotional to K (constant) 8
ReplyDelete35 = K8
therefore K = 35/8
so K = 4.375
1.b. the value of F when the spring has streched 11 inches: F is propotional to K 11
ReplyDeletetherefore F = k 11
so F = 4.375 (11)
F = 48.125
@ purple rain for pt1......how is it dat you just replaced the 8 in the equation with 11 and also the question stated that F = 35 when the spring has streched 8 inches not 11 inches
ReplyDeletea) d*t
ReplyDeleteimplies d= kt , in this case k=s where s is speed.
therefore d= st
d= st implies
s= d/t
t= 2hrs d= 115 miles
therefore s= 115/2
= 57.5
the units are given by distance/time
= miles /hr
therefore the speed of the automobile is
57.5 miles/hr
how far suggests distance therefore how far the car travels in 7 hours refers to the distance covered
therefore d= st
d= 57.5(7)
d= 402.5 miles
what are some ways variation may be applied in everyday life.
ReplyDeletek is the constant of proportionality, so F is proportional to x: giving F=kx.
ReplyDeleteSince F=35, and the spring stretches 8inches,then putting this into the equation F=kx gives 35=k8 and making k the subject of the formula gives k=35/8. therefore k=4.375.
The value of F when the spring is stretched 11inches:since since F=kx and k=4.375 and x=11,
ReplyDeletethen F=4.375x11
F=48.125
the first example the constant would be the distance stretched to the distance of the spring at rest and part
ReplyDelete(b) since f=35 at 8
so 8k=35
k=35/8
therefore 11k=35/8 * 11
= 48.125
For Example 1.
ReplyDelete(a)Let S= stretch of the spring
Fα S
F= kS (where k is the constant proportionality of the spring)
when F= 35 and S= 8
35=k(8)
k= 35/8
therefore the constant proportionality is 35/8 or 4.375
Continuation from first question with spring.
ReplyDelete(b) when the string has stretched 11 inches
S= 11
Therefore F= 4.375 x 11
F= 48.125
For Question 2.
ReplyDelete(a) distance d varies directly with time t.
dα t
d= kt is the equation that relates d and t.
(b) when d= 115miles and t= 2hours
ReplyDelete115= k2
k= 115/2
k= 57.5 is the constant proportionality
The units, however, is in miles per hour.
In order for the units on the right to be equal to that of the left…
k= 57.5miles/hrs
(c) Using the equation
ReplyDeleted= kt
when k= 57.5 and t= 7
d= 57.5miles/hrs x 7hrs
d= 402.5 miles
For question 3.
ReplyDeleteIf the sides of a square doubles then the Perimeter of that particular square also doubles.
Take for example a square of side x has Perimeter P.
P= 4x, if x doubles then
P= 4(2x)
P= 8x
This means that the Perimeter of a square varies directly as the side.
From the equation P= 4x, we can also say that the constant proportionality is 4
Because Pα S (let S= side) and k is the constant proportionality
P= kS
P= 4x is of the form P= kS
Therefore k=4
Question 4.
ReplyDeletea varies as the square of b
a α b^2
when b= 7,a= 4 the constant proportionality is k
a= kb2
4=k(7^2)
4= k (49)
4/49 = k
When b= 35
a= 4/49 x 352
a= 4/49 x 1225
a= 100
Therefpre the value of a when b=35 is 100
Second part of Question 4.
ReplyDeletea varies as the square of b
a α b^2
when b= 20,a= 32 the constant proportionality is k
a= kb^2
32=k(20^2)
32= k (400)
32/400 = k
k= 0.08
When b= 15
a= 0.08 x 152
a= 0.08 x 225
a= 18
therefore the value of a when b= 15 is 18
1.a
ReplyDeleteF-kx
35=k8
k=35/8
k=4.375
1.b
ReplyDeleteF=kx
F=4.375*11
F=48.125
3.
ReplyDeletewhen side= x
perimeter= 4x
when side doubles= 2x
perimeter doubles= 8x
a = b^2
ReplyDeletewhen a=4
b=7
a=kb^2
4=k(7)^2
4=k49
k=49/4=12.25
when b=35
a=kb^2
a=(12.25)(35)^2
a=15006.25
F=k 35
ReplyDelete35= k 8
35= k 11
k= 35/11
k= 3.18
question 1.
d=kt
ReplyDelete115 miles= k 2hrs
115km= k 0.002km
115=k 0.002 (7)
k= 0.014
k= 115/0.014= 8214.3
question 2.
for number 4
ReplyDeletea α b²
when a=4
b=7
a=kb²
4=k(7)²
4=k49
k=49/4=12.25
therfore when b=35
a=kb²
a=(12.25)(35)²
a=15006.25
a varies as the square of b
ReplyDeletea α b^2
when b= 7,a= 4 the constant proportionality is k
a= kb2
4=k(7^2)
4= k (49)
4/49 = k
When b= 35
a= 4/49 x 352
a= 4/49 x 1225
a= 100
Therefpre the value of a when b=35 is 100
i dont understand no. 5!!!!!
ReplyDeleteexample question
ReplyDeletea)F=kl
F/l=k
k=F/l
= 35/8
k=4.375
constant of proportionality
example question
ReplyDeleteb)F=kl
=(4.375)(11)
= 48.125
the value of F when the spring has stretched 11 inches is 48.125
F=kx and k=4.375 and x=11,
ReplyDeletethen F=4.375x11
F=48.125
question a
ReplyDeletelet e represent the extention
let F represent the force
F is proportional to e
F=ke
when F=35, e=8
35=k8
35/8=k
4.375=k
part b
ReplyDeletewhen e=11
F=4.375(11)
F=48.125
question a
ReplyDeleted is proportional to t
d=kt
part b
ReplyDeleted=kt
115=k2
115/2=k
km/hr=k
k=kmhr^-1
part c
ReplyDeleted=kt
d=57.5(7)
d=402.5 km
question 3
ReplyDeleteperimeter is proportional to total length of the sides of the square.
so if the 2 sides double in length the perimeter will double as well