4.6 "Variation"
y = kx "y varies directly as x" (or directly proportional)
(k is a constant)
"y varies inversely as x" (or inversely proportional)
Express the statement as a formula involving k. Then find k.
(ex) s varies directly as t. If t=10, then s=18
(ex) w varies directly as z and inversely
as the square root of u.
If z=2 and u=9, then w=6
(ex) q is inversely proportional to the
sum of x and y. If x=0.5 and y=0.7, then q=1.4
(ex) r is directly proportional to the product of s and v and inversely
proportional to the cube of p.
If s=2, v=3, & p=5, then r=40.
Hook's law states that the force F required to stretch a spring x units beyond its natural length is directly proportional to x.
a. Express F as a function of x by means of a formula that involves a constant of proportionality k.
b. A weight of 4 pounds stretches a certain
spring from its natural length of 10 inches
to a length of 10.3 inches.
Find the value of k in part (a).
c. What weight will stretch the spring in part (b)
to a length of 11.5 inches?
d. Sketch a graph of the relationship between F and x for x > 0.
F (lbs)
30
20
10
1
2
x
3 (in)
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The intensity of illumination, I, from a source of light varies inversely as the square
of the distance, d, from the source.
a. Express I in terms of d and a constant
of variation k.
b. A searchlight has an intensity of 1,000,000
candle­power at a distance of 50 feet. Find
the value of k in part (a).
I
(candlepower)
1,500,000
c. Sketch a graph of the relationship between
I and d for d > 0.
1,000,000
500,000
d. Approximate the intensity of the searchlight in part (b) at a distance of 1 mile.
50
100
dist.
150 (ft)
A circular cylinder is sometimes used in physiology as a simple representation of a human limb.
a. Express the volume V of a cylinder in terms of its length L and the square of its
circumference C.
b. The formula obtained in part (a) can be used to approximate the volume of a limb
from length and circumference measurements. Suppose the (average) circumference of a human forearm is 22 cm and the average length is 27 cm.
Approximate the volume of the forearm to the nearest cm2.
It is known in physics that the range R of a projectile is directly proportional to the square of its velocity v.
a. Express R as a function of v by means of a formula that involves a constant of
proportionality k.
b. A motorcycle daredevil has made a jump of 150 feet. If the speed coming off the ramp was 70 mi/hr, find the value of k in part (a).
c. If the daredevil can reach a speed of 80 mi/hr coming off the ramp and maintain
proper balance, estimate the possible length of the jump.
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Coulomb's law in electrical theory states that the force F of attraction between two oppositely charged particles varies directly as the product of the magnitudes Q1 and Q2 of the charges and inversely as the square of the distance d between the particles.
a. Find a formula for F in terms of Q1, Q2, d, and a constant of variation k.
b. What is the effect of reducing the distance between the particles by a factor of 1/4?
The ideal gas law states that the volume V that a gas occupies is directly proportional to the product of the number n of moles of gas and the temperature T (in K) and is inversely proportional to the pressure P (in atmospheres).
a. Express V in terms of n, T, P, and a constant of proportionality k.
b. What is the effect on the volume if the number of moles is doubled and both the temperature and the pressure are reduced by a factor of one­half?
Trout Population. Suppose 200 trout are caught, tagged, and released in a lake's general population. Let T denote the number of tagged fish that are recaptured when a sample of n trout are caught at a later date. The validity of the mark­recapture method for estimating the lake's total trout population is based on the assumption that T is directly proportional to n. If 10 tagged trout are recovered from a sample of 300, estimate the total trout population of the lake. .
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