39 ## VARIATIONThe constant of proportionality THE SUBJECT OF VARIATION is more properly the subject of arithmetic, because it rests squarely on the concept of ratio. See Skill in Arithmetic, Lesson 17. Direct variation We say that a quantity That means that if Let the initial values of
Since problems typically ask for the value of
Example 1.
Formally,
2 : 10 = 7 : 10 is five times 2. Therefore,
Example 2.
That is, 8 : 13 = 4 : Alternately, 4 is half of 8. Therefore,
Problem 1. To see the answer, pass your mouse over the colored area.
The value of
Problem 2. When
6 : 9 = 42 :
Alternately, 42 is
Or, 9 is The constant of proportionality When
The circumference C of a circle, for example, varies directly as the diameter D. The constant of proportionality is called π. C = πD. That constant has been the subject of investigation for over 2500 years. In scientific problems, the constant of proportionality is determined by experiment. In what is called Hooke's Law, the force
In other words, the greater the stretch Example 3. a) For a given spring,
35 = Therefore,
b) What is the value of
Therefore, when
In general, if
That implies:
Therefore, When
See the following problem. Problem 3. a) The distance
b) The units on the right must be same as those of
The units of In this example, the constant of proportionality is the constant c) How far has the car traveled after 7 hours?
As for Problems 1 and 2 above, while we
Problem 4. Prove:
If Problem 5. If the side of a square doubles, how will the perimeter change?
The perimeter will also double, because the perimeter Problem 6. If the diameter of a circle changes from 6 cm to 12 cm, how will the circumference change?
The circumference will also double, because the circumference varies as the diameter. Problem 7. If the diameter of a circle changes from 6 cm to 9 cm, how will the circumference change? In going from 6 cm to 9 cm, the diameter has increased one and a half times; that is the ratio of 9 to 6. Therefore, the circumference will also increase one and a half times.
Problem 8. The circumference What number is the constant of proportionality?
Therefore the perimeter of the square is equal to 4D.
Varies as the square A quantity If Algebraically: If
then
Thus if
Problem 9.
In going from 7 to 35,
Problem 10.
In going from 20 to 15,
Theorem. If This is easily proved if we write the ratios in fractional form.
Therefore, on squaring both sides: This implies
This means that Problem 11. 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
The side of the circumscribed square is equal to the diameter
But b) If the radius of a circle changes from 6 cm to 12 cm, how will the
In going from 6 cm to 12 cm, the radius has doubled, that is, it has changed by a factor of 2. The area therefore will change by a factor of 2 c) What is the constant of proportionality that relates the area
π. Example 4. The surface area of a sphere. The surface area of a sphere is proportional to the surface area of the circumscribed cube. Now, each face of the cube is a square whose side is equal to the diameter In other words, the surface area Do you know what the constant of proportionality is?
π.
Problem 12. Show that the surface area of a sphere varies as the square of its radius. Write the equation that relates the surface area
Since Section 2: Varies inversely. Varies as the inverse square. Please make a donation to keep TheMathPage online. Copyright © 2019 Lawrence Spector Questions or comments? E-mail: themathpage@yandex.com Private tutoring available. |