Lesson 3 · from §115 of the 1910 text
Problems in Variation
Laws stated as variations
The theory of variation is really involved in proportion, but this fact is not obvious to the beginner. Hence it is necessary to make clear the meaning of the terms used in variation, and to show how proportion is applied to the solution of problems in variation. It is doubly necessary that the student himself make this application in many cases, otherwise he will not readily grasp numerous relations in physics, in chemistry, and in astronomy; for many important laws of these sciences are often stated in the form of a variation. In connection with these laws many problems arise which require for their solution clear notions of the principles of variation. With a knowledge of proportion only, the student would often find the laws vague and the problems difficult.
A method for every problem
- Decide how the quantities vary: directly, inversely, jointly, as a square, a square root, and so on.
- Write the variation as an equation with a constant: y = Kx, y = \dfrac{K}{x}, y = Kx^2, y = Kxz, \ldots
- Use the pair (or set) of values that is given to find K.
- Use K to find the value asked for.
Or, equally, write the proportion between two sets of values, as in the examples of the last lesson, and solve it.
Example. The intensity of light varies inversely as the square of the distance from the source. A page 3 feet from a lamp is moved to 6 feet. Then I = \dfrac{K}{d^2}, so I_1 : I_2 = \dfrac{1}{3^2} : \dfrac{1}{6^2} = 36 : 9 = 4 : 1. At 3 feet the page is 4 times as bright.
Think about it. Give concrete illustrations of direct, inverse, and joint variation different from those given in this book. (The Teachers' Edition suggests, among others: the weights of spheres of the same material vary as the cubes of their diameters; and the pressure of sunlight on a planet varies directly as the square of the planet's diameter and inversely as the square of its distance from the sun.)
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
The weight of any object below the surface of the earth varies directly as its distance from the center of the earth. An object weighs 100 pounds at the surface of the earth. What would be its weight (a) 1000 miles below the surface (radius of the earth = 4000 miles)? (b) 2000 miles below the surface? (c) at the center of the earth?
If a wagon wheel 4 feet 8 inches in diameter makes 360 revolutions in going a certain distance, how many revolutions will a wheel 5 feet in diameter make in going the same distance?
The distance which sound travels varies directly as the time. A man measures with a stop watch the time between the sight of the smoke from a hunter's gun and the sound of its report. When the hunter was one mile distant, the time was 4\frac25 seconds. How far off was the hunter when the observed time was 2 seconds? (Give the distance in feet.)
When the volume of air in a bicycle pump is 24 cubic inches, the pressure on the handle is 30 pounds. Later, when the volume of air is 20 cubic inches, the pressure is 36 pounds. Assume that a proportion exists here, determine whether it is direct or inverse, and find the volume of the air when the pressure is 48 pounds.
The distance (in feet) through which a body falls from rest varies as the square of the time in seconds. If a body falls 16 feet in one second, how far will it fall in 6 seconds?
The intensity (brightness) of light varies inversely as the square of the distance from the source of the light. A reader holds his book 4 feet from a lamp, and later 6 feet distant. At which distance does the page appear brighter? How many times as bright?
A lamp shines on the page of a book 9 feet distant. Where must the book be held so that the page will receive four times as much light? twice as much light?
The weight of an object above the surface of the earth varies inversely as the square of its distance from the center of the earth. An object weighs 100 pounds at the surface of the earth. What would it weigh (a) 1000 miles above the surface? (b) 2000 miles above the surface? (c) 4000 miles?
The area of a circle varies as the square of its radius. The area of a certain circle is 154 square inches and its radius is 7 inches. Find the radius of a circle whose area is 594 square inches.
The weight of a sphere of given material varies directly as the cube of its radius. Two spheres of the same material have radii 2 inches and 6 inches respectively. The first weighs 6 pounds. Find the weight of the second.
The time required by a pendulum to make one vibration varies directly as the square root of its length. If a pendulum 100 centimeters long vibrates once in one second, find the time of one vibration of a pendulum 64 centimeters long.
Find the length of a pendulum which vibrates once in 2 seconds. (Use the facts of Exercise 22.)
The pressure of wind on a flat (plane) surface varies jointly as the area of the surface and the square of the wind's velocity. The pressure on one square foot is .9 pounds when the wind is moving at the rate of 15 miles per hour. Find the velocity of the wind when the pressure on one square yard is 18 pounds.