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V A R I A T I O N

By Miss Osbourne

THE CLOSER YOU ARE…

T H E L O U D E R I T I S!

The loudness of music is measured in decibels (dB).

The closer you are to the source of the music, the louder the music sounds. Why is this?

There is an INDIRECT RELATIONSHIP between distance and sound. The shorter the distance away from the source the greater the sound.

+D E F I N I T I O N S

Variation refers to a function that relates the values of one variable to those of other variables.

Two types of variation are DIRECT and INVERSE (Indirect) variation.

DIRECT VARIATION describes the relationship where one variable is positively related to another variable. That is as one variable increases the other will also increase. Likewise if one variable decreases the other variable will also decrease.

Thus y = kx tells you y varies directly as x. Further, y is directly proportional to x.

+D E F I N I T I O N S

INVERSE VARIATION(Indirect) describes the relationship where one variable is negatively related to the other variable. That is as one variable increases the other will decrease. Likewise if one variable decreases the other variable increases.

Thus y = k/x tells you y varies inversely as x. Further, y is inversely proportional to x. Notice x is on the “bottom”.

The value of k is called the constant of variation. It specifically describes the relationship between the variables.

+A P P L I C A T I O N O F V A R I A T I O N

If values of x and y are known then you can solve for the constant of variation, k. 1. Consider the variation table below.

By observation we notice as x increases so does y.

That is y = kx.

A direct variation relationship exists.

1 = k 18 = k 2

Thus, k = 1 ÷ 1 = 1 Or k = 8 ÷ 2 = 4However k is not constant.

Let’s try another relationship.

1 * 1 = 1 2 * 2= 4 but 2 * 2* 2 = 83 * 3 * 3 = 27 and 4 * 4 * 4 = 64

Therefore y = kx3 and k = 1.

x 1 2 3 4 5

y 1 8 27 64 125

Similarly, if the constant of variation, k and one other variable (for instance, y) is known then we can solve for the unknown variable, x; and vice versa.

2. Consider the circumference of the face of a steel pan (a circle). The circumference of a circle varies directly as its diameter.

Let C rep. circumference and d rep. diameter.

So far we know a direct variation relationship exists. That is, C = kd

Based on previous knowledge of circles, the constant of variation, k = π Therefore the equation will be C = π d

S O L U T I O N S(i) n = k t

(ii) n = k t Substitute known values for n and t. 56 = k (16) Solve for k. 56 ÷ 16 = k 3 . 5 = k

(iii) Now that the value of k is known the variation equation is n = 3. 5 t Substitute t = 84 into the equation t = 3. 5 (84) t = 294

3. The length of a note, n is increased by playing a particular note for a longer time period, t. That is, the variable n varies directly as t. When n is 16, t is 56.(i) Write a variation model for this situation. Use k as the constant of variation.(ii) Solve for the constant of variation.(iii) Find the value of z when w is 84.

4. The loudness of sound measured in decibels (dB) varies inversely as the square of the distance between the listener and the source of the sound. If the loudness of the sound is 17.92 dB at a distance of 10ft from a speaker, what is the decibel level 20 ft from the speaker?

Finally we use this information to find the loudness of the sound 20ft away from the speaker. Substitute k = 1,792 and d = 20 into the equation.

l = 1,792/202

l = 4.48 dB

The first step is to WRITE OUT the type of relationship.

The information provided tells us its an inverse relationship, that isl = k/d2

where l rep. the loudness of sound and d the distance from the speaker

Secondly, we have the values for l and d. 17. 92 = l and 10 = d respectively.

Since we have this information we substitute the values into the equation to find a value for k.

17.92 = k/102

17.92 x 102 = k1,792 = k

5. The speed of a race boat in still water varies directly as the square root of the length of the boat.(a)If a 16 ft. Boat can travel 6.2 mph in still water, find an equation that

relates the speed of the boat to its length.(b)Find the speed of a 25 ft boat.

Finally we use this information to find the speed of a 25ft boat. Substitute k = 1. 55 and l = 25 into the equation.

s = 1.55 x √25s = 7.75 mph

The first step is to WRITE OUT the type of relationship.

The information provided tells us it’s a direct relationship, that iss = k√l

Where s rep. the speed of the raceboat and l rep. the length of the boat

Secondly, we have the values for s and l. 6.2 = s and 16 = l respectively.

Since we have this information we substitute the values into the equation to find a value for k.6. 2 = k√166. 2 ÷ √16 = k6. 2 ÷ 4 = k1.55 = k