Today, we will discus on a topic called 'direct variation.The term direct variation is a variation that shows a relationship between two or more quantities.In real life, if a cup of rice is 200 naria, what would be the cost of 2 , 3 ,4 ....... x cups? .This tutorial also prepay you about your WAEC and JAMB Examination.
This means, as the cups are increasing, the cost is also increasing at the same rate, We call this 'Direct Variation'
If we make Y the numbers of rice, and y the cost of rice, state the ratio of the numbers of cups the cost is constant,then the number is said to represent x .Therefore, y and x are called quantities.
Equation connecting the two quantities is
x
-------- =k or c = kx
y
Where k is the constant...
We say, when the graph of x is plotted against y, a straight line is obtained.
Examples
1. If u varies directly as V, and V = 15 when U = 9 , find
i. the relation between U and V .
ii. U when V is 4.
Solution
i. U ∝ V
U = KV
9 = 15K
9 3
K = ----- = -----
15 5
ii. When V = 4 , then 3/5u = 4
5 x 4 20
u = ------- = ------
5 5
y = 4
This means our y is 4, v means varies.
2.The value of e varies directly as the square of f. when e = 3 and f = 2; find
i. the relation between e and f
ii. e when f = 3
iii. f when e =12
Solution
e ∝ f²
e = k f²
3 = k ( f)²
= k (2)²
3 = 4k
3
k ---------f²
4
i. The relationship between e and f is :
3
e ---------f²
4
Note if a = 2, solve a²
= (a)²
= (2)² means 2² then we have 4,because 2 x 2 = 4
ii. Lets find e if f = 3
our e goes like this:
3
= e ---------f²
4
3
= e ---------(3)²
4
3 x 9
= -----------
4
27
e = --------------
4
3
e = 6 -------------
4
iii. f when e =12
3
e = ---------f² --- remember our e =12
4
12 x 4 = 3f²
f² = 4 x 4 ----------square root 14 is 4
f =4
Now lets solve WAEC question on Direct variation
y varies directly as square of x.If y = 98, when x = 7 , calculate y, when x = 5
Solution
y varies directly as square of x means:
y ∝ x²
y = kx² --------y is 98 so..
98 = k (x)² --------- x is 7 so...
= k(7)² ------ means 7 x 7 = 49
98= 49k
look for a common divisor that will divide 98 and 49 without a remainder. The only prime number is 7.So...
= 98/7 = 14 also 49/7 = 7
14 =7x
y = 14/7
y = 2
calculate y, when x = 5
y = 2 (x)²
y = 2 x (5)²
= 2 x 25 ----- 5 x 5 = 25
y = 50
JAMB focus question
If x - 3 is directly proportional to the square of y, and x = 5 when y =2, find x when y = 6
remember our x = 5 , so..
x - 3 ∝ y²
5 - 3 = ky² ------ x is 5
2 = k ( 2)²
2 = 4k
2 1
x = ------------ = ----------y² ---- relationship between x and y
4 2
1
x= ----------y²
2
1 x (6)² 36
---------------- = ----------
1 x 2 2
x = 18
Please if yu have any question, you are free to put them in the comment box
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