Answer:
a = 1/10
Step-by-step explanation:
You want to know the value of 'a' when x=1 and y=4 if its value is 10 when x=5 and y=2, and it varies directly with x² and inversely with y².
Variation
The value of 'a' is proportional to x²/y². If the original value of x²/y² is 5²/2² = 25/4, and the new value is 1²/4² = 1/16, that means the value of 'a' is multiplied by ...
(1/16)/(25/4) = 1/100
So, the new value of 'a' is ...
a = 10 × 1/100
a = 1/10
Alternate solution
The equation for 'a' can be written as ...
[tex]a=k\dfrac{x^2}{y^2}[/tex]
Solving for k gives ...
[tex]k=a\dfrac{y^2}{x^2}=10\left(\dfrac{2}{5}\right)^2=1.6[/tex]
Using the new values of x and y, we have ...
[tex]a=1.6\dfrac{1^2}{4^2}=\dfrac{1.6}{16}\\\\\boxed{a=0.1}[/tex]
