Question

X=a+b+4, a is directly Propotional to y² and b is inversely propotional to y, when Y= 2₁ X=18 and y=1, X=-3, Find X when y=4​

Answer 1

x = 81

Explanation:

If a is directly proportional to y²:

`\boxed{a \propto y^2 \implies a=ky^2}`

If b is inversely proportional to y:

`\boxed{b \propto (1)/(y) \implies b=(m)/(y)}`

Given equation:

`x=a+b+4`

Substitute the derived expressions for a and b into the given equation:

`\implies x=ky^2+(m)/(y)+4`

Substitute the given values of x and y into the equation to create two equations in terms of k and m.

Given that x = -3 when y = 1:

`\implies -3=k+m+4`

Given that x = 18 when y = 2:

`\implies 18=4k+(m)/(2)+4`

Rewrite the first equation to isolate k:

`\implies k=-m-7`

Substitute the expression for k into the second equation and solve for m:

`\implies 18=4(-m-7)+(m)/(2)+4`

`\implies 18=-4m-28+(m)/(2)+4`

`\implies -4m+(m)/(2)=42`

`\implies -8m+m=84`

`\implies -7m=84`

`\implies m=-12`

Substitute the found value of m into the expression for k and solve for k:

`\implies k=-(-12)-7`

`\implies k=5`

Substitute the found values of k and m into the equation to create an equation for x in terms of y:

`\boxed{x=5y^2-(12)/(y)+4}`

Finally, to find the value of x when y = 4, substitute y = 4 into the equation and solve for x:

`\implies x=5(4)^2-(12)/(4)+4`

`\implies x=5(16)-3+4`

`\implies x=80-3+4`

`\implies x=81`