Question 956610: Given that y is directly proportional to the cube of x and inversly proportional to the square of z, if x decreases by 35.9% and z increases by 5.8%, by what factor does y change?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your equation is:
y = k * x^3 / z^2
when x decreased by 35.9%, then the value of x becomes x - .359 * x which is equal to .641 * x.
when z increased by 5.8%, then the value of z becomes z + .058 * z which is equal to 1.058 * z.
the new equation becomes:
y = k * (.641 * x)^3 / (1.058 * z)^2
this becomes:
y = (k * .641^3 * x^3) / (1.058^2 * x^2)
this is the same as:
y = (.641^3 / 1.058^2) * k * (x^3 / z^2)
simplify to get:
y = (.2352896118 * k * x^3) / z^2
y changes by a factor of .2352896118.
let's do an example to see if this holds true.
let x = 9 and let z = 5
when this happens, y is equal to 87.48
you get:
87.48 = k * 9^3 / 5^2
solve for k to get k = 3.
k is the constant of variation and so never changes.
now let y = k * (.641 * x)^3 / (1.058 * z)^2
when x = 9 and when z = 5 and when k = 3, this equation becomes:
y = (3 * (.641 * 9)^3) / (1.058 * 5)^2
this becomes:
y = (3 * 5.769^3) / 5.29^2 which becomes:
y = 20.58313524
the original value of y was 87.48
the new value of y is 20.58313524
the new value of y divided by the original value of y is equal to 20.58313524 / 87.48 which is equal to .2352896118.
the new value of y is equal to .2352896118 times the original value of y.
i believe that's what you are looking for.
that's the factor that the value of y was changed by.
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