I've asked this once, but didn't get a good answer. When you answer please explain in detail.
y varies directly as and inversely as the square of z, and y= 105 when x= 14 and z= 5
The answer in the book says y=187.5(x/z^2) please explain why this is the answer
y varies directly as and inversely as the square of z, and y= 105 when x= 14 and z= 5
The answer in the book says y=187.5(x/z^2) please explain why this is the answer
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Fortunately, I have x-ray vision, so I can see the invisible "x" that is between "y varies directly as" and "and".
y varies directly as x -> This means when x varies, y varies as Cx for some constant C.
Then if x doubles, y doubles, and if x is multiplied by 1/10, y is multiplied by 1/10, etc.
y varies inversely as the square of z -> This means when z varies, y varies as C/z^2 for some constant C.
Then if z doubles, y is multiplied by 1/4, and if z is multiplied by 1/10, y is multiplied by 100 for example.
Combine the constants. If y varies directly as x and inversely as z^2, then the relation is y = Cx/z^2 for some constant z.
For example if x is multiplied by 4 and z is divided by 2, then y stays the same. Or if x is multiplied by 3 and z is multiplied by 7, y is multiplied by 3/49.
Given y=105 when x=14 and z=5, just solve for C to find out what constant it is:
y = C x / z^2
105 = C (14/25)
Solve for C : multiply by 25 and divide by 14
C = 25 x 105 / 14
Factor 105 = 5 x 21 and then simplify 21/14 = 3/2
C = 25 x 5 x 3/2 = 125 x 3 / 2 = 375 / 2 = 187.5
It is the answer because the 187.5 (x/z^2) has been fully explained.
y varies directly as x -> This means when x varies, y varies as Cx for some constant C.
Then if x doubles, y doubles, and if x is multiplied by 1/10, y is multiplied by 1/10, etc.
y varies inversely as the square of z -> This means when z varies, y varies as C/z^2 for some constant C.
Then if z doubles, y is multiplied by 1/4, and if z is multiplied by 1/10, y is multiplied by 100 for example.
Combine the constants. If y varies directly as x and inversely as z^2, then the relation is y = Cx/z^2 for some constant z.
For example if x is multiplied by 4 and z is divided by 2, then y stays the same. Or if x is multiplied by 3 and z is multiplied by 7, y is multiplied by 3/49.
Given y=105 when x=14 and z=5, just solve for C to find out what constant it is:
y = C x / z^2
105 = C (14/25)
Solve for C : multiply by 25 and divide by 14
C = 25 x 105 / 14
Factor 105 = 5 x 21 and then simplify 21/14 = 3/2
C = 25 x 5 x 3/2 = 125 x 3 / 2 = 375 / 2 = 187.5
It is the answer because the 187.5 (x/z^2) has been fully explained.