Question

never fully understood these problems. Please help!
y varies directly as the square of x and inversely as the square root of z. what is the effect on y if x is halved and z is quadrupled?

Answer 1

i know there is supposed to be a k somewhere in the equation but so far i have \[yx ^{2}/\sqrt{z}\]

Answer 2

ok, best to look at this in parts.
start with the statement "y varies directly as the square of x"
do you know what that means?

Answer 3

it means multiply right?

Answer 4

it "means" y changes as some constant times "x squared"
we would normally write this as:\[y=kx^2\]where k is some constant.

Answer 5

understand so far?

Answer 6

ohh ok. got it so far.

Answer 7

so would it be?
y=\[kx ^{2}/\sqrt{z}\]

Answer 8

good, now lets look at the second part:
"y varies inversely as the square root of z"

this would imply that y changes as some constant divided by the square root of z, and we would write this as:\[y=\frac{k_2}{\sqrt{z}}\]where \(k_2\) is just some other constant.

Answer 9

yes - you have it right, if we put these two statements together, then we get:\[y=\frac{kx^2}{\sqrt{z}}\]where k is some constant

Answer 10

now what would happen to y if x is halved?

Answer 11

we would take the square root since its \[x ^{2}\] right?

Answer 12

no, you need to replace the 'x' in the equation with 'x/2' and see what you get

Answer 13

sorry this is the part i get confused…ok so we do \[(x/2)/\sqrt{4}\]?

Answer 14

but I'm not understanding what we do with the y

Answer 15

ok, let me try to explain it to you step-by-step...

Answer 16

ok thank you.'

Answer 17

we have:\[y=\frac{kx^2}{\sqrt{z}}\]now lets replace x with x/2 to get a new y as follows:\[y_{new}=\frac{k(x/2)^2}{\sqrt{z}}=\frac{kx^2/4}{\sqrt{z}}=\frac{kx^2}{4\sqrt{z}}\]now notice that this is a quarter of the original y. So halving x has reduced y by a quarter.

Answer 18

does that make sense?

Answer 19

its still a little bit confusing to me but since you showed me step by step i understand it better now. Thank you for your help!

Answer 20

no problems my friend.
now lets see what happens to y if z is also quadrupled...

Answer 21

we know that if we halve x we end up with:\[y_{new}=\frac{kx^2}{4\sqrt{z}}\]now lets replace z with 4z and see what happens to y now:\[y_{new}=\frac{kx^2}{4\sqrt{4z}}=\frac{kx^2}{4\times2\sqrt{z}}=\frac{kx^2}{8\sqrt{z}}\]now compare this to the original y which was given by:\[y=\frac{kx^2}{\sqrt{z}}\]how has y changed when we halved x and quadrupled z?

Answer 22

it becomes ⅛ right? since before z was quadrupled it was ¼ and the square root of 4 is 2 so we multiply 2 and 4

Answer 23

perfect! I think you are starting to understand this better now. :)

Answer 24

good! i feel like I am too! thank you so much for your help!

Answer 25

yw :)