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Airplanes are often used to drop water on forest fires in an effort to stop the spread of the fire. The time t it takes the water to travel from height h to the ground can be derived from the equation h=1/2 * gt^2 where g is the acceleration due to gravity (32feet/second^2). Determine the equation that will give time as a function of height. Please don't just put the equation. Instead, I'd love to learn how to do this so explain please :)

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Whenever we have two variables and one depends on the other, that is for instance "x" and "y". And one of these variables depends on the other and follows a certain pattern which is express mathematically, for instance: \[y=x^3+5x\] We can see that we will acquire different values of "y" as "x" changes values, so we denote it as \(y=f(x)\), this means that "y" is a variable that depends on the value that "x" takes. So we can deduce from this simple observation, that the "y" variable is dependant while the "x" variable is independant, this is what is called a "functional" relationship and also we can call it function "f" that is dependant on the value of "x": \[f(x)=\phi (x)\] Now that we have established what a function is, in a very compressed matter... We can observe the function given on the excercise: \[h=\frac{ 1 }{ 2 }g.t^2\] By the definition I stated to you, we can see that this mathematical pattern is expressed as a function of "t", so, we can express it as \(h= f(t) \). But what happens if we want to change the roles? Well, let's begin by working the grounds, since we want to make the variable "t" as dependant variable and "h" as independant variable, we will deduce that "t" will be a function "h": \(t=f(h)\) so, we can replace it: \[h=\frac{ 1 }{ 2 }g.t^2 \rightarrow h=\frac{ 1 }{ 2 }g.f(h)^2\] so, if we solve for "f(h)" we will obtain the function in question: \[h=\frac{ 1 }{ 2 }g.f(h)^2 \rightarrow f(h)^2=2h.g\] \[f(h)=\sqrt{2h.g}\]
oh, sorry, I messed up on one step: \[h=\frac{ 1 }{ 2 }g.f(h)^2 \rightarrow 2h=gf(h)^2\] \[f(h)^2=\frac{ 2h }{ g }\] \[\therefore f(h)=\sqrt{\frac{ 2h }{ g }}\]
Shouldn't we plug in 32 for g?

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Other answers:

You can do it if so desire, but since "g" is a constant, it does not hurt to leave it just as "g".
Thank you very much :)
if you have any further questions, ask away.
Okay :) let me look through it :)
This next one just requires me to find out how many seconds it will take for water to hit the ground, so let me try it first :)
Using the formula you explained how to get, I plugged in 1024 for the height and 32 for g, and I got that it takes 8 seconds. :)
I'll open a new post for the next one :)
okay, tag me.

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