Quadratic Equation

The flight of an aircraft is represented by the equation h(t) = -10t^2 + 500t +9050 where h is the height in metres and t is the time in seconds. What is the maximum altitude of the plane.

Please explain thank you!

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`h(t) = -10t^2+500t+9050`

`First take -10 out of the expression. h(t) = -10(t^2-50t-905)`

Now we have to complete a square in the left side.

Look at following expression.

`(t-a)^2 = t^2-2at+a^2`

Consider` t^2-50t-905 ` with the right side of the above. Compare the component of t in both equations.

`-2a = -50`

` a = 25`

`(t-25)^2 = t^2-50t+25^2 = t^2-50t+625`

But what we want is t^2-50t-905.

So we write;

`t^2-50t-905 = (t^2-50t+625)-625-905`

`t^2-50t-905 = (t-25)^2-1530`

So;

`h(t) = -10[(t-25)^2-1530]`

`h(t) = -10(t-25)^2+1530`

We know that `(t-25)^2 >=0` always.

Then `-10(t-25)^2<=0`

So h(t) will have a maximum when -`10(t-25)^2 ` has lesser negative value or the maximum value.

-10(t-25)^2<=0

maximum here is 0. all others are negative.

So h(t) will be maximum when `-10(t-25)^2 = 0`

*maximum h(t) = 1530*

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