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`
`h(t) = -10[(t-25)^2-1530]`
`h(t) = -10(t-25)^2+1530`
We know that `(t-25)^2 >=0` always.
So h(t) will have a maximum when -`10(t-25)^2 ` has lesser negative value or the maximum value.
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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