Respuesta :
h (t) = -4t^2 + 24t = -4 (t² - 6t)
= -4 [(t - 3)² - 3²]
= -4 (t - 3)² + 36
max (3 ; 36)
(t - 3)² = t² - 2*t*3 + 3² = t² - 6t + 9
so it's why t² - 6t = (t - 3)² - 9
The maximum height of the projectile is 36 feet.
It is given that the projectile on a planet in our solar system at a given time t in seconds is represented by the function:
[tex]\rm h(t) = -4t^2+24t[/tex] can be written as a function:
[tex]\rm h(t) = a(t-h)^2+k\\[/tex]
It is required to determine the maximum height of the projectile.
What are maxima and minima?
Maxima and minima of a function are the extrema within the range, in other words, the maximum value of a function at a certain point is called maxima and the minimum value of a function at a certain point is called minima.
The given function:
[tex]\rm h(t) = -4t^2+24t[/tex] for the maximum height of the projectile differentiate the given function with respect to 't' and equate it to zero.
[tex]\rm h(t) = -4t^2+24t\\[/tex]
[tex]\frac{\mathrm{d} }{\mathrm{d} x} [h(t)] = \frac{\mathrm{d} }{\mathrm{d} x}(-4t^2+24t)\\[/tex]
[tex]\frac{\mathrm{d} }{\mathrm{d} x} [h(t)] = \frac{\mathrm{d} }{\mathrm{d} x}(-4t^2)+\frac{\mathrm{d} }{\mathrm{d} x}(24t)\\[/tex]
[tex]\frac{\mathrm{d} }{\mathrm{d} x} [h(t)] = -8t+24\\[/tex]
[tex]\frac{\mathrm{d} }{\mathrm{d} x} [h(t)] = 0 \Rightarrow-8t+24=0\\[/tex]
[tex]\rm -8t+24=0\\\rm t= 3 \ sec[/tex]
Here, [tex]\rm \frac{\mathrm{d}^2 }{\mathrm dx^2} \leq 0[/tex]
It means that at t=3, we get the maximum height of the projectile.
[tex]\rm h(3)= -4(3)^2+24(3)\\\rm h(3) = -36+72\\\rm h(3)= 36 \ Feet[/tex]
Thus, the maximum height of the projectile is 36 feet.
Know more about the maxima and minima here:
brainly.com/question/6422517
