Question

Part 2

Answer 1

Answer:

$\displaystyle f(t) = -16t^2 + 192t; Maximum$

Step-by-step explanation:

Maximum Value → Parabola opens downward [−A]

Minimum Value → Parabola opens upward [A]

See graph above

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Answer 2

Answer:

The maximum value of f(t) is 576, the maximum height of the firework.

Step-by-step explanation:

h = -16t² + v₀t + h₀

Data:

v₀ = 192 ft/s

h₀ = 0

Calculation:

h = -16t² + 192t  

This is the equation of a parabola.

We must solve the equation to find the maximum height.  

The coefficient of t² is negative, so the parabola opens downward, and the vertex is a maximum.

Complete the square

(a) Divide both sides by -16 to make the coefficient of  t² equal to 1.

(-1/16)h = t² - 12t  

(b) Square half the coefficient of t

(-12/2)² = (-6)² = 36

(c) Add and subtract it on the right-hand side

(-1/16)h = t² - 12t  + 36 - 36  

(d) Write the first three terms as the square of a binomial

(-1/16)h = (t - 6)² - 36

(e) Multiply both sides by -16

h = -16(t - 6)² + 576

You have converted your equation to  the vertex form of a parabola:

y = a(t - h)² + k = 0,

where (h, k) is the vertex.

h = 6 and k = 576, so the vertex is at (6, 576).

The vertex represents the maximum height of the firework.

The Figure below shows that your firework reaches a maximum height of 576 ft  6 s after ignition.