Part 2

this activity will you meet these educational goals:

you will create a function to model a fireworks show and examine the attributes of the function.

you’re in charge of planning a fireworks show. the company you hire proposes using fireworks called mortar fireworks. these fireworks are placed in a tube that sits on the ground or a flat surface and are shot from the tube with an initial velocity that propels them into the sky.

mortar fireworks have two fuses that are lit at the same time. the first fuse burns fastest and causes the initial force that launches the firework into the sky. the second fuse takes longer to burn. when the second fuse reaches the middle of the firework, the firework explodes and we see the light show in the sky. this second fuse does not add any extra propulsion to the firework while it’s in the air.

you plan to have the company light the fireworks from the ground. based on information provided by the company, you’ve determined that the fireworks will have an initial velocity of 192 feet/second.

the formula for the vertical motion of an object is h = -16t2 + v0t + h0, where h is the height of the object, h0 is the initial, or starting, height, v0 is the initial velocity, and t is the time in seconds.

using the equation representing the height of the firework (h = -16t2 + v0t + h0), algebraically determine the extreme value of f(t) by completing the square and finding the vertex. interpret what the value represents in this situation.

Step-by-step explanation:

Maximum Value → Parabola opens downward [−A]

Minimum Value → Parabola opens upward [A]

See graph above

I am joyous to assist you anytime.

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.

step-by-step explanation:

30% of $420 is 0.30($420) = $126 and $420 + $126 = $546.

or, 130% of $420 is 1.3($420) = $546.

notice that this is not the $600 original list price with which we started.

why doesn't this work?

the method does not work because 30% of $420 is not the same amount as the 30% of $600. it may seem obvious when stated this way, but it is a very common error made on problems of this type. we cannot just use the same percent and get back to the original amount because we are taking that percent of a different value.

to work problems of this type correctly, it is generally best to write an equation for the situation and then solve.

example: since (100% – 30%)(original price) = (sale price)

70% of the   (original price) is $420.

0.7x   = 420

equation image indicator

x = 600

the original cost was $600.

we may also solve this problem using the two-step method. working the problem in two steps requires us to use more algebra skills. we have to know that x = 1x (identity property of multiplicationtext annotation indicator) and that 1x – 0.3x = 0.7x (subtracting like terms). then we solve the problem as follows:

(original price) – 30%(original price) = sale price

x – 0.3x = 420

0.7x = $420

equation image indicator

x = 600

the original cost was $600.  

note that with both of the above methods we do obtain the correct original amount.

example: after a 5% pay raise, hermione is earning $22,680 per year. what was she earning before the pay raise?

105% of (original salary) = current salary

1.05x = 22,680

equation image indicator

x = 21,600

hermione earned $21,600 per year before the pay raise.

this is an example so maybe you will get how to at least do it!

i believe it is c.y = 9x

rise over run. 9 units up over 1 unit right or 9/1