Distance in Feet 1. Earlier this week the famous "Live Cannonball” - Gunter Money was shot from a cannon at the Austin Expo.
Gunter flew to a height of 70 feet and traveled a distance of 160 feet only to land perfectly in a safety net.
Assume (0, 0) was the starting point for Gunter's flight and that he landed in the net at (160, 0).
(a) Use matrices to find the equation for Gunter Money's path.
(b) Write the equation for Gunter Money's path in vertex form.
US 11:50
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  a) y = -7/640x^2 +7/4x

  b) y = -7/640(x -80)^2 +70

Step-by-step explanation:

a) Given the equation in standard form:

  y = ax^2 +bx +c

we can use the points (0, 0), (80, 70), and (160, 0) to find the values of a, b, c.

  a·0 +b·0 +c = 0

  a·80^2 +b·80 +c = 70

  a·160^2 +b·160 +c = 0

The first equation tells us that c=0, so we can write a reduced matrix for the last two equations. The augmented matrix is ...

A matrix solver finds the solution to be ...

  a = -0.0109375 = 7/640, b = 1.75 = 7/4

The standard-form equation for Money's path is ...

  y = -7/640x^2 +7/4x

__

b) Factoring out -7/64, we have ...

  y = -7/640(x^2 -160x)

adding and subtracting the square of the x-coefficient, we have ...

  y = -7/640(x^2 -160x +6400) +70

  y = -7/640(x -80)^2 +70

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!

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