Question

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
acer

Answer 1

9514 1404 393

Answer:

a) y = -7/640x^2 +7/4x

b) y = -7/640(x -80)^2 +70

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 ...

`\left[\begin{array}cc6400&80&70\\25600&160&0\end{array}\right]`

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