Organizers are launching fireworks over the top of the Washington Monument

Organizers are launching fireworks over the top of the WashingtonMonument. Due to costs they would like to find the smallest shell possible that will reach the correct height.

The monument is 555 ft tall. The initial velocities of each size shell are given below.

size - 2 inches
initial velocity ft/s - 117.5

size - 3 inches
initial velocity ft/s - 144

size - 4 inches
initial velocity ft/s - 166

size - 5
initial velocity ft/s - 186

size - 6
initial velocity ft/s - 203.6

The height of the firework t seconds after it is launched is given by
h = -16t^2 + vt where v is the initial velocity.

Use the discriminant to determine the smallest size shell that will reach over 555 feet. Explain.


What is the equation for the discriminant for this problem?