Solving for Minimum Angle for Home Run

[jazz20]

Homework Statement

the center field fence in a ball park is 10ft high and 400 feet from home plate. the ball is hit 3 ft above ground. it leaves the bat at an angle of theta degrees with the horizontal speed of 147.67ft/second. find the minimum angle at which the ball must leave the bat in order for hit to be a home run.
the path of the projectile is modeled by the parametric equations:
x=v₀cos(theta)t
y=3+(v₀sin(theta)t-15t^2

Homework Equations

The Attempt at a Solution

substitute 400 in for x and 147.67 for v₀to solve for t
t=400/(146.67cos(theta)
then i took t and put it into the equation for y
y=3+(147.67sin(theta)(400/(146.67cos(theta))-16(400/(146.67cos(theta))^2


Did I do this correctly? If so, can some one help me with the math to solve for theta?

 

[rl.bhat]

1.

The Attempt at a Solution

substitute 400 in for x and 147.67 for v₀to solve for t
t=400/(146.67cos(theta)
then i took t and put it into the equation for y
y=3+(147.67sin(theta)(400/(146.67cos(theta))-16(400/(146.67cos(theta))^2

You can rewrite the equation as
y = 3 + 400*tanθ - 16*400^2*sec^2θ/146.67^2 [ 1/cosθ = secθ]
Put sec^2θ = 1 + tan^2θ,and solve the quadratic for tanθ.