(a)

The horizontal distance that a projectile will travel in the air (ignoring air resistance) is .

Where is the initial velocity of the projectile, is the angle of elevation,

is acceleration due to gravity ( ).

Find the angle of elevation .

Consider .

Substitute , , and in .

The general solution of is , where is an integer.

.

Find the angles on the interval .

If , then .

If , then .

Therefore, the angle of elevation and .

(b)

To find the maximum distance first find the critical points.

Consider .

Differentiate the above function with respect to .

Equate to zero.

The general solution of is , where is an integer.

.

Find the angles on the interval .

If , then .

If , then .

Therefore, the solution is on the interval .

Consider .

Differentiate the above function with respect to .

Substitue in .

By the second derivative test, the maximum distance occures at . 

Consider .

Substitue , and in .

Therefore, the maximum distance that can throw the ball is .

(c)

The function is .

Substitue   and in .

.

Draw the coordinate plane.

Use a graphing utility graph the function in the interval .

Graph :

.

(d)

Graph of :

Observe the above graph :

At , the distance that can throw the ball is .

At , the distance that can throw the ball is .

The maximum distance occurs at .

The maximum distance that can throw the ball is .

(a)

The angle of elevation and .

(b)

The maximum distance that can throw the ball is .

(c)

Graph of :

(d)

Observe the above graph :

At , the distance that can throw the ball is .

At , the distance that can throw the ball is .

The maximum distance occurs at .

The maximum distance that we can throw the ball is .