An astronaut on the moon throws a baseball upward. the astronaut is 6 ft, 6 in. tall, and the initial velocity of the ball is 30 ft per sec. the height s of the ball in feet is given by the equation s equals negative 2.7 t squared plus 30 t plus 6.5, where t is the number of seconds after the ball was thrown. complete parts a and b.

89.803 ft

Explanation:

The complete question is written below:

An astronaut on the moon throws a baseball upward. The astronaut is 6ft, 6 in. tall and the initial velocity of the ball is 30 ft per second. The height s of the ball in feet is given by the equation,s=-2.7t^2+30t+6.5, where t is the number of seconds after the ball is thrown.

The ball will never reach a height of 100ft. How can this be determined algebraically?

We have the following equation that expresses the height as a function of time:

(1)

Now, if we wan to find the maximum height the baseball reaches and prove it is less than 100 ft, we firstly have to find the time the whole parabolic movement lasts and then find which is the time it takes the baseball to reach its maximum height.

So, if we want to calculate , this is fulfilled when , when the baseball hits the ground:

(2)

This is a quadratic equation of the form , and we have to use the quadratic formula if we want to find  :

 (3)

Where , ,

Substituting the known values and choosing the positive result of the equation:

 (4)

Then we can calculate :

(5)

 (6)

Substituting (6) in (1):

(7)

(8) This is the maximum height the baseball reaches, as we can see it is less than 100 ft