a baseball throwing machine throws a baseball straight up with an initial velocity of 160 ft/sec from a…

Explanation:

Step1: Recall the kinematic - height formula

The general formula for the height (h(t)) of an object in vertical - motion under the influence of gravity is (h(t)=-16t^{2}+v_{0}t + h_{0}), where (v_{0}) is the initial velocity and (h_{0}) is the initial height.

Step2: Identify the given values

We are given that (v_{0}=160) ft/sec and (h_{0} = 45) ft.

Step3: Write the height - equation

Substitute (v_{0}=160) and (h_{0}=45) into the formula (h(t)=-16t^{2}+v_{0}t + h_{0}), we get (h(t)=-16t^{2}+160t + 45).

Step4: For part (b), find the time to reach the maximum height

The function (h(t)=-16t^{2}+160t + 45) is a quadratic function in the form (y = ax^{2}+bx + c) with (a=-16), (b = 160), and (c = 45). The time (t) at which the maximum of a quadratic function (y=ax^{2}+bx + c) occurs is given by (t=-\frac{b}{2a}). Substitute (a=-16) and (b = 160) into (t =-\frac{b}{2a}), we have (t=-\frac{160}{2\times(-16)}=\frac{160}{32}=5) seconds.

Step5: Find the maximum height

Substitute (t = 5) into the height - function (h(t)=-16t^{2}+160t + 45). (h(5)=-16\times5^{2}+160\times5 + 45=-16\times25+800 + 45=-400+800 + 45=445) ft.

Answer:

(a) The equation is (h(t)=-16t^{2}+160t + 45) (b) The maximum height is 445 ft and it takes 5 seconds to reach that height.