Question 51417
<pre><font size = 4 color = "green"><b>please help...thankyou 
3)Suppose a baseball is shot up from the ground straight 
up with an initial velocity of 32 feet per second. A 
function can be created by expressing distance above the 
ground, s, as a function of time, t. This function is 
s = -16t² + v<sub>O</sub>t + s<sub>O</sub> 
<sub>O</sub>
[16 represents 1/2g, the gravitational pull due to gravity 
(measured in feet per second²). v<sub>O</sub> is the initial velocity 
(how hard do you throw the object, measured in feet per second).

a) What is the function that describes this problem?
Answer: 

>>...initial velocity of 32 feet per second...<<

>>...v0 is the initial velocity (how hard do you throw the 
object, measured in feet per second)...<<

Translation:  v<sub>O</sub> = 32

>>...baseball is shot up from the ground...<<

>>...If you are standing on the ground, then s<sub>O</sub> = 0...<<

Translation:  s<sub>O</sub> = 0

>>...This function is s = -16t² + v<sub>O</sub>t + s<sub>O</sub>...<<

Translation: s = -16t² + 32t + 0

             s = -16t² + 32t  

Compare that to

          f(x) = ax² + bx

and we see that s = f(x), t = x, a = -16 and b = 32.

The vertex is the point which has coordinates

( -b/(2a), f(-b/(2a)) ) 

( -(32)/(2·-16), f(-b/(2a)) )

( 1, f(1) )

f(1) = -16(1)² + 32(1) = -16 + 32 = 16

So the vertex is the point (1, 16)

This means that the maximum height of 16 feet is obtained when 
the elapsed time = 1 second. 

b)The ball will be how high above the ground after 1 second?

             s = -16t² + 32t  

             s = -16(1)² + 32(1) = -16 + 32 = 16

c)How long will it take to hit the ground?

It is on the ground when s = 0


             s = -16t² + 32t

             0 = -16t² + 32t

    16t² - 32t = 0

Factor out 16t on the left:

    16t(t - 2) = 0
  
setting 16t = 0 gives t = 0

stting (t - 2) = 0 gives t = 2.

So the ball is on the ground at the start, when t = 0, and
again after the ball goes up and falls back down to the ground
2 seconds later.     


d)What is the maximum height of the ball?

             s = -16t² + 32t  

Compare that to

          f(x) = ax² + bx

and we see that s = f(x), t = x, a = -16 and b = 32.

The vertex is the point which has coordinates

( -b/(2a), f(-b/(2a)) ) 

( -(32)/(2·-16), f(-b/(2a)) )

( 1, f(1) )

What time will the maximum height be attained?

f(1) = -16(1)² + 32(1) = -16 + 32 = 16

So the vertex is the point (1, 16)

This means that the maximum height of 16 feet is obtained when 
the elapsed time = 1 second.  Note that this was the same time
as we were asked to use in part (b), and the 16 feet obtained
in part (b) was the actual maximum height.

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4)John has 300 feet of lumber to frame a rectangular patio
(the perimeter of a rectangle is 2 times length plus 2 
times width). He wants to maximize the area of his patio 
(area of a rectangle is length times width). What should 
the dimensions of the patio be, and show how the maximum 
area of the patio is calculated from the algebraic equation. 
Show clearly the algebraic steps which prove your dimensions 
are the maximum area which can be obtained. Use the vertex 
form to find the maximum area.

Answer: 
Show work in this space.

        _____L______
       |            |
       |            |W
       |____________|

>>...the perimeter of a rectangle is 2 times length 
plus 2 times width...<<

Translation:  P = 2L + 2W 


>>...John has 300 feet of lumber...<<

Translation:  P = 300

So 2L + 2W = 300
        2W = 300 - 2L
         W = 150 - L
>>...He wants to maximize the area of his patio (area of a 
rectangle is length times width)...<<

Translation: He wants to maximize A, where

        A = LW
        
Substitute (150 - L) for W:

        A = L(150 - L)

        A = 150L - L²

Arrange right side in descending powers of L.

        A = -L² + 150L

Compare that to

     f(x) = ax² + bx

and we see that y = f(x), L = x, a = -1 and b = 150.

The vertex is the point which has coordinates

( -b/(2a), f(-b/(2a)) ) 

( -(150)/(2·-1), f(-b/(2a)) )

( 75, f(75) )

f(75) = -1(75)² + 150(75) = -1(5625) + 11250 = 5625

So the vertex is the point (75, 5625)

This means that the maximum area pf 5625 square feet is obtained when 
the length L = 75.  Substituting 75 for L in

         W = 150 - L  
         W = 150 - 75
         W = 75

This means that the maximium area of 5625 square feet is obtained when
the Length and the width are both 75 feet, which means it is square.
My, that's a big patio!

Edwin</pre>