Question

HELP ASAPPPPPPPPPP 25 POINTSSSSSS

Answer 1

(1).
`h(t)=-4(t-1)^2+36`

(2). 1 minute

Explanation:

• They want you to factorize the equation in such a way that the vertex appears as a number in the equation; and you do that by using a method called completing the square
• Here is our equation:
  `h(t)=-4t^2+8t+32`
• We factor it by completing the square: But first remember this:
• A quadratic equation has the general form
  `f(x)=ax^2+bx+c`
• Where a and b are the numbers before x squared and x respectively, and c is the number without an x, and f(x) is the value dependent on x
• In this case x is t
• So the steps are as follows

1. Equate the equation to zero:
   `-4t^2+8t+32=0`
2. Divide each term by the (a) of the equation in this case is it -4, and we get:
   `t^2-2t-8=0`
3. Then take the new (c) to the other side of the equation, in the case we add 8 to both sides to get:
   `t^2-2t=8`
4. Now the tricky part, you have to add to both sides of the equation the square of half of the coefficient of t or number before t, not t squared just t and you get:
   `t^2-2t+(-1)^2=8+(1)^2`
5. Now the left side is in the square form, or it just means when you factor the left side, you get it as the square of a certain single term, in this case we get:
   `(t-1)^2=8+1`
6. When we simplify we get:
   `(t-1)^2=9`
7. Now any equation in this form, will give you the vertex when you equate the term in parenthesis to zero, and simplify:
   `t-1=0,t=1`
8. 
   `t=1` is the value of the t or time at the vertex
9. To write the equation again, multiply every term with the (a) you used to get:
   `h(t)=-4(t-1)^2+36` , and this is the equation for #(1)

• Now here is why we needed to get the vertex; the vertex tells us at what point the height either reaches its maximum/highest level, or where it reaches its minimum/lowest level
• So since the time (t) at the vertex is 1, in order to find the height at this time, just plug it into the equation:
• 
  `h(1)=-4((1)-1)^2+36\\h(1)=-4(0)+36\\h(1)=36` So that's the height at the vertex
• Now it can either be the maximum/highest height or the minimum/lowest height, in order to know this we check as follows
• Remember the (a) we used to factor the equation? -4, if the (a) value of a quadratic function is less than 0, then it is a maximum equation, mean whatever vertex you get will be the point where the equation reaches its biggest value.
• So at a height of 36 meters, and a time of 1 minute, the craft reaches its highest point.