Question

(09.06 HC) The function H(t) = −16t2 + 112t + 24 shows the height H(t), in feet, of a cannon ball after t seconds. A second cannon ball moves in the air along a path represented by g(t) = 5 + 3.2t, where g(t) is the height, in feet, of the object from the ground at time t seconds. Part A: Create a table using integers 4 through 7 for the 2 functions. Between what 2 seconds is the solution to H(t) = g(t) located? How do you know? (6 points) Part B: Explain what the solution from Part A means in the context of the problem. (4 points)

Answer 1

Answer and Explanation :

Given : The function $H(t) = -16t^2 + 112t + 24$ shows the height H(t), in feet, of a cannon ball after t seconds. A second cannon ball moves in the air along a path represented by $g(t) = 5 + 3.2t$, where g(t) is the height, in feet, of the object from the ground at time t seconds.

To find :

Part A - Create a table using integers 4 through 7 for the 2 functions. Between what 2 seconds is the solution to H(t) = g(t) located?

Solution :

We create a table of H(t) and g(t) at t values from 4 to 7.

t           $H(t) = -16t^2 + 112t + 24$             $g(t) = 5 + 3.2t$

4          $-16(4)^2+112(4)+24=216$           $5+3.2(4)=17.8$

5          $-16(5)^2+112(5)+24=184$           $5+3.2(5)=21$

6          $-16(6)^2+112(6)+24=120$           $5+3.2(6)=24.2$

7          $-16(7)^2+112(7)+24=24$            $5+3.2(7)=27.4$

To find the points at which H(t)=g(t) we plot the graph of the equations and points which shows the intersection and interval.

Refer the attached figure below.

The intersection point is (6.97,27.305).

So, The interval in which this value lie is from 6 to 7.

Therefore, Between 6 and 7, 2 seconds is the solution to H(t) = g(t) located.

Part B - Explain what the solution from Part A means in the context of the problem.

Solution : Part A shows you the point where both the equation coincide and give you the interval in which both the equations are equal.  

Answer 2

PART A:

The table of h(t) and g(t) for the value of x between 4 and 7 inclusive are shown below.

The solution for h(x)=g(x) lies between x = 4 and x = 5, this is becuase the range of value given by h(4) and h(5) and g(4) and g(5). 

The graph is shown in picture 2 to confirm this. The two functions intersect each other between x = 4 and x = 5

PART B: 

The point of intersection is the point where the cannon balls will collide.