Question

The function H(t) = −16t2 + 96t + 80 shows the height H(t), in feet, of a projectile after t seconds. A second object moves in the air along a path represented by g(t) = 31 + 32.2t, 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 2 through 5 for the 2 functions. Between what 2 seconds is the solution to H(t) = g(t) located? How do you know? Part B: Explain what the solution from Part A means in the context of the problem.

Answer 1

Answer and Explanation :

Given : The function
`H(t)=-16t^2+96t+80` shows the height H(t), in feet, of a projectile after t seconds.

A second object moves in the air along a path represented by
`g(t) = 31 + 32.2t`, 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 2 through 5 for the 2 functions. Between what 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 :

For Part A -

Substitute the value of t from 2 to 5 in the equation H(t) and g(t)

t
`H(t)=-16t^2+96t+80`
`g(t) = 31 + 32.2t`

2
`-16(2)^2+96(2)+80=208`
`31 + 32.2(2)=95.4`

3
`-16(3)^2+96(3)+80=224`
`31 + 32.2(3)=127.6`

4
`-16(4)^2+96(4)+80=208`
`31 + 32.2(4)=159.8`

5
`-16(5)^2+96(5)+80=160`
`31 + 32.2(5)=192`

To determine the interval in which H(t)=g(t), we have to plot the equation and the table points and see the intersection.

Refer the attached figure below.

The intersection point of both the equations are (4.647,180.62)

This point lies between t at 4 to 5.

Therefore, H(t)=g(t) between t=4 to t=5.

For Part B -

The Part A shows that the projectile destroyed along the path are same in interval [4,5] and gives the solution of equation.

Answer 2

A.
t H(t) g(t)
2 208 95.4
3 224 127.6
4 208 159.8
5 160 192

The solution for H(t) = g(t) is in between 4 and 5 seconds.

B.
Between 4 and 5 seconds, the projectile and the second object will have the same.