The time it will take for the object to traveled 74 ft is:
[tex]t=2\ seconds[/tex]
The formula which represents the distance traveled by a falling objectin time t is given by:
[tex]d=5t+16t^2[/tex]
Now, we are asked to find the time t required by the object in order to travel the distance of 74 ft.
i.e. we are asked to find t when d=74 ft.
i.e.
[tex]16t^2+5t=74\\\\16t^2+5t-74=0[/tex]
Now, we know that any quadratic equation of the type:
[tex]ax^2+bx+c=0[/tex]
is solved using the quadratic formula:
[tex]x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
Here,
[tex]a=16\ ,\ b=5\ and\ c=-74[/tex]
Hence, on putting these value in the formula we have:
[tex]t=-2.313\ and\ t=2[/tex]
Since, the time can't be negative.
Hence, we have:
[tex]t=2[/tex]
We have the height equation for a falling equation, we want to use it to find how long takes to travel a given distance.
We will find that the falling object needs to travel for 2 seconds to travel 74ft.
Here the height function, in ft, is:
d = 5*t + 16*t^2
We want to find the value of t, such that d = 74ft
Then we only need to solve:
74 = 5*t + 16*t^2
We can rewrite this as:
-16*t^2 - 5*t + 74 = 0.
This is a quadratic equation, so we can use Bhaskara's formula to get the solutions:
[tex]t = \frac{- (-5) \pm \sqrt{(-5)^2 - 4*(-16)*74} }{2*(-16)} \\\\t = \frac{ 5 \pm 69 }{-32}[/tex]
Then the two solutions are:
For how the problem is defined, the solution that is relevant is the positive one.
We can conclude that the object needs to travel for 2 seconds to travel 74ft.
If you want to learn more, you can read:
https://brainly.com/question/2925460
