math help asap!

Question

solve the problem and explain briefly.

Answer 1

"In context" nearly always means that we're dealing with a real-world problem and that some of the purely mathematical aspects of the equation don't make sense in real life.

Graph the equation, first of all.

Note that the curve extends into quadrants I, III, and IV. But wait! That would mean that the rocket can keep going after it hits the ground (not to mention being underground before it starts). It also suggests that the x (time) axis can be negative. But that makes no sense either.

So all you are really being asked here is What range of values for x (i.e., the domain of the function) makes sense. Since there's nothing going on before launch x≥0. Since nothing happens after the rocket returns to earth after 4 seconds, then x ≤4. Write this as a 3-part inequality like this: 0≤x≤4 or in interval notation as [0,4].

Best wishes!

Answer 2

Great quesiton.

So -16t² + 64t is the equation of a parabola. Essentially, it will initially get really large, stop at a point, and curve downwards. Because the height is represented by this equation, think about it this way:

The height can only be 0 or positive, right? In the real-world the height will not be negative since the toy rocket is not going underground, it will simply hit the ground.

So, we need to find when the parabola intersects the x-axis because that is when h=0.

So, set the height to 0 and solve for x.

0 = -16t² + 64t

0 = -16t(t-4)

From this, we can conclude t = 0 or 4 since that makes the right hand side = 0 and satisfy our equation.

So, the domain is 0 < x < 4 since the rocket is at or above the ground between these points. It is not negative, so this domain works!

Check this equation out on Desmos for a visual demonstration. It may make it easier to understand.