Arocket is launched upward so that its distance, in feet, above the ground after t seconds is represented by the function below. what is its maximum height? h(t) = -14t^2 +420t a) 13,020 ft b) 5684 ft c) 3150 ft d) 1740 ft e) none

The maximun height that the rocket will reach is 3150 ft.

Step-by-step explanation:

This problem requieres that we maximize the function. For this we need the function to be of the form:

We already have that covered, as the height of the rocket is represented by:

We start calculating the derivative of h(t), wich is:

We need to remember that the derivative of a function represents the slope of said function at a given point. The maximum value of the function will have a slope equal to zero.

So we find the value in wich the derivative equals zero:

This means that 15 seconds is the time at wich the rocket will reach it's maximum height.

We replace this value in the original equation to solve the problem:

With this we can conclude that the maximum height the rocket will reach is 3150 ft.

the range of a function is the range of y-values for which it is defined, or exists. given that we are only looking at the y-values, we can already discount the first two answers as they deal with the domain of the function (range of x-values).

now, what we need to do is find the smallest and highest y-values for which the function is defined. if you'd like, you could take a ruler and start at the very bottom of the graph, find the lowest point, then move the ruler higher and higher until you reach the highest point. doing so either in your head or with an actual ruler, we can see that the lowest y-value is -5 and the highest y-value is -1.

the question now remains as to whether they are included in the range (fourth answer) or excluded (third answer). the key here is to recognise whether they are marked with an open or a closed circle. since both (-2, -5) and (2, -1) are marked with a closed circle, they are points that exist on the graph and are thus included in the range (note that if they were marked with an open circle like (4, -2) then that would signify that there is a break in the graph at that point).

thus, the fourth answer, (y | -5 ≤ y ≤ -1), is correct.

also note that usually it is good practice to scan your eye inbetween the highest and lowest point for any breaks in the graph - we do have one here at (4, -2), however -2 is still included in the range because there is another point that is included in the graph with the same y-value, ie. point (1, -2).

i hope that but if you require any further explanation don't hesitate to comment below.

111

step-by-step explanation: