If the power of the rocket handler is given by P = [(t – 11)(t² + 1) + 5(7t – 5)]e¬0.5t Where t is the time of flight: What is the value of p as t gets large [end of flight], t → ∞ ii. At what values of t, p is zero. iii. Find the minimum value of p. iv. i. Find the maximum value of p. Sketch the graph of p against t for t20.0 v.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![In the fourth month of your apprenticeship you were asked to
demonstrate your engineering mathematics capabilities through
dedicated tasks that should be submitted in a well written report to be
reviewed by your supervisors and easily presented to sales and public
relation departments.
Task 1
If the power of the rocket handler is given by
P = [(t – 11)(t² + 1) + 5(7t – 5)]e¬0.5t
Where t is the time of flight:
i.
ii. At what values of t, p is zero.
iii.
What is the value of p as t gets large [end of flight], t → ∞
Find the minimum value of p.
iv.
Find the maximum value of p.
Sketch the graph of p against t for t20.0
vi.
v.
Analytically, determine the intervals when P is increasing and
when it is decreasing through the equation and compare it to
the graph you sketched.
vii. Does P have an inflection point, if yes find it.
viii. Find the maximum and minimum energy points in the time
interval (0, 5) sec. (Hint: energy = S p(t)dt)
ix. Find the energy between t=1 sec and t=4 sec.
Task 2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0655a5d1-dcfa-44da-85e4-2120d213c0db%2Faede989f-a94c-47ee-a7fa-8289cc314cc2%2F03l4g7d_processed.png&w=3840&q=75)
![Scenario:
Nanosatellite space launches could significantly benefit from an
electrically powered launch complex, based on an electromagnetic coil
launcher. A research group at Lockheed Martin Company conducted
several experiments and studies to estimate the required launcher
parameters and some fixed facility issues. One of these studies is based
on electromagnetic launch, or electromagnetic gun technology, which is
constrained to a coaxial geometry to take advantage of the efficiency of
closely-coupled coils. A baseline configuration for analysis considers a
payload mass of 10 kg, launch velocity of 6 km/s, a second stage solid
booster for orbital insertion, and a payload fraction of about 0. 1. The
launch facility is envisioned as an inclined track, 1 -2 km in length,
mounted on a hillside at 25 degrees aimed in the orbital inclination of
interest. The launcher energy and power requirements fall in the range
of 2000 MJ and 2 MW electric. This energy would be supplied by 400
modules of energy storage and magnetic coils. With a prime power
generator of 2 MW, a launch rate of some 200 satellites per day is
possible. The launch requires high acceleration, so the satellite package
must be hardened to launch acceleration on the order of 1000 gee.
Parametric evaluations compare performance parameters for a launcher
length of 1-2 km, exit velocity of 4-8 km/s, and payloads of 1 - 100 kg. The
EM launch complex could greatly reduce the amount of fuels handling,
reduce the turn-around time between launches, allow more concurrence
in launch preparation, reduce the manpower requirements for launch
vehicle preparation and increase the reliability of launch by using more
standardized vehicle preparations. Most importantly, such a facility
could reduce the cost per launch and could give true launch-on-demand
capability for nanosatellites.
You were offered an apprenticeship at design department at Lockheed
Martin Company to verify the results of its nanosatellite space launchers
[https://www.lockheedmartin.com/en-us/index.html].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0655a5d1-dcfa-44da-85e4-2120d213c0db%2Faede989f-a94c-47ee-a7fa-8289cc314cc2%2Fbxdwrux_processed.png&w=3840&q=75)
Step by step
Solved in 6 steps with 6 images
