# Population The resident population P (in millions) of the United States from 2000 through 2013 can be modeled by P = -0.00232t^{3} + 0.0151y^{2} + 2.83t + 281.8, 0 leq t leq 13, where t = 0 corresponds to 2000. (Source: U.S. Census Bureau) Make a conjecture about the maximum and minimum populations of the United States from 2000 to 2013. Analytically find the maximum and minimum populations over the interval. The brief paragrah while comparing a conjecture with the minimum population was 281.8 million in 2000 and the maximum population was 316.1 million in 2013. We need to calculate: The absolute extrema of the popullation P = -0.00232t^{3} + 0.0151y^{2} + 2.83t + 281.8, 0 leq t leq 13 over the closed interval [0, 13].

Question
Comparing two groups
Population The resident population P (in millions) of the United States from 2000 through 2013 can be modeled by $$P = -0.00232t^{3} + 0.0151y^{2} + 2.83t + 281.8, 0 \leq t \leq 13,$$ where $$t = 0$$ corresponds to 2000.
(Source: U.S. Census Bureau)
Make a conjecture about the maximum and minimum populations of the United States from 2000 to 2013.
Analytically find the maximum and minimum populations over the interval.
The brief paragrah while comparing a conjecture with the minimum population was 281.8 million in 2000 and the maximum population was 316.1 million in 2013.
We need to calculate: The absolute extrema of the popullation $$P = -0.00232t^{3} + 0.0151y^{2} + 2.83t + 281.8, 0 \leq t \leq 13$$ over the closed interval [0, 13].

2021-03-08
Used formula:
The derivative of power function given by
$$\frac{d}{dx} (x^{n}) = nx^{n - 1}$$
Procedure to find the extrema of the continuous function f on closed interval [a, b].
Step 1: Find the derivative of the function f.
Step 2: Find the critical points of f in the open interval (a, b).
Step 3: Determine the value of f at each of the critical numbers in the open interval (a,b).
Step 4: Determine the value of f at each of the end-points a and b.
Step 5: The least of these values is the minimum and the greatest is the maximum.
Calculation:
Consider the function $$P = P = -0.00232t^{3} + 0.0151y^{2} + 2.83t + 281.8,$$
Step 1. Determine first derivative of the function $$P = P = -0.00232t^{3} + 0.0151y^{2} + 2.83t + 281.8, 0$$
$$P = -0.0069t^{2} + 0.03t + 2.83$$
Step 2. Find out the critical points of P. It occurs when $$P = 0$$ or P underfined.
$$P = 0$$
$$-0.0069t^{2} + 0.03t + 2.83 = 0$$
$$t=\frac{-0.03 \pm \sqrt{0.03}^{2}-4(-0.0069)(2.83)}{2(-0.0069)}$$
$$=\frac{-0.03 \pm \sqrt{0.0009 + 0.078108}}{-0.0138}$$
$$=\frac{-0.03 \pm \sqrt{0.079008}}{-0.0138}$$
$$=\frac{-0.03 \pm 0.281}{-0.0138}$$
Further solving,
$$t =\frac{-0.03 \pm 0.281}{-0.0138}$$
$$= -18.19$$
And,
$$t =\frac{-0.03 - 0.281}{-0.0138}$$
$$= 22.53$$
This gives,
$$t = -18.19, 22.53$$
For end-point $$t = 0,$$
Substitute $$t = 0$$ in the function $$P = -0.00232t^{3} + 0.0151y^{2} + 2.83t + 281.8,$$
$$P = -0.0023(0)^{3} + 0.015(0)^{2} + 2.83(0) + 281.8,$$
$$= 281.8$$
For end-point $$t = 13,$$
Substitute $$t = 13$$ in the function $$P = -0.0023^{3} + 0.015^{2} + 2.83t + 281.8,$$
$$P = -0.0023(13)^{3} + 0.015(13)^{2} + 2.83(13) + 281.8,$$
$$= -0.0023 (2197) + 0.015 (169) + 3.83(13) + 281.8$$
$$= -5.531 + 2.535 + 36.79 + 281.8$$
$$= 316.1$$
Use all the information to form the table as,
$$\begin{array}{|c|c|} \hline t-value & s=0 & r=13 \\ \hline P & 281.8 & 316.1 \\ \hline Conclusion & Minimum & Maximum \\\hline \end{array}$$
From the table, it can be concluded that the minimum population was 281.8 million in 2000 and the maximum population was 316.1 million in 2013.

### Relevant Questions

Population The resident population P (in millions) of the United States from 2000 through 2013 can be modeled by $$P = -0.00232t^{3} + 0.0151y^{2} + 2.83t + 281.8, 0 \leq t \leq 13,$$ where $$t = 0$$ corresponds to 2000.
(Source: U.S. Census Bureau)
Make a conjecture about the maximum and minimum populations of the United States from 2000 to 2013.
Analytically find the maximum and minimum populations over the interval.
The brief paragrah while comparing a conjecture with the minimum population was 281.8 million in 2000 and the maximum population was 316.1 million in 2013.
The population P (in thousands) of Tallahassee, Florida, from 2000 through 2014 can be modeled by $$P = 150.9e^{kt},$$ where t represents the year, with $$t = 0$$ corresponding to 2000. In 2005, the population of Tallahassee was about 163,075.
(a) Find the value of k. Is the population increasing or decreasing? Explain.
(b) Use the model to predict the populations of Tallahassee in 2020 and 2025. Are the results reasonable? Explain.
(c) According to the model, during what year will the — populates reach 200,000?

The U.S. Census Bureau publishes information on the population of the United States in Current Population Reports. The following table gives the resident U.S. population, in millions of persons, for the years 1990-2009. Forecast the U.S. population in the years 2010 and 2011

$$\begin{array}{|c|c|} \hline \text{Year} & \text{Population (millions)} \\ \hline 1990 & 250 \\ \hline 1991 & 253\\ \hline 1992 & 257\\ \hline 1993 & 260\\ \hline 1994 & 263\\ \hline 1995 & 266\\ \hline 1996 & 269\\ \hline 1997 & 273\\ \hline 1998 & 276\\ \hline 1999 & 279\\ \hline 2000 & 282\\ \hline 2001 & 285\\ \hline 2002 & 288\\ \hline 2003 & 290\\ \hline 2004 & 293\\ \hline 2005 & 296\\ \hline 2006 & 299\\ \hline 2007 & 302\\ \hline 2008 & 304\\ \hline 2009 & 307\\ \hline \end{array}$$

a) Obtain a scatterplot for the data.

b) Find and interpret the regression equation.

c) Mace the specified forecasts.

The unstable nucleus uranium-236 can be regarded as auniformly charged sphere of charge Q=+92e and radius $$\displaystyle{R}={7.4}\times{10}^{{-{15}}}$$ m. In nuclear fission, this can divide into twosmaller nuclei, each of 1/2 the charge and 1/2 the voume of theoriginal uranium-236 nucleus. This is one of the reactionsthat occurred n the nuclear weapon that exploded over Hiroshima, Japan in August 1945.
A. Find the radii of the two "daughter" nuclei of charge+46e.
B. In a simple model for the fission process, immediatelyafter the uranium-236 nucleus has undergone fission the "daughter"nuclei are at rest and just touching. Calculate the kineticenergy that each of the "daughter" nuclei will have when they arevery far apart.
C. In this model the sum of the kinetic energies of the two"daughter" nuclei is the energy released by the fission of oneuranium-236 nucleus. Calculate the energy released by thefission of 10.0 kg of uranium-236. The atomic mass ofuranium-236 is 236 u, where 1 u = 1 atomic mass unit $$\displaystyle={1.66}\times{10}^{{-{27}}}$$ kg. Express your answer both in joules and in kilotonsof TNT (1 kiloton of TNT releases 4.18 x 10^12 J when itexplodes).
In the figure below, the rolling axle, 1.43 m long, is pushed along horizontal rails at a constant speed v = 3.36 m/s.

A resistor R = 0.325 ohm is connected to the rails at points a and b, directly opposite each other. (The wheels make good electrical contact with the rails, and so the axle, rails, and R form a closed-loop circuit. The only significant resistance in the circuit is R.) There is a uniform magnetic field B = 0.0850 T vertically downward. Calculate the induced current I in the resistor and what horizontal force F is required to keep the axle rolling at constant speed?
A wagon with two boxes of Gold, having total mass 300 kg, is cutloose from the hoses by an outlaw when the wagon is at rest 50m upa 6.0 degree slope. The outlaw plans to have the wagon roll downthe slope and across the level ground, and then fall into thecanyon where his confederates wait. But in a tree 40m from thecanyon edge wait the Lone Ranger (mass 75.0kg) and Tonto (mass60.0kg). They drop vertically into the wagon as it passes beneaththem. a) if they require 5.0 s to grab the gold and jump out, willthey make it before the wagon goes over the edge? b) When the twoheroes drop into the wagon, is the kinetic energy of the system ofthe heroes plus the wagon conserved? If not, does it increase ordecrease and by how much?
Two light sources of identical strength are placed 8 m apart. An object is to be placed at a point P on a line ? parallel to the line joining the light sources and at a distance d meters from it (see the figure). We want to locate P on ? so that the intensity of illumination is minimized. We need to use the fact that the intensity of illumination for a single source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source.
We will now add support for register-memory ALU operations to the classic five-stage RISC pipeline. To offset this increase in complexity, all memory addressing will be restricted to register indirect (i.e., all addresses are simply a value held in a register; no offset or displacement may be added to the register value). For example, the register-memory instruction add x4, x5, (x1) means add the contents of register x5 to the contents of the memory location with address equal to the value in register x1 and put the sum in register x4. Register-register ALU operations are unchanged. The following items apply to the integer RISC pipeline:
a. List a rearranged order of the five traditional stages of the RISC pipeline that will support register-memory operations implemented exclusively by register indirect addressing.
b. Describe what new forwarding paths are needed for the rearranged pipeline by stating the source, destination, and information transferred on each needed new path.
c. For the reordered stages of the RISC pipeline, what new data hazards are created by this addressing mode? Give an instruction sequence illustrating each new hazard.
d. List all of the ways that the RISC pipeline with register-memory ALU operations can have a different instruction count for a given program than the original RISC pipeline. Give a pair of specific instruction sequences, one for the original pipeline and one for the rearranged pipeline, to illustrate each way.
Hint for (d): Give a pair of instruction sequences where the RISC pipeline has “more” instructions than the reg-mem architecture. Also give a pair of instruction sequences where the RISC pipeline has “fewer” instructions than the reg-mem architecture.
Indicate true or false for the following statements. If false, specify what change will make the statement true.
a) In the two-sample t test, the number of degrees of freedom for the test statistic increases as sample sizes increase.
b) When the means of two independent samples are used to to compare two population means, we are dealing with dependent (paired) samples.
c) The $$\displaystyle{x}^{{{2}}}$$ distribution is used for making inferences about two population variances.
d) The standard normal (z) score may be used for inferences concerning population proportions.
e) The F distribution is symmetric and has a mean of 0.
f) The pooled variance estimate is used when comparing means of two populations using independent samples.
g) It is not necessary to have equal sample sizes for the paired t test.
The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
Suspect was Armed:
Black - 543
White - 1176
Hispanic - 378
Total - 2097
Suspect was unarmed:
Black - 60
White - 67
Hispanic - 38
Total - 165
Total:
Black - 603
White - 1243
Hispanic - 416
Total - 2262
Give your answer as a decimal to at least three decimal places.
a) What percent are Black?
b) What percent are Unarmed?
c) In order for two variables to be Independent of each other, the P $$(A and B) = P(A) \cdot P(B) P(A and B) = P(A) \cdot P(B).$$
This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
If answer c is "significantly different" than answer d, then that means that there could be a different percentage of unarmed people being shot based on race. We will check this out later in the course.
Let's compare the percentage of unarmed shot for each race.
e) What percent are White and Unarmed?
f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
Why is that?
This is because there are more white people in the United States than any other race and therefore there are likely to be more white people in the table. Since there are more white people in the table, there most likely would be more white and unarmed people shot by police than any other race. This pulls the percentage of white and unarmed up. In addition, there most likely would be more white and armed shot by police. All the percentages for white people would be higher, because there are more white people. For example, the table contains very few Hispanic people, and the percentage of people in the table that were Hispanic and unarmed is the lowest percentage.
Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
The correct way to compare is "conditional probability". Conditional probability is getting the probability of something happening, given we are dealing with just the people in a particular group.
g) What percent of blacks shot and killed by police were unarmed?
h) What percent of whites shot and killed by police were unarmed?
i) What percent of Hispanics shot and killed by police were unarmed?
You can see by the answers to part g and h, that the percentage of blacks that were unarmed and killed by police is approximately twice that of whites that were unarmed and killed by police.
j) Why do you believe this is happening?
Do a search on the internet for reasons why blacks are more likely to be killed by police. Read a few articles on the topic. Write your response using the articles as references. Give the websites used in your response. Your answer should be several sentences long with at least one website listed. This part of this problem will be graded after the due date.
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