Answer:
[tex]z=\frac{46.5-46.7}{\frac{1.1}{\sqrt{150}}}=-2.23[/tex]
The p value would be given by:
[tex]p_v =2*P(z<-2.23)=0.0257[/tex]
For this case since th p value is lower than the significance level of0.05 we have enough evidence to reject the null hypothesis and we can conclude that the true mean for this case is significantly different from 46.7 MPG
Step-by-step explanation:
Information given
[tex]\bar X=46.5[/tex] represent the mean
[tex]\sigma=1.1[/tex] represent the population standard deviation
[tex]n=150[/tex] sample size
[tex]\mu_o =46.7[/tex] represent the value to verify
[tex]\alpha=0.05[/tex] represent the significance level for the hypothesis test.
z would represent the statistic
[tex]p_v[/tex] represent the p value
Hypothesis to test
We want to test if the true mean for this case is 46.7, the system of hypothesis would be:
Null hypothesis:[tex]\mu = 46.7[/tex]
Alternative hypothesis:[tex]\mu \neq 46.7[/tex]
Since we know the population deviation the statistic is given by:
[tex]z=\frac{\bar X-\mu_o}{\frac{\sigma}{\sqrt{n}}}[/tex] (1)
Replacing we got:
[tex]z=\frac{46.5-46.7}{\frac{1.1}{\sqrt{150}}}=-2.23[/tex]
The p value would be given by:
[tex]p_v =2*P(z<-2.23)=0.0257[/tex]
For this case since th p value is lower than the significance level of0.05 we have enough evidence to reject the null hypothesis and we can conclude that the true mean for this case is significantly different from 46.7 MPG
1. Growth of Functions (11 points) (1) (4 points) Determine whether each of these functions is O(x 2 ). Proof is not required but it may be good to try to justify it (a) 100x + 1000 (b) 100x 2 + 1000
Answer:
See explanation
Step-by-step explanation:
To determine whether each of these functions is [tex]O(x^2)[/tex], we apply these theorems:
A polynomial is always O(the term containing the highest power of n)Any O(x) function is always [tex]O(x^2)[/tex].(a)Given the function: f(x)=100x+1000
The highest power of n is 1.
Therefore f(x) is O(x).
Since any O(x) function is always [tex]O(x^2)[/tex], 100x+1000 is [tex]O(x^2)[/tex].
[tex](b) f(x)=100x^ 2 + 1000[/tex]
The highest power of n is 2.
Therefore the function is [tex]O(x^2)[/tex].
Answer:
i think its 2000
Step-by-step explanation:
Find the remainder when f(x)=2x3−x2+x+1 is divided by 2x+1.
Step-by-step explanation:
it can be simply done by using remainder theorem.
The two-way table shows the medal count for the top-performing countries in the 2012 Summer Olympics. A 5-column table has 5 rows. The first column has entries United notes, China, Russia, Great Britain, Total. The second column is labeled Gold with entries 46, 38, 24, 29, 137. The third column is labeled Silver with entries 29, 27, 26, 17, 99. The fourth column is labeled Bronze with entries 29, 23, 32, 19, 103. The fifth column is labeled Total with entries 104, 88, 82, 65, 339. Which statement is true?
Which statement is true?
The probability that a randomly selected silver medal was awarded to Great Britain is StartFraction 17 Over 99 EndFraction. The probability that a randomly selected medal won by Russia was a bronze medal is StartFraction 32 Over 103 EndFraction. The probability that a randomly selected gold medal was awarded to China is StartFraction 88 Over 137 EndFraction. The probability that a randomly selected medal won by the United States was a silver medal is StartFraction 104 Over 339 EndFraction.Answer:
(A)The probability that a randomly selected silver medal was awarded to Great Britain is 17/99.
Step-by-step explanation:
The table is given below:
[tex]\left|\begin{array}{l|c|c|c|c|c} &Gold&Silver & Bronze &Total\\United States &46 & 29 & 29 & 104\\China & 38 & 27 & 23 & 88\\Russia & 24 & 26 & 32 &82\\Great Britain & 29 & 17 & 19 & 65\\&&&&&\\Total &137 & 99 & 103 & 339\end{array}\right[/tex]
We calculate the probabilities given in the statements.
(A) The probability that a randomly selected silver medal was awarded to Great Britain
= 17/99
(B)The probability that a randomly selected medal won by Russia was a bronze medal
=32/82
(C)The probability that a randomly selected gold medal was awarded to China
=38/137
(D)The probability that a randomly selected medal won by the United States was a silver medal
=29/104
We can see that only the first statement is true.
Answer: A. The probability that a randomly selected silver medal was awarded to Great Britain is 17/99.
Step-by-step explanation:
I got it right on edge
Which expressions are equivalent to -3(2w+6)-4
Answer:
B is the answer
Step-by-step explanation:
-3(2w+6)-4
-6w-18-4
-6w-22
Answer:
B = 2(−3w + (−11)) is the answer.Step-by-step explanation:
-3(2w + 6) - 4
1. Distribute
= -3*2w = -6w
= -3 * 6 = -18
= -6w -18
2. Simplify like terms
= -18 - 4
= -22
3. Place variables and numbers together
= -6w - 22
-6w -22 is the answer.So, B is the answer.Explanation:
2 * -3w = -6w
2*-11 = -22
Place them together and you get the answer!
Write an equation in slope-intercept form of the line that passes through the point (-6,-5) with slope 6.
Answer:
y=6x+31
Step-by-step explanation:
Since we are given a point and a slope, we can use the slope-intercept formula.
[tex]y-y_{1} =m(x-x_{1})[/tex]
where (x1,y1) is a point on the line and m is the slope.
The point given is (-6,-5) and the slope is 6.
x1= -6
y1= -5
m=6
[tex]y--5=6(x--6)[/tex]
A negative number subtracted from another number, or two negative signs, becomes a positive.
[tex]y+5=6(x+6)[/tex]
We want to find the equation of the line, which is y=mx+b (m is the slope and b is the y-intercept). Therefore, we must get y by itself on one side of the equation.
First, distribute the 6. Multiply each term inside the parentheses by 6.
[tex]y+5=(6*x)+(6*6)[/tex]
[tex]y+5=6x+36[/tex]
Subtract 5 from both sides, because it is being added on to y.
[tex]y+5-5=6x+36-5[/tex]
[tex]y=6x+36-5[/tex]
[tex]y=6x+31[/tex]
The equation of the line is y=6x+31
For 9$ a shoekeeper buys 13 dozen pencils.However 3 dozen broke in transit. At what price per dozen must the shoekeeper sell the remaining pencils to make back 1/3 of the whole cost
Answer: $0.30 per dozen.
Step-by-step explanation:
Given: The whole cost of 13 dozen pencils = $9
If 3 dozen broke in transit, remaining dozens of pencils = 13-3 = 10 dozens
Also, Selling price of theses 10 dozens = [tex]\dfrac{1}{3}\times\text{whole cost}[/tex] [given]
[tex]=\dfrac{1}{3}\times9=\$3[/tex]
Then, the selling price of each dozen = [tex]\dfrac{\$3}{10}=\$0.30[/tex]
Hence, the shopkeeper sells the remaining pencils at $0.30 per dozen.
An interior angle of a regular polygon has a measure of 108°. What type of polygon is it?
Answer:
Polygon is pentagon
Step-by-step explanation:
In a regular polygon each angle is equal.
In a regular polygon Each angle of polygon is given by (2n-4)90/n
where n is the number of sides of the polygon
given
An interior angle of a regular polygon has a measure of 108°.
(2n-4)90/n = 108
=> 180n - 360 = 108n
=> 180n-108n= 360
=> 72n = 360
=> n = 360/72 = 5
Thus, polygon has 5 sides
and we know that regular polygon which has 5 sides is called pentagon.
Thus, Polygon is pentagon
In a study of 205 adults, the average heart rate was 75 beats per minute. Assume the population of heart rates is known to be approximately normal, with a standard deviation of 8 beats per minute. What does a margin of error of 1.1 for the 95% confidence interval of the average beats per minute mean? There is a 95% chance that the population mean is between 67 and 83 beats per minute. There is a 95% chance that the population mean is between 73.9 and 76.1 beats per minute. There is a 5% chance that the population mean is less than 75 beats per minute. There is a 5% chance that the population mean is more than 75 beats per minute.
Answer:
There is a 95% chance that the population mean is between 73.9 and 76.1 beats per minute.
Step-by-step explanation:
i have the test
There is a 95% chance that the population mean is between 73.9 and 76.1 beats per minute.
Calculation of margin of error:Since
The average heart rate was 75 beats per minute.
The standard deviation is 8 beats per minute
And, there is the study of 205 adults
Now the following formula is to be used
Since
[tex]x \pm z \frac{\sigma}{\sqrt{n} }[/tex]
Here
z = 1.96 at 95% confidence interval
So,
[tex]= 75 \pm 1.96 \frac{8}{\sqrt{205} } \\\\= 75 - 1.96 \frac{8}{\sqrt{205} } , 75 + 1.96 \frac{8}{\sqrt{205} }[/tex]
= 73.9 ,76.1
Hence, the above statement should be true.
Learn more about standard deviation here: https://brainly.com/question/20529928
The dimensions of a closed rectangular box are measured as 96 cm, 58 cm, and 48 cm, respectively, with a possible error of 0.2 cm in each dimension. Use differentials to estimate the maximum error in calculating the surface area of the box.
Answer:
161.6 cm²Step-by-step explanation:
Surface Area of the rectangular box = 2(LW+LH+WH)
L is the length of the box
W is the width of the box
H is the height of the box
let dL, dW and dH be the possible error in the dimensions L, W and H respectively.
Since there is a possible error of 0.2cm in each dimension, then dL = dW = dH = 0.2cm
The surface Area of the rectangular box using the differentials is expressed as shown;
S = 2{(LdW+WdL)+(LdH+HdL)+(WdH+HdW)]
Also given L = 96cm W = 58cm and H = 48cm, on substituting this given values and the differential error, we will have;
S = 2{(96*0.2+58*0.2) + (96*0.2+48*0.2)+(58*0.2+48*0.2)}
S = 2{19.2+11.6+19.2+9.6+11.6+9.6}
S = 2(80.8)
S = 161.6 cm²
Hence, the surface area of the box is 161.6 cm²
Jina wants to measure the width of a river. She marks off two right triangles, as shown in the figure. The base of the larger triangle has a length of 56m, and the base of the smaller triangle has a length of 26m. The height of the smaller triangle is 20.9m. How wide is the river? Round your answer to the nearest meter.
Answer:
width of a river = 45m
Step-by-step explanation:
ration and proportion
let x = width of a river
x 20.9 m
------ = --------
56 m 26 m
x = (20.9 * 56) / 26
x = 45 m
therefore the width of a river is 45 m
A manufacturer of chocolate chips would like to know whether its bag filling machine works correctly at the 418 gram setting. It is believed that the machine is underfilling the bags. A 9 bag sample had a mean of 413 grams with a standard deviation of 20. A level of significance of 0.1 will be used. Assume the population distribution is approximately normal. Is there sufficient evidence to support the claim that the bags are underfilled?
Answer:
No. At a significance level of 0.1, there is not enough evidence to support the claim that the bags are underfilled (population mean significantly less than 418 g.)
Step-by-step explanation:
This is a hypothesis test for the population mean.
The claim is that the bags are underfilled (population mean significantly less than 418 g.)
Then, the null and alternative hypothesis are:
[tex]H_0: \mu=418\\\\H_a:\mu< 418[/tex]
The significance level is 0.1.
The sample has a size n=9.
The sample mean is M=413.
As the standard deviation of the population is not known, we estimate it with the sample standard deviation, that has a value of s=20.
The estimated standard error of the mean is computed using the formula:
[tex]s_M=\dfrac{s}{\sqrt{n}}=\dfrac{20}{\sqrt{9}}=6.6667[/tex]
Then, we can calculate the t-statistic as:
[tex]t=\dfrac{M-\mu}{s/\sqrt{n}}=\dfrac{413-418}{6.6667}=\dfrac{-5}{6.6667}=-0.75[/tex]
The degrees of freedom for this sample size are:
[tex]df=n-1=9-1=8[/tex]
This test is a left-tailed test, with 8 degrees of freedom and t=-0.75, so the P-value for this test is calculated as (using a t-table):
[tex]\text{P-value}=P(t<-0.75)=0.237[/tex]
As the P-value (0.237) is bigger than the significance level (0.1), the effect is not significant.
The null hypothesis failed to be rejected.
At a significance level of 0.1, there is not enough evidence to support the claim that the bags are underfilled (population mean significantly less than 418 g.)
Which of the following equation is equivalent toY=2x+3? A. Y - 3 = 2(x-1) B. Y - 2x=3 C. Y - 3 = 2(x+1) D. Y + 2x = 3
Answer:
the answer is b
Step-by-step explanation:
A smaller square of side length 17 feet is cut out of a square board. What is the approximate area (shaded region) of the remaining board in square feet?
Answer:
The area of the remaining board is (x² - 289) sq. ft.
Step-by-step explanation:
Let the sides of the bigger square board be, x feet.
It is provided that a smaller square of side length 17 feet is cut out of the bigger square board.
The area of a square is:
[tex]Area=(side)^{2}[/tex]
Compute the area of the bigger square board as follows:
[tex]A_{b}=(side_{b})^{2}=x^{2}[/tex]
Compute the area of the smaller square board as follows:
[tex]A_{s}=(side_{s})^{2}=(17)^{2}=289[/tex]
Compute the area of the remaining board in square feet as follows:
[tex]\text{Remaining Area}=A_{b}-A_{s}[/tex]
[tex]=[x^{2}-289]\ \text{square ft.}[/tex]
Thus, the area of the remaining board is (x² - 289) sq. ft.
Which of the following values are in the range of the function graphed below? check all that apply.
A. 0
B. -4
C. 2
D. 1
E. -1
F. 4
Answer:
1
Step-by-step explanation:
The range is the output values
The only output value is y=1
The range is 1
Find sets of parametric equations and symmetric equations of the line that passes through the two points (if possible). (For each line, write the direction numbers as integers.) (0, 0, 25), (10, 10, 0)
Answer:
a)Parametric equations are
X= -10t
Y= -10t and
z= 25+25t
b) Symmetric equations are
(x/-10) = (y/-10) = (z- 25)/25
Step-by-step explanation:
We were told to fin two things here which are ; a) the parametric equations and b) the symmetric equations
The given two points are (0, 0, 25)and (10, 10, 0)
The direction vector from the points (0, 0, 25) and (10, 10, 0)
(a,b,c) =( 0 -10 , 0-10 ,25-0)
= < -10 , -10 ,25>
The direction vector is
(a,b,c) = < -10 , -10 ,25>
The parametric equations passing through the point (X₁,Y₁,Z₁)and parallel to the direction vector (a,b,c) are X= x₁+ at ,y=y₁+by ,z=z₁+ct
Substitute (X₁ ,Y₁ ,Z₁)= (0, 0, 25), and (a,b,c) = < -10 , -10 ,25>
and in parametric equations.
Parametric equations are X= 0-10t
Y= 0-10t and z= 25+25t
Therefore, the Parametric equations are
X= -10t
Y= -10t and
z= 25+25t
b) Symmetric equations:
If the direction numbers image and image are all non zero, then eliminate the parameter image to obtain symmetric equations of the line.
(x-x₁)/a = (y-y₁)/b = (z-z₁)/c
CHECK THE ATTACHMENT FOR DETAILED EXPLANATION
22424+72346*823456-4
Answer:
5.9573
Step-by-step explanation:
Answer:
59573770196 -- that is the answer
Step-by-step explanation:
Mark me as brainliest
The Westwood Warriors basketball team wants to score more points. To get better at scoring points the team is trying to improve its offensive strategies. Some opponents primarily use a zone defense, while others primarily use a man-to-man defense. When the Warriors play against teams that use a zone defense they score an average of 67 points per game with a standard deviation of 8 points per game. When they used a new offensive strategy against this defense, they scored 77 points. What is the Z-score of this value
Answer:
It is better for the warriors to use man-to-man defense.
Step-by-step explanation:
The complete question is: The Westwood Warriors basketball team wants to score more points. To get better at scoring points the team is trying to improve its offensive strategies. Some opponents primarily use a zone defense, while others primarily use a man-to-man defense. When the Warriors play against teams that use a zone defense they score an average of 67 points per game with a standard deviation of 8 points per game. When they play against teams that use a man-to-man defense they score an average of 62 points per game with a standard deviation of 5 points per game.
Since the Warriors started using their improved offensive strategies they have played two games with the following results.
Against the McNeil Mavericks
Maverick defense: zone
Warrior points: 77
Against the Round Rock Dragons
Dragon defense: man-to-man
Warrior points: 71
What is the Z-score of these values?
We are given that when the Warriors play against teams that use a zone defense they score an average of 67 points per game with a standard deviation of 8 points per game. When they play against teams that use a man-to-man defense they score an average of 62 points per game with a standard deviation of 5 points per game.
We have to find the z-scores.
Finding the z-score for the zone defense;Let X = points score by warriors when they use zone defense
The z-score probability distribution for the normal distribution is given by;
Z = [tex]\frac{X-\mu}{\sigma}[/tex] ~ N(0,1)
where, [tex]\mu[/tex] = mean score = 67 points
[tex]\sigma[/tex] = standard deviation = 8 points
It is stated that the Warriors scored 77 points when they used zone defense, so;
z-score for 77 = [tex]\frac{X-\mu}{\sigma}[/tex]
= [tex]\frac{77-67}{8}[/tex] = 1.25
Finding the z-score for the zone defense;Let X = points score by warriors when they use man-to-man defense
The z-score probability distribution for the normal distribution is given by;
Z = [tex]\frac{X-\mu}{\sigma}[/tex] ~ N(0,1)
where, [tex]\mu[/tex] = mean score = 62 points
[tex]\sigma[/tex] = standard deviation = 5 points
It is stated that the Warriors scored 71 points when they used man-to-man defense, so;
z-score for 71 = [tex]\frac{X-\mu}{\sigma}[/tex]
= [tex]\frac{71-62}{5}[/tex] = 1.8
So, it is better for the warriors to use man-to-man defense.
The trip is 375 miles and the train usually travels at a speed of 230mph. How long will it take them to travel.
Answer:
97.5 minutes or 1.63 hours
Step-by-step explanation:
1. Find the amount of time it takes to travel 1 mile
[tex]\frac{60}{230}[/tex] = 0.26 minutes
2. Multiply the distance by the time it takes to travel 1 mile
375 · 0.26 = 97.5 minutes
To convert to hours, divide by 60 because there are 60 minutes in 1 hour.
97.5 ÷ 60 = 1.63 hours
what is 3/5 of 1800
Answer:
1080
Step-by-step explanation:
first do 3 times 1800, because they are both the numerators. Then divide that number, which is 5400, by the denominator: 5. You will get 1080.
6th grade math :D help me please :)
Answer:
B
Step-by-step explanation:
In order to combine like terms, they must share the same variable. We can't combine things 9y and 3p because they contain two different variables. On the other hand, 7r and r work because there is only one. 7r and r combine to 8r+2
Please help me I’ll mark brainliest
pleassseeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
Answer:
The answer is 0.4
Step-by-step explanation:
2/5 is equal to 0.4
Answer:
0.4
Step-by-step explanation:
start with 2/5.
multipy the 5 by 20 to get 100, and the 2 by 20 to get 40.
so, now your fraction should look like this:
40/100
then, shift the decimal in 40 over 2 spots to get 0.40, or 0.4
hope this helps :)
Refer to the Trowbridge Manufacturing example in Problem 2-35. The quality control inspection proce- dure is to select 6 items, and if there are 0 or 1 de- fective cases in the group of 6, the process is said to be in control. If the number of defects is more than 1, the process is out of control. Suppose that the true proportion of defective items is 0.15. What is the probability that there will be 0 or 1 defects in a sam- ple of 6 if the true proportion of defects is 0.15
Answer:
77.64% probability that there will be 0 or 1 defects in a sample of 6.
Step-by-step explanation:
For each item, there are only two possible outcomes. Either it is defective, or it is not. The probability of an item being defective is independent of other items. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
The true proportion of defects is 0.15
This means that [tex]p = 0.15[/tex]
Sample of 6:
This means that [tex]n = 6[/tex]
What is the probability that there will be 0 or 1 defects in a sample of 6?
[tex]P(X \leq 1) = P(X = 0) + P(X = 1)[/tex]
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{6,0}.(0.15)^{0}.(0.85)^{6} = 0.3771[/tex]
[tex]P(X = 1) = C_{6,1}.(0.15)^{1}.(0.85)^{5} = 0.3993[/tex]
[tex]P(X \leq 1) = P(X = 0) + P(X = 1) = 0.3771 + 0.3993 = 0.7764[/tex]
77.64% probability that there will be 0 or 1 defects in a sample of 6.
A class of 30 music students includes 13 who play the piano, 15 who play the guitar, and 9 who play both the piano and the guitar. How many students in the class play neither instrument?
Answer: 2
Step-by-step explanation:
As given, out of 30 students, 15 play guitar and 13 play piano, thats 28.
Among these, 9 play both the guitar and the piano.
That means, only 2 remaining students play neither instrument. (30-15-13)
Suppose you have two six-sided dice where each side is equally likely to land face up when rolled.
Required:
a. What is the probability that you will roll doubles?
b. What is the probability that you will roll a sum of four?
c. Are these empirical or a theoretical probabilities?
i. Empirical
ii. Theoretical
Answer:a. ii.
A. Is Theoretical because there is no real way of knowing what you will roll.
Answer:
a. 0.17
b. 0.08
c. theoretical
Step-by-step explanation:
The length of a rectangle is 5M more than twice the width and the area of the rectangle is 63M to find the dimension of the rectangle
Answer:
width = 4.5 m
length = 14 m
Step-by-step explanation:
okay so first you right down that L = 5 + 2w
then as you know that Area = length * width so you replace the length with 5 + 2w
so it's A = (5 +2w) * w = 63
then 2 w^2 + 5w - 63 =0
so we solve for w which equals 4.5 after that you solve for length : 5+ 2*4.5 = 14
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━━━━━━━☆☆━━━━━━━
▹ Answer
0.25 = 1/4 because 25/100 = 1/4
▹ Step-by-Step Explanation
0.25 to a fraction → 25/100
25/100 = 1/4
Therefore, this statement is true. (0.25 = 1/4 because 25/100 = 1/4)
Hope this helps!
- CloutAnswers ❁
Brainliest is greatly appreciated!
━━━━━━━☆☆━━━━━━━
The following chart represents the record low temperatures recorded in Phoenix for April-November. Select the answer below that best describes the mean and the median of the data set (round answers to the nearest tenth). A graph titled Phoenix Low Temperatures has month on the x-axis and temperature (degrees Fahrenheit) on the y-axis. April, 32; May, 40; June, 50; July, 61; August, 60; September, 47; October, 34; November, 25. a. The mean is 43.5°F, and the median is 43.6°F. b. The mean is 60.5°F, and the median is 60.5°F. c. The mean is 60°F, and the median is 61°F. d. The mean is 43.6°F, and the median is 43.5°F.
Answer:
d. The mean is 43.6°F, and the median is 43.5°F.
Step-by-step explanation:
Hello!
The data corresponds to the low temperatures in Phoenix recorded for April to November.
April: 32ºF
May: 40ºF
June: 50ºF
July: 61ºF
August: 60ºF
September: 47ºF
October: 34ºF
November: 25ºF
Sample size: n= 8 months
The mean or average temperature of the low temperatures in Phoenix can be calculated as:
[tex]\frac{}{X}[/tex]= ∑X/n= (32+40+50+61+60+47+34+25)/8= 43.625ºF (≅ 43.6ºF)
The Median (Me) is the value that separates the data set in two halves, first you have to calculate its position:
PosMe= (n+1)/2= (8+1)/2= 4.5
The value that separates the sample in halves is between the 4th and the 5th observations, so first you have to order the data from least to greatest:
25; 32; 34; 40; 47; 50; 60; 61
The Median is between 40 and 47 ºF, so you have to calculate the average between these two values:
[tex]Me= \frac{(40+47)}{2} = 43.5[/tex] ºF
The correct option is D.
I hope this helps!
Answer:
it is d
Step-by-step explanation:
Use the Laplace transform to solve the given initial-value problem.
y' + 3y = f(t), y(0) = 0
where f(t) = t, 0 ≤ t < 1 0, t ≥ 1
Answer:
The solution to the given Initial - Value - Problem is [tex]y(t) = \frac{-1}{9} + \frac{1}{3}t + \frac{1}{9}e^{-3t} - [\frac{-1}{9} + \frac{1}{3}t - \frac{2}{9}e^{-3(t-1)}]u(t-1)[/tex]
Step-by-step explanation:
y' + 3y = f(t).................(1)
f(t) = t when 0 ≤ t < 1
f(t) = 0 when t ≥ 1
Step 1: Take the Laplace transform of the LHS of equation (1)
That is L(y' + 3y) = sY(s) + 3Y(s) = Y(s)[s + 3]..............(*)
Step 2: Get an expression for f(t)
For f(t) = t when 0 ≤ t < 1
f₁(t) = t (1 - u(t - 1)) ( there is a time shift of the unit step)
For f(t) = 0 when t ≥ 1
f₂(t) = 0(u(t-1))
f(t) = f₁(t) + f₂(t)
f(t) = t - t u(t-1)................(2)
Step 3: Taking the Laplace transform of equation (2)
[tex]F(s) = \frac{1}{s^2} - e^{-s} ( \frac{1}{s^2} + \frac{1}{s})[/tex]...............(**)
Step 4: Equating * and **
[tex]Y(s) [s + 3]=\frac{1}{s^2} - e^{-s} ( \frac{1}{s^2} + \frac{1}{s}) \\Y(s) = \frac{1}{s^2(s+3)} - e^{-s} ( \frac{1}{s^2(s+3)} + \frac{1}{s(s+3)})[/tex].......................(3)
Since y(t) is the solution we are looking for we need to find the Inverse Laplace Transform of equation (3) by first breaking every fraction into partial fraction:
[tex]\frac{1}{s^2 (s+3)} = \frac{-1}{9s} + \frac{1}{3s^2} + \frac{1}{9(s+3)}[/tex]
[tex]\frac{1}{s (s+3)} = \frac{1}{3s} + \frac{1}{3(s+3)}[/tex]
We can rewrite equation (3) by representing the fractions by their partial fractions.
[tex]Y(s) = \frac{-1}{9s} + \frac{1}{3s^2} + \frac{1}{9(s+3)} - e^{-s} [\frac{-1}{9s} + \frac{1}{3s^2} + \frac{1}{9(s+3)} + \frac{1}{3s} + \frac{1}{3(s+3)}]\\Y(s) = \frac{-1}{9s} + \frac{1}{3s^2} + \frac{1}{9(s+3)} - e^{-s}[\frac{2}{9s} + \frac{1}{3s^2} - \frac{2}{9(s+3)}][/tex]................(4)
step 5: Take the inverse Laplace transform of equation (4)
[tex]y(t) = \frac{-1}{9} + \frac{1}{3}t + \frac{1}{9}e^{-3t} - u(t-1)[\frac{2}{9} + \frac{1}{3}(t-1) - \frac{2}{9}e^{-3(t-1)}][/tex]
Simplifying the above equation:
[tex]y(t) = \frac{-1}{9} + \frac{1}{3}t + \frac{1}{9}e^{-3t} - [\frac{-1}{9} + \frac{1}{3}t - \frac{2}{9}e^{-3(t-1)}]u(t-1)[/tex]
The Laplace transform is use to solve the differential equation problem.
The solution for the given initial-value problem is,
[tex]y(t)=\dfrac{-1}{9}+\dfrac{-1}{3}t+\dfrac{1}{9}e^-3t-\left[\dfrac{-1}{9}+\dfrac{-1}{3}t+\dfrac{2}{9}e^{-3(t-1)}\right]u(t-1)[/tex]
Given:
The given initial value problem is [tex]y' + 3y = f(t)[/tex].
Consider the left hand side of the given equation.
[tex]y'+3y[/tex]
Take the Laplace transform.
[tex]L(y' + 3y) = sY(s) + 3Y(s) \\L(y' + 3y) = Y(s)[s + 3][/tex]
Consider the right hand side and get the expression for [tex]f(t)[/tex].
[tex]f(t) = t[/tex] when 0 ≤ t < 1
From time shift of the unit step
[tex]f_1(t) = t (1 - u(t - 1))[/tex]
For f(t) = 0 when t ≥ 1
Now,
[tex]f_2(t) = 0(u(t-1))f(t) = f_1(t) + f_2(t)f(t) = t - t u(t-1)[/tex]
Take the Laplace for above expression.
[tex]F(s)=\dfrac{1}{s^2}-e^{-s}\left(\dfrac{1}{s^2}+\dfrac{1}{s}\right)[/tex]
Now, the equate the above two equation.
[tex]Y(s)\left[s+3\right ]=\dfrac{1}{s^2}-e^{-s}\left(\dfrac{1}{s^2}+\dfrac{1}{s}\right)\\Y(s)=\dfrac {1}{(s^2(s+3))}-e^{-s}\left(\dfrac{1}{(s^2(s+3))}+\dfrac{1}{s(s+3)\right)}[/tex]
Find the inverse Laplace for the above equation.
[tex]\dfrac{1}{(s^2(s+3))}=\dfrac{-1}{9s}+\dfrac{1}{3s^2}+\dfrac{1}{9(s+3)}\\\dfrac{1}{(s(s+3))}=\dfrac{1}{3s}+\dfrac{1}{3(s+3)}[/tex]
Calculate the partial fraction of above equation.
[tex]Y(s)=\dfrac{-1}{9s}+\dfrac{1}{3s^2}+\dfrac{1}{9(s+3)}-e^{-s}\left[\dfrac{-1}{9s}+\dfrac{1}{3s^2}+\dfrac{1}{9(s+3)}+\dfrac{1}{3s}+\dfrac{1}{3(s+3)}\right]\\Y(s)=\dfrac{2}{9s}+\dfrac{1}{3s^2}+\dfrac{1}{9(s+3)}-e^{-s}\left[\dfrac{2}{9s}+\dfrac{1}{3s^2}-\dfrac{2}{9(s+3)}\right][/tex]
Take the inverse Laplace of the above equation.
[tex]y(t)=\dfrac{-1}{9}+\dfrac{-1}{3}t+\dfrac{1}{9}e^-3t-\left[\dfrac{-1}{9}+\dfrac{-1}{3}t+\dfrac{2}{9}e^{-3(t-1)}\right]u(t-1)[/tex]
Thus, the solution for the given initial-value problem is,
[tex]y(t)=\dfrac{-1}{9}+\dfrac{-1}{3}t+\dfrac{1}{9}e^-3t-\left[\dfrac{-1}{9}+\dfrac{-1}{3}t+\dfrac{2}{9}e^{-3(t-1)}\right]u(t-1)[/tex]
Learn more about what Laplace transformation is here:
https://brainly.com/question/14487937
If C(x) = 16000 + 600x − 1.8x2 + 0.004x3 is the cost function and p(x) = 4200 − 6x is the demand function, find the production level that will maximize profit. (Hint: If the profit is maximized, then the marginal revenue equals the marginal cost.)
Answer:
Quantity that will maximize profit=1000
Step-by-step explanation:
Assume quantity=x
Revenue=price*quantity
=(4200-6x)x
=4200x-6x^2
Marginal revenue(MR) =4200-12x
Cost(x)= 16000 + 600x − 1.8x2 + 0.004x3
Marginal cost(MC) =600-3.6x+0.012x^2
Marginal cost=Marginal revenue
600-3.6x+0.012x^2=4200-12x
600-3.6x+0.012x^2-4200+12x=0
0.012x^2-8.4x-3600=0
Solve the quadratic equation using
x= -b +or- √b^2-4ac/2a
a=0.012
b=-8.4
c=-3600
x=-(-8.4) +or- √(-8.4)^2- (4)(0.012)(-3600) / (2)(0.012)
= 8.4 +or- √70.56-(-172.8) / 0.024
= 8.4 +or- √70.56+172.8 / 0.024
= 8.4 +or- √243.36 / 0.024
= 8.4 +or- 15.6/0.024
= 8.4/0.024 +15.6/0.024
= 350+650
x=1000
OR
= 8.4/0.024 -15.6/0.024
= 350 - 650
= -300
x=1000 or -300
Quantity that maximises profits can not be negative
So, quantity that maximises profits=1000