Answer:
The break-even point is when x is equal to 3.75
Step-by-step explanation:
At the break-even point, total cost function is equal to the total revenue function. In that regard, break-even is when;
C = 12x + 30 is equal to R = 20x.
thus, 12x + 30 = 20x
then, 12x - 12x + 30 = 20x - 12x
therefore, 30 = 8x
then, 30/8 = 8x/8
finally, x = 15/4 or 3.75
A stuffed animal business has a total cost of production C=12x+30 and a revenue function R=20x, the Break even point is (3.75,75)
Given :
A stuffed animal business has a total cost of production C=12x+30 and a revenue function R=20x.
Break even point occurs when revenue = cost
R=C
Replace the expression and solve for x
[tex]R=C\\20x=12x+30\\20x-12x=30\\8x=30\\divide \; by \; 8\\x=\frac{15}{4}\\x=3.75[/tex]
Now we find out y using Revenue
[tex]R= 20x\\R=20(3.75)\\R=75[/tex]
So y is 75
Break even point is (3.75,75)
Learn more : brainly.com/question/15281855
In a recent year, the Better Business Bureau settled 75% of complaints they received. You have been hired by the Bureau to investigate complaints this year involving computer stores. You plan to select a random sample of complaints to estimate the proportion of complaints the Bureau is able to settle. Suppose your sample size is 113. What is the probability that the sample proportion will be at most 2 percent more than the population proportion
Answer:
what
Step-by-step explanation:
Each limit represents the derivative of some function f at some number a. State such an f and a in each case.
lim √9 + h - 3 / h
h-->0
Answer:
a = 0f(h) = [tex]\frac{\sqrt{9+h} - 3}{h}[/tex]limit of the function is 1/6Step-by-step explanation:
The general form representing limit of a function is expressed as shown below;
[tex]\lim_{h \to a} f(h)[/tex] where a is the value that h will take and use in the function f(h). It can be expressed in words as limit of function f as h tends to a. Comparing the genaral form of the limit to the limit given in question [tex]\lim_{h \to 0} \frac{\sqrt{9+h} - 3}{h}[/tex], it can be seen that a = 0 and f(h) = [tex]\frac{\sqrt{9+h} - 3}{h}[/tex]
Taking the limit of the function
[tex]\lim_{h \to 0} \frac{\sqrt{9+h} -3}{h}\\= \frac{\sqrt{9+0}-3 }{0}\\= \frac{0}{0}(indeterminate)[/tex]
Applying l'hopital rule
[tex]\lim_{h \to 0} \frac{\frac{d}{dh} (\sqrt{9+h} - 3)} {\frac{d}{dh} (h)}\\= \lim_{h \to 0} \frac{1}{2} (9+h)^{-1/2} /1\\=\frac{1}{2} (9+0)^{-1/2}\\= \frac{1}{2} * \frac{1}{\sqrt{9} } \\= 1/2 * 1/3\\= 1/6[/tex]
Write each of these statements in the form "if p, then q" in English.
[Hint: Refer to the list of common ways to express conditional statements provided in this
section.]
a. [2pt] I will remember to send you the address only if you send me an e-mail message.
If you send me an e-mail message, then I will remember to send you the address.
b. [2pt] To be a citizen of this country, it is sufficient that you were born in the United
States.
If you were born in the United States, then you can be a citizen of the United States.
c. [2pt] If you keep your textbook, it will be a useful reference in your future courses.
If you keep your textbook, then it will be a useful reference in your future courses.
d. [2pt] The Red Wings will win the Stanley Cup if their goalie plays well.
If the Red Wings’s goalie plays well, they will win the Stanley Cup.
e. [2pt] That you get the job implies that you had the best credentials.
If you get the job, then you had the best credentials.
f. [2pt] The beach erodes whenever there is a storm.
If there is a storm, then beach erodes.
g. [2pt] It is necessary to have a valid password to log on to the server.
If you can log on to the server, then you must have a valid password.
h. [2pt] You will reach the summit unless you begin your climb too late.
If you begin your climb too late, then you will not reach the summit
Answer:
A. If you send me an e-mail message, then I will remember to send you the address.
B. If you were born in the United States, then you can be a citizen of the United States.
C. If you keep your textbook, then it will be a useful reference in your future courses.
D. If the Red Wings’s goalie plays well, then they will win the Stanley Cup.
E. If you get the job, then you had the best credentials.
F. If there is a storm, then beach erodes.
G. If you can log on to the server, then you must have a valid password.
H. If you begin your climb too late, then you will not reach the summit
Step-by-step explanation:
The statement which is written in the form, "if p, then q" symbolized by p-->q is known as a conditional statement. p is known as the hypothesis, while q is the conclusion. In such statements, the hypothesis and the conclusion may or may not have a correlation. But for the statement to be true, the conclusion must be true even if the hypothesis is wrong.
In statements A, the condition for sending the address is if I receive an email message.
In statement B, the condition for being a citizen is if one is born in the United States.
In statement C, the condition for my books to be a useful reference is if I keep them.
In statement D, the condition for winning the Stanley Cup by the Red Wings is if their Goalie plays well.
In statement E, the condition I get the job is if I had the best credentials.
In statement F, the condition for the beach eroding is if there's a storm.
In statement G, the only way I can log on to the server is if I have the correct password. In each of these statements, the conclusions correlates with the hypothesis. The conditions in the hypothesis need to be met for the conclusion to be true.
Statement H gives a conclusion which depends on not meeting the conditions of the hypothesis which is that if I don't start climbing the mountain early (if I start late), I will not reach the summit.
if you’re good with set theory and word problems in math 30 please help with questions 42 and 43 !! real answers only !!
Answer: 42) 2, 3 43) 144
Step-by-step explanation:
42)
1. n(P) = 8 + 5 = 13 (not 8) This statement is False.
2. n(Q but not P) = 9 This statement is True!
3. n(neither P nor Q) = 2 This statement is True!
4. n(Q') --> (n (not Q) = 8 + 2 = 10 (not 8) This statement is False.
5. P ∪ Q = 8 + 9 - 5 = 12 (not 5) This statement is False.
43)
Fill in the Venn Diagram as follows (from left to right):
M only = 55 --> 89 - (17 + 4 + 13) = 55
M ∩ E = 17 --> 30 - 13 = 17
E only = 5 --> 46 - (17 + 13 + 11) = 5
B ∩ M = 4 --> 17 - 13 = 4
M ∩ E ∩ B = 13 --> given
B ∩ E = 11 --> 24 - 13 = 11
E only = 35 --> 63 - (13 + 4 + 11) = 35
(M ∪ E ∪ B)' = 4 --> given
Total = 144
Ellen is opening. Cookie shop eat a local middle school . She randomly surgery’s students to determine The types of cookies they would buy at the cookie shop the results of the surveys are below based on the survey which statement is not true
Complete Question:
Ellen is opening. Cookie shop eat a local middle school . She randomly surveys students to determine The types of cookies they would buy at the cookie shop the results of the surveys are below based on the survey. Which statement is not true?
A) If 160 students buy a cookie, approximately 18 students would buy a sugar cookie.
B) 22½% of the students surveyed would buy a sugar cookie.
C) If 240 students were to purchase a cookie from the store, approximately 66 students will purchase a peanut butter cookie.
D) Half of the students prefer chocolate chips or oatmeal raisin cookies.
(Table showing the results of her survey is in the attachment below)
Answer:
A) If 160 students buy a cookie, approximately 18 students would buy a sugar cookie.
Step-by-step Explanation:
STEP 1:
In order to ascertain which of the statements given in the options that is NOT TRUE, let's express the given number of students that would buy the different types of cookies in their PROPORTIONS AND PERCENTAGES in the survey result.
Thus:
Type of Cookie==> no of students => Proportion (no of students of a cookie type ÷ total no of students) => % (each proportion for a cookie type)
Chocolate Chip==> 25 => 0.3125 (25÷80) => 31.25% (0.3125 × 100)
Oatmeal Raisin==> 15 => 0.1875 (15÷850) => 18.75% (0.1875 × 100)
Peanut Butter== 22 => 0.275 (22÷80) => 27.5% (0.275 × 100)
Sugar ==> 18 => 0.225 (18÷80) => 22.5% (0.225 × 100)
STEP 2:
Next step is to consider each statement given in the question to see if they are true or not.
==>OPTION A: If 160 students buy a cookie, approximately 18 students would buy a sugar cookie.
To find out if this is true, multiply the proportion of students who would be buy a sugar cookie (0.255) by 160 = 0.255 × 160 = 36.
If 160 buy a cookie, approximately 36 students would buy a sugar.
Option A IS NOT TRUE.
OPTION B: 22½% of the students surveyed would buy a sugar cookie.
From our calculation in STEP 2, we have:
Sugar ==> 18 => 0.225 (18÷80) => 22.5% (0.225 × 100).
Option B is TRUE. 22.5% of the students would go for sugar cookie.
OPTION C: If 240 students were to purchase a cookie from the store, approximately 66 students will purchase a peanut butter cookie.
Number of students that would buy the peanut butter if 240 students were to get a cookie = proportion of students that opt for peanut butter cookie in the survey (0.275) × 240 = 0.275 × 240 = 66.
Option C is TRUE.
OPTION D: D) Half of the students prefer chocolate chips or oatmeal raisin cookies.
Proportion of students who prefer chocolate chips or oatmeal raisin cookies = 0.3125 + 0.1785 = 0.491
This is approximately 0.5 = ½ of the total number of students.
Option D is TRUE.
Therefore, we can conclude that the statement, "A) If 160 students buy a cookie, approximately 18 students would buy a sugar cookie" is NOT TRUE.
Justin and Hayley conducted a mission to a new planet, Planet X, to study arm length. They took a random sample of 100 Planet X residents. Then they calculated a 95% confidence interval for the mean arm length.
Answer:
The correct answer is I am 95% sure that this interval includes the population mean arm length.
Note: Kindly find an attached image or copy of the complete question to this solution below:
Sources: The complete question was researched from Quizlet and Course hero sites.
Step-by-step explanation:
Solution
From the given question, i will say that I am 95 % confident interval that this interval include the population mean arm length is correct because using sample mean and confidence level we can obtain an interval which gives us the certain level of confidence that population mean is within this interval and here we are 95% confident that population mean is within this interval.
Find the slope of a line perpendicular to the graph of the equation. y = –8
Answer:
undefined
Step-by-step explanation:
y = -8 is a horizontal line with a slope of zero
A line that is perpendicular is a vertical line, which has a slope of undefined
Find the domain and the range of the relation. Determine whether the relation is a function.
{(-1,0),(7,0),(-1,9),(-6,-9)}
Answer:
In a relation h(x) = y, the values of x are the domain and the values of y are the range.
Here the domain is:
D = {-6, -1, 7}
The range is:
R = {-9, 0 , 9}
Now, a relation f(x) = y is a function only if for every x in the domain we have only and only one value in the range.
Here we can see that for the value -1 in the domain we have two different values in the range:
(-1, 0) and (-1, 9)
So this can not be a function.
(If you want to take the variable as y, you also have that the value y = 0 leads to two different values in x, so this cant be a function either)
Given that the area of a rectangle is 36 square cm and its length is 12 cm. Find the
width of the rectangle.
Answer:
is 3
Step-by-step explanation:
because to find the area for a ractangle you have to multiply LxW and 12x3=36
Please help! Correct answer only, please! Which of the following is one of the cheapest routes to pass through each vertex once starting and ending with Vertex "A" and using the Nearest Neighbor Algorithm. A. ABDCA, $890 B. ACDBA, $900 C. ABCDA, $960 D. None of the Above
Answer: c) ABCDA, $960
Step-by-step explanation:
The nearest Neighbor Algorithm states to choose the next vertex based only on the weights of the neighbor of that vertex.
Starting at A: Options are B = 220, C = 240, D = 310
Choose B because it has the smallest value.
From B: Options are C = 200, D = 210
Choose C because it has the smallest value.
From C: There is only one option --> D = 230 (we cannot choose A because it was our starting point and we haven't touched every vertex, yet).
From D: We touched all of the vertices so return to the starting point, A = 310
A → B → C → D → A --> 220 + 200 + 230 + 310 = 960
Notice that if we looked at the entire circuit first, this is NOT the optimum path. But this is the result using the Nearest Neighbor Algorithm.
find the next two terms in this sequence. 5,-5,10,-10,15,?,?
Answer:
I think the answer is -15
A camp counselor and six campers are to be seated along a picnic bench. In how many ways can this be done if the counselor must be seated in the third seat and a camper who has a tendency to engage in food fights must sit to the counselor's immediate right?
Answer:
This can be done in 120 ways.
Step-by-step explanation:
Number of arrangments:
The number of arrangments of n elements is given by the following formula:
[tex]A_{n} = n![/tex]
In this question:
7 elements.
Two conditions: Counselor in the third sear, and the camper who has the tendency to engage in food fights in the fourth.
For the other 5 seats, the other 5 campers can be arranged in:
[tex]A_{5} = 5! = 120[/tex]
Ways.
This can be done in 120 ways.
Assume that in a statistics class the probability of receiving a grade of A equals .30 and the probability of receiving a grade of B equals .30. The probability that a randomly selected student from this class will receive either an A or a B equals.
a. .09
b. .6
c. .9
d. .3
Answer:
Answer D is correct
Use the Law of Cosines to determine the length of the third side of the triangle.
4.1 cm
350
6.7 cm
a 1 = 4.9cm
b. 1 = 5.3cm
C.
1 = 5.5cm
d. 1 = 4.1cm
Corrected Question
Given a triangle with two sides 4.1 cm and 6.7cm and an Included angle of [tex]35^\circ[/tex]. Use the Law of Cosines to determine the length of the third side of the triangle.
Answer:
(D) 4.1cm
Step-by-step explanation:
Given a triangle with two sides 4.1 cm and 6.7 cm and an included angle of [tex]35^\circ[/tex].
Using Law of Cosines
[tex]a^2=b^2+c^2-2bc\cos A\\a^2=4.1^2+6.7^2-2*4.1*6.7\cos 35^\circ\\a^2=16.6958\\a=\sqrt{16.6958}\\a=4.09\\a \approx 4.1$ cm[/tex]
The correct option is D.
Dan buys a car for £2100.
It depreciates at a rate of 2.2% per year.
How much will it be worth in 6 years?
Give your answer to the nearest penny where appropriate.
Answer:
£472.92
Step-by-step explanation:
£2100(0.78)^6
Find the volume of a right circular cone that has a height of 8.1 in and a base with a diameter of 7.6 in. Round your answer to the nearest tenth of cubic inch
Answer:
[tex]V=122.5in^3[/tex]
Step-by-step explanation:
The volume of a right circular cone is given by:
[tex]V=\frac{\pi r^2h}{3}[/tex]
where r is the radius of the circle and h is the height of the cone, and [tex]\pi[/tex] is a constant [tex]\pi=3.1416[/tex].
According to the problem the height is:
[tex]h=8.1 in[/tex]
and we don't have the radius but we have the diameter, which is useful to find it. We just divide the diameter by 2 to find the radius:
[tex]r=\frac{d}{2}=\frac{7.6in}{3}=3.8in[/tex]
Now, we can find the volume by substituting all the known values:
[tex]V=\frac{\pi r^2h}{3}[/tex]
[tex]V=\frac{(3.1416)(3.8in)^2(8.1in)}{3} \\\\V=\frac{(3.1416)(14.44in^2)(8.1in)}{3} \\\\V=\frac{367.454in^3}{3} \\\\V=122.485[/tex]
Rounding the volume to the nearest tenth of cubic inch we get:
[tex]V=122.5in^3[/tex]
John Calipari, head basketball coach for the national champion University of Kentucky Wildcats, is the highest paid coach in college basketball with an annual salary of million (USA Today, March 29, 2012). The following sample shows the head basketball coach's salary for a sample of schools playing NCAA Division basketball. Salary data are in millions of dollars.
University Coach's Salary University Coach's Salary
Indian 2.2 Syracuse 1.5
Xavier .5 Murry State .2
Texas 2.4 Florida State 1.1
Connecticut 2.7 South Dekota State .1
West Virginia 2.0 Vermont .2
A. Use the sample mean for the 10 schools to estimate the population mean annual salary for head basketball coaches at colleges and universities playing NCAA Division 1 basketball (to 2 decimal).
B. Use the data to estimate the population standard deviation for the annual salary for head basketball coaches (to 4 decimals).
C. what is the 95% confidence interval for the population variance (to 2 decimals)?
D. what is the 95% confidence interval for the population standard deviation (to 2 decimals)?
Answer:
A) sample mean = $1.36 million
B) standard deviation = $0.9189 million
C) variance confidence interval = ($0.40 million, $2.81 million)
D) standard deviation confidence interval = ($1.93 million , $0.79 million)
*since the sample size is very small, the confidence interval is not valid.
Step-by-step explanation:
samples:
$2.7 million$2.4 million$2.2 million$2 million$1.5 million$1.5 million$0.5 million$0.5 million$0.2 million$0.1 millionsample mean = $1.36 million
the standard deviation:
$2.7 million - $1.36 million = 1.34² = 1.7956$2.4 million - $1.36 million = 1.04² = 1.0816$2.2 million - $1.36 million = 0.84² = 0.7056$2 million - $1.36 million = 0.64² = 0.4096$1.5 million - $1.36 million = 0.14² = 0.0196$1.5 million - $1.36 million = 0.14² = 0.0196$0.5 million - $1.36 million = -0.86² = 0.7396$0.5 million - $1.36 million = -0.86² = 0.7396$0.2 million - $1.36 million = -1.16² = 1.3456$0.1 million - $1.36 million = -1.26² = 1.5876total $8.444 million / 10 = $0.8444 millionvariance 0.8444
standard deviation = √0.8444 = 0.9189
in order to calculate the confidence interval for the population variance we are going to use a chi-square distribution with 2.5% on each tail ⇒ table values 2.7004 and 19.023 enclose 95% of the distribution.
[(n - 1) x variance] / 2.7004 = (9 x 0.8444) / 2.7004 = 2.81
[(n - 1) x variance] / 19.023 = (9 x 0.8444) / 19.023 = 0.40
95% confidence interval = mean +/- 1.96 standard deviations/√n:
$1.36 million + [(1.96 x $0.9189 million)/√10] = $1.36 million + $0.57 million = $1.93 million
$1.36 million - $0.57 million = $0.79 million
Find the focus and directrix of the parabola y = {(x + 2)2 – 3.
Answer:
focus: (-2, -2.75)directrix: y = -3.25Step-by-step explanation:
For focus-to-vertex distance "p", the equation of a parabola with vertex (h, k) can be written as ...
y = 1/(4p)(x -h)^2 +k
Comparing this to your equation, we see that ...
1/(4p) = 1
h = -2
k = -3
Solving for p, we find ...
1/(4p) = 1
1/4 = p . . . . . multiply by p
The parabola opens upward, so this means the focus is 1/4 unit above the vertex, and the directrix is 1/4 unit below the vertex.
focus: (-2, -2.75)directrix: y = -3.25Answer:
The focus is at (–2,–212) and the directrix is at y = –312.
Step-by-step explanation:
Find the focus and directrix of the parabola y=12(x+2)2−3.
got the answer right in the test.
In a village
The number of houses and the number of flats are in the ratio 9:5
The number of flats and the number of bungalows are in the ratio 10:3
There are 30 bungalows in the village.
How many houses are there in the village?
Note: please make sure your final answer clear by writing ... houses
The number of houses are 180, and the number of flats are 100.
It is given that the number of houses and the number of flats ratio is 9:5 the number of flats and the number of bungalows ratio is 10:3.
It is required to find the number of houses in the village if the number of bungalows is 30.
What is a fraction?Fraction number consists of two parts one is the top of the fraction number which is called the numerator and the second is the bottom of the fraction number which is called the denominator.
The ratio of the number of houses and the number of flats:
= 9:5 and
The ratio of the number of flats and the number of bungalows :
=10:3
It means we can write the ratio of the number of houses and the number of flats = 18:10
And the ratio of the number of:
Houses : Flats : Bungalows = 18:10:3
But the number of bungalows are 30.
Then the ratios are:
180:100:30
Thus, the number of houses are 180, and the number of flats are 100.
Learn more about the fraction here:
brainly.com/question/1301963
Adrian is training for a 5K race. She
ran 5.5 miles the first week, 7.25
miles the second week and 10 miles
the third week. On the average, how
many miles did she run per week?
Round to the nearest hundredth.
PLZ HELP PLZ!!!!!
To get this we divide the total miles she ran for the weeks by the no of weeks she ran.
Total miles =5.5 +7.25+10= 22.25 miles altogether.
no of weeks =3.
Average miles = 22.75 /3=7.583333≅7.58(nearest hundredth)
Function c(x) = 5x
If your input was 2, what is your output?
Find the point-slope equation for the line
that passes through the points (15, 10)
and (16, 15). Use the first point in your
equation.
y - [?] = [](x - [ ])
Answer:
y-10=5(x-15)
Step-by-step explanation:
Slope =5
15-10/16-15=5
Then plug in 10 as the y1 and 15 as y2.
A board of directors consists of 10 people, in how many ways can a chief executive officer, director, a treasurer, and a secretary be selected?
Answer:
The correct answer to the following question will be "5040".
Step-by-step explanation:
Given:
The number of directors,
n = 10
and they select on 4 peoples, then
Number of ways will be:
⇒ [tex]10_{P}_{4}[/tex]
⇒ [tex]\frac{10!}{10-4!}[/tex]
⇒ [tex]\frac{10!}{6!}[/tex]
⇒ [tex]\frac{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1!}{6\times 5\times 4\times 3\times 2\times 1!}[/tex]
⇒ [tex]5040[/tex]
Evaluating and solving functions
Use the function f(x)=240(0.7)^x to answer the following questions. Round your nearest answers to two decimal places.
Evaluate f(7) f(7)=
Determine x when f(×)=120
Answer:
Evaluate f(7)
[tex] f(7) = 240(0.7)^7 = 19.765[/tex]
Determine x when f(×)=120
[tex] 120 = 240 (0.7)^x[/tex]
We can derive both sides by 240 and we got:
[tex] 0.5 = 0.7^x[/tex]
Now we can apply natural log on both sides and we got:
[tex] ln(0.5)= x ln(0.7)[/tex]
And if we solve for the value of x we got:
[tex] x =\frac{ln(0.5)}{ln(0.7)}= 1.943[/tex]
And then the value of x = 1.943
Step-by-step explanation:
We have the following function given:
[tex]f(x) = 240(0.7)^x [/tex]
Evaluate f(7)
And we want to find [tex] f(7) [/tex] so we just need to replace x=7 and we got:
[tex] f(7) = 240(0.7)^7 = 19.765[/tex]
Determine x when f(×)=120
And for the second part we want to find a value of x who satisfy that the function would be equal to 120 and we can set up this:
[tex] 120 = 240 (0.7)^x[/tex]
We can derive both sides by 240 and we got:
[tex] 0.5 = 0.7^x[/tex]
Now we can apply natural log on both sides and we got:
[tex] ln(0.5)= x ln(0.7)[/tex]
And if we solve for the value of x we got:
[tex] x =\frac{ln(0.5)}{ln(0.7)}= 1.943[/tex]
And then the value of x = 1.943
Which set of ordered pairs represents a function?
A {(2, 7), (2, 8), (3, 8)}
B {(3, 2), (3, 3), (3, 4)}
C {(4, 1), (5, 1), (4,4)}
D {(5, 6), (8, 6), (9, 6)}
does anyone know the answer
Answer:
Its D
Step-by-step explanation:
becasue the X is not repeating. the other are.
(Bonus) A rectangular box has its edges changing length as time passes. At a par-ticular instant, the sides have lengthsa= 150 feet,b= 80 feet, andc= 50 feet.At that instant,ais increasing at 100 feet/sec,bis decreasing 20 feet/sec, andcisincreasing at 5 feet/sec. Determine if the volume of the box is increasing, decreasing,or not changing at all, at that instant.
Answer:
the volume of the box is increasing
dV = +310,000 ft^3/s
Step-by-step explanation:
Volume of a rectangular box with side a,b and c can be expressed as;
V = abc
The change in volume dV can be expressed as;
dV = d(abc)/da + d(abc)/db + d(abc)/dc
dV = bc.da + ac.db + ab.dc ......1
Given:
a= 150 feet,
b= 80 feet, and
c= 50 feet
ais increasing at 100 feet/sec,bis decreasing 20 feet/sec, andcisincreasing at 5 feet/sec
da = +100 feet/s
db = -20 feet/s
dc = +5 feet/s
Substituting the values into the equation 1;
dV = (80×50×+100) + (150×50×-20) + (150×80×+5)
dV = +400000 - 150000 + 60000 ft^3/s
dV = +310,000 ft^3/s
Since dV is positive, the volume of the box is increasing at that instant.
Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function. y′+y=7+δ(t−3),y(0)=0. y′+y=7+δ(t−3),y(0)=0. Find the Laplace transform of the solution. Y(s)=L{y(t)}=Y(s)=L{y(t)}= Obtain the solution y(t)y(t).
Answer:
a. [tex]\mathbf{Y(s) = L \{y(t)\} = \dfrac{7}{s(s+1)}+ \dfrac{e^{-3s}}{s+1}}[/tex]
b. [tex]\mathbf{y(t) = \{7e^t + e^3 u (t-3)-7\}e^{-t}}[/tex]
Step-by-step explanation:
The initial value problem is given as:
[tex]y' +y = 7+\delta (t-3) \\ \\ y(0)=0[/tex]
Applying laplace transformation on the expression [tex]y' +y = 7+\delta (t-3)[/tex]
to get [tex]L[{y+y'} ]= L[{7 + \delta (t-3)}][/tex]
[tex]l\{y' \} + L \{y\} = L \{7\} + L \{ \delta (t-3\} \\ \\ sY(s) -y(0) +Y(s) = \dfrac{7}{s}+ e ^{-3s} \\ \\ (s+1) Y(s) -0 = \dfrac{7}{s}+ e^{-3s} \\ \\ \mathbf{Y(s) = L \{y(t)\} = \dfrac{7}{s(s+1)}+ \dfrac{e^{-3s}}{s+1}}[/tex]
Taking inverse of Laplace transformation
[tex]y(t) = 7 L^{-1} [ \dfrac{1}{(s+1)}] + L^{-1} [\dfrac{e^{-3s}}{s+1}] \\ \\ y(t) = 7L^{-1} [\dfrac{(s+1)-s}{s(s+1)}] +L^{-1} [\dfrac{e^{-3s}}{s+1}] \\ \\ y(t) = 7L^{-1} [\dfrac{1}{s}-\dfrac{1}{s+1}] + L^{-1}[\dfrac{e^{-3s}}{s+1}] \\ \\ y(t) = 7 [1-e^{-t} ] + L^{-1} [\dfrac{e^{-3s}}{s+1}][/tex]
[tex]L^{-1}[\dfrac{e^{-3s}}{s+1}][/tex]
[tex]L^{-1}[\dfrac{1}{s+1}] = e^{-t} = f(t) \ then \ by \ second \ shifting \ theorem;[/tex]
[tex]L^{-1}[\dfrac{e^{-3s}}{s+1}] = \left \{ {{f(t-3) \ \ \ t>3} \atop {0 \ \ \ \ \ \ \ \ \ t <3}} \ \ \ \right.[/tex]
[tex]L^{-1}[\dfrac{e^{-3s}}{s+1}] = \left \{ {{e^{(-t-3)} \ \ \ t>3} \atop {0 \ \ \ \ \ \ \ \ \ t <3}} \ \ \ \right.[/tex]
[tex]= e^{-t-3} \left \{ {{1 \ \ \ \ \ t>3} \atop {0 \ \ \ \ \ t<3}} \right.[/tex]
= [tex]e^{-(t-3)} u (t-3)[/tex]
Recall that:
[tex]y(t) = 7 [1-e^{-t} ] + L^{-1} [\dfrac{e^{-3s}}{s+1}][/tex]
Then
[tex]y(t) = 7 -7e^{-t} +e^{-(t-3)} u (t-3)[/tex]
[tex]y(t) = 7 -7e^{-t} +e^{-t} e^{-3} u (t-3)[/tex]
[tex]\mathbf{y(t) = \{7e^t + e^3 u (t-3)-7\}e^{-t}}[/tex]
PLEASE HELP!!! Find the equation of the line passing through the point (6,3) that is perpendicular to the line 4x−5y=−10. Enter your answers below. Use a forward slash (i.e. "/") for fractions (e.g. 1/2 for 12). Solution Step 1: Find the slope of the line 4x−5y=−10. Use a forward slash (i.e. "/") for all fractions (e.g. 1/2 for 12). m= _____ What would the perpendicular slope be? m= _____ Step 2: Use the slope to find the y-intercept of the perpendicular line. b= ____ Step 3: Write the equation of the line that passes through the point (6,3) that is perpendicular to the line 4x−5y=−10 y= ____ x+ Answer
Linear equations are typically organized in slope-intercept form:
[tex]y=mx+b[/tex]
m = slopeb = y-interceptPerpendicular lines have slopes that are negative reciprocals.
Example: 2 and -1/2Example: 3/4 and -4/3SolutionWe're given:
Perpendicular to [tex]4x-5y=-10[/tex]Passes through (6,3)1) Determine the slope
Let's first rearrange this equation into slope-intercept form:
[tex]4x-5y=-10\\-5y=-4x-10\\\\y=\dfrac{4}{5}x+2[/tex]
Notice how [tex]\dfrac{4}{5}[/tex] is in the place of m in y = mx + b. This is the slope of the give line.
Since perpendicular lines are negative reciprocals, we know the slope of the other line is [tex]-\dfrac{5}{4}[/tex]. Plug this into y = mx + b:
[tex]y=-\dfrac{5}{4}x+b[/tex]
2) Determine the y-intercept
We're also given that the line passes through (6,3). Plug this point into our equation and solve for b:
[tex]y=-\dfrac{5}{4}x+b\\\\3=-\dfrac{5}{4}(6)+b\\\\b=3+\dfrac{5}{4}(6)\\\\b=\dfrac{21}{2}[/tex]
Plug this back into our original equation:
[tex]y=-\dfrac{5}{4}x+\dfrac{21}{2}[/tex]
Answer[tex]y=-\dfrac{5}{4}x+\dfrac{21}{2}[/tex]
A trip took you 5 hours and you traveled 283.5 miles. If you averaged 49 mph
for the first part of the trip and 60 mph for the 2nd part, how long did you drive
at each rate?
Answer:
1.5 hr and 3.5 hr
Step-by-step explanation:
Time in the first part= x
Time in the second part= 5 - x
49x+60(5-x)=283.549x+300-60x=283.511x=300-283.511x= 16.5x= 16.5/11x= 1.5 hrand5-x= 5-1.5= 3.5 hrEnter the y-coordinate of the solution. Round to the nearest tenth. 5x+2y=7 -2x+6y=9
Answer:
59/34
Step-by-step explanation:
5x+2y=7
-2x+6y=9
Multiply the top equation by 3:
15x+6y=21
Subtract the second equation from the first:
17x=12
x=12/17
Plug this back into one of the other equations to find y:
5(12/17)+2y=7
60/17+2y=7
2y=59/17
y=59/34
Hope this helps!