Answer:
No down payment = $73 267; 10 % down payment = $81 408
Step-by-step explanation:
1. With no down payment
The formula for a maximum affordable loan (A) is
A = (P/i)[1 − (1 + i)^-N]
where
P = the amount of each equal payment
i = the interest rate per period
N = the total number of payments
Data:
P = 500
APR = 2.83 % = 0.0283
t = 15 yr
Calculations:
You are making monthly payments, so
i = 0.0283/12 = 0.002 358 333
The term of the loan is 15 yr, so
N = 15 × 12 = 180
A = (500/0.002 3583)[1 − (1 + 0.002 3583)^-180]
= 212 014(1 - 1.002 3583^-180)
= 212 014(1 - 0.654 424)
= 212 014 × 0.345 576
= 73 267
You can afford to spend $73 267 on a home.
2. With a 10 % down payment
Without down payment, loan = 73 267
With 10 % down payment, you pay 0.90 × new loan
0.90 × new loan = 73 267
New loan = 73267/0.90 = 81 408
With a 10 % down payment, you can afford to borrow $81 408 .
Here’s how it works:
Purchase price = $81 408
Less 10 % down = -8 141
Loan = $73 267
And that's just what you can afford.
A barber’s chair with a person in it weighs 2118.0 N. The output plunger of a hydraulic system begins to lift the chair when the barber’s foot applies a force of 63.0 N to the input piston. Neglect any height difference between the plunger and the piston. What is the ratio of the radius of the plunger to the radius of the piston?
Answer:5.8
Step-by-step explanation:
Given
weight of chair with person on it =2118 N
Barber applies a force of 63 N
Using pascal's law
we know pressure around both sides is equal
i.e. [tex]P_{chair}=P_{piston}[/tex]
pressure=[tex]\frac{force}{area}[/tex]
[tex]P_{chair}[/tex]=[tex]\frac{2118}{\pi \dot r_{plunger}^2}[/tex]
[tex]P_{piston}[/tex]=[tex]\frac{63}{\pi \dot r_{piston}^2}[/tex]
substituting values
[tex]\left [\frac{r_{plunger}^2}{r_{piston}^2}\right ][/tex]=[tex]\frac{2118}{63}[/tex]
[tex]\left [\frac{r_{plunger}}{r_{piston}}\right ][/tex]=[tex]\left [ \frac{2118}{63}\right ]^{0.5}[/tex]
[tex]\left [\frac{r_{plunger}}{r_{piston}}\right ][/tex]=5.798
Find an equation of the line that contains the following pair of points. (3,2) and (1,3) The equation of the line is (Simnlif vour ancwor Llco.intogore or fractione.for anu.numborc.in.the aquatio
Answer: [tex]x+2y-7=0[/tex]
Step-by-step explanation:
We know that the equation of a line passing through points (a,b) and (c,d) is given by :-
[tex](y-b)=\dfrac{d-b}{c-a}(x-a)[/tex]
Then , the equation of a line passing through points (3,2) and (1,3) is given by :-
[tex](y-2)=\dfrac{3-2}{1-3}(x-3)\\\\\Rightarrow\ (y-2)=\dfrac{1}{-2}(x-3)\\\\\Rightarrow\ -2(y-2)=(x-3)\\\\\Rightarrow\ -2y+4=x-3\\\\\Rightarrow\ x+2y-7=0[/tex]
Hence, the equation of a line passing through points (3,2) and (1,3) is : [tex]x+2y-7=0[/tex]
Suppose that in a senior college class of 500 students itis found that 210 smoke, 258 drink alcoholic beverages, 216 eatbetween meals, 122 smoke and drink alcoholic beverages, 83 eatbetween meals and drink alcoholic beverages, 97 smoke and eatbetween meals, and 52 engage in all three of these bad healthpractices. If a member of this senior class is selected at random,find the probability that the studenta.) smokes but does not drink alcoholic beverages;b.) eats between meals and drinks alcoholic beverages but doesnot smoke;c.) neither smokes nor eats between meals.Please show all steps to solve, not in books listed
Answer:
a)Smoke but doesn't drink alcoholic beverages
P=0.176 or 17.6%
b)eats between meals and drinks alcoholic beverages but doesn't smoke
P=0.062 or 6.2%
c.) neither smokes nor eats between meals
P=0.342 or 34.2%
Step-by-step explanation:
Using a Venn diagram with three circles, one for each bad health habit. Let us [tex]a[/tex] for the students that only smoke, [tex]b[/tex] for only drink alcoholic beverages, and [tex]c[/tex] for only eat between meals. Following this logic, [tex]d[/tex] represents smoke and drink alcoholic beverages but not eat between meals, [tex]e[/tex] drink alcoholic beverages and eat between meals but not smoke,[tex]f[/tex] eat between meals and smoke but not drink alcoholic beverages. Finally [tex]g[/tex] for having all the three, this means [tex]g=52[/tex].
To find [tex]f[/tex], subtract 52 (the three problems) from 97 because this last number represents all the students that smoke and eat between meals, including the students that have the three bad habits. The same goes for 'd' and 'e'.
[tex]f=97-52=45[/tex]
[tex]e=83-52=31[/tex]
[tex]d=122-52=70[/tex]
To find [tex]a[/tex], subtract [tex]e[/tex], [tex]f[/tex], and [tex]g[/tex] from the total of smokers. This is because [tex]e[/tex], [tex]f[/tex], and [tex]g[/tex] represent smoke and at least another bad habit and [tex]a[/tex] represents only smoking.
[tex]a=210-d-f-g=210-70-45-52=43[/tex]
The same goes for [tex]b[/tex] and [tex]c[/tex].
[tex]b=258-d-e-g=258-70-31-52=105[/tex]
[tex]c=216-e-f.-g=216-31-45-52=88[/tex]
Adding all letters and subtract from the total to see if there is any healthy student:
[tex]500-a+b+c+d+e+f+g =500-43+105+88+70+31+45+52=500- 434=66[/tex]
a)Smoke but doesn't drink alcoholic beverages
This will be 'a' (only smokes) and 'f' ( smokes and eats between meals but doesn't drink) divided by the total of students.
[tex]P=(43+45)/500=0.176[/tex]
b)eats between meals and drinks alcoholic beverages but doesn't smoke
This probability is 'e' divided by the total of students.
[tex]P=31/500=0.062[/tex]
c.) neither smokes nor eats between meals
This will be 'b' (only drinks) plus the healthy students (66) divided by the total of students.
[tex]P=(105+66)/500=0.342[/tex]
The probability a randomly selected student smokes but does not drink is 0.176, eats between meals and drinks alcohol but does not smoke is 0.062, and neither smokes nor eats between meals is 0.286.
Explanation:To solve this, we need to break down the information into three categories: those who smoke (S), those who drink alcoholic beverages (D), and those who eat between meals (E). We’re told that there are 210 smokers, 258 drinkers, and 216 who eat between meals. We also know that some students fall into multiple categories.
a) Find those who smoke but do not drink. We know that 122 students both smoke and drink, so the number who smoke but do not drink is 210 (total smokers) - 122 (smokers and drinkers) = 88. The probability is then 88 out of 500, or 0.176.
b) To find those who eat between meals and drink alcoholic beverages but do not smoke, we subtract those who do all three (52) from those who eat between meals and drink (83) to get 31. The probability is then 31 out of 500, or 0.062.
c) To find those who neither smoke nor eat between meals, we subtract those who do either from the total. We add together the numbers who smoke, drink, or eat between meals (210+258+216), then subtract off twice the number who do two activities (122+83+97) since they were counted twice in the first total, then add back in the number who do all three (52) since they were subtracted too many times. Subtracting this from 500 gives us the number who do neither. So, 500 - ((210+258+216)-(2*(122+83+97))+52) = 143. The probability is then 0.286.
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8. Show all computations for the following. Do not use a calculator.
(a) How many bit strings are there of length six which are palindromes? Explain
(b) How many bit strings are there of length 6 which contain exactly three 1’s? Explain
Answer:
a= 8 ways
b. 20
Step-by-step explanation:
Palindrome of length 6 means first three digits must be same as the last three in reverse. For example 123321 is palindrome of six digits.
There 2 bits 0 and 1
a.So each of first three digits can be filled in 2 ways
therefore, 2*2*2= 8 ways
number of different palindromes of 6 digits will be 8
b. In a 6 digit a palindrome there Are 6 spaces in which 3 spaces are to be filled with 1's
this cab be done in
[tex]_{6}^{3}\textrm{C}= \frac{6!}{3!\times3!}[/tex]
= 20
Exercise 1.4.4: Solve y' + xy = x.
Answer:
Step-by-step explanation:
Let's rewrite this with dy/dx in place of y', since they mean the same thing. But to solve a differential we will need to take the antiderivative by separation to find the general solution.
[tex]\frac{dy}{dx}+xy=x[/tex]
The goal is to get the x stuff on one side and the y stuff on the other side by separation. But we have an xy term there that we need to be able to break apart. So let's get everything on one side separate from the dy/dx and take it from there.
[tex]\frac{dy}{dx}=x-xy[/tex]
Now we can factor out the x:
[tex]\frac{dy}{dx}=x(1-y)[/tex]
And now we can separate:
[tex]\frac{dy}{(1-y)}=x dx[/tex]
Now we solve by taking the antiderivative of both sides:
[tex]\int\ {\frac{1}{1-y}dy }=\int\ {x} \, dx[/tex]
On the left side, the antiderivative of the derivative of y cancels out, and the other part takes on the form of the natural log, while we follow the power rule backwards on the right to integrate x:
[tex]ln(1-y)=\frac{1}{2}x^2+C[/tex]
That's the general solution. Not sure what your book has you solving for. Some books solve for the constant, C. Some solve for y when applicable. I'm leaving it like it is.
If you lived in a hobbit house and had a round window that needed? trim, how much trim would be needed to go around a window with a radius of 4.2? feet?
Answer:
To find out how much trim you would need for the window, you would need to calculate the Circumference.
Step-by-step explanation:
Circumference is the length around a circle and is calculated using the following formula.
[tex]C = 2\pi r[/tex]
In which C is the circumference and r is the radius. Since we already know the radius all we have to do is plug it into the equation and solve for C.
[tex]C = 2\pi (4.2)[/tex]
[tex]C = 26.389 ft[/tex]
Answer: The window has a Circumference of 26.389 feet so you would need that amount of trim to fit the area around the window.
Hope my answer would be a great help for you.
If you have more questions feel free to ask here at Brainly.
Approximately 27 feet of trim would be needed to go around a window with a radius of 4.2 feet.
Explanation:To calculate the amount of trim needed to go around a round window with a radius of 4.2 feet, we can use the formula for the circumference of a circle. The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle. In this case, the radius is 4.2 feet. Plugging in the value of the radius into the formula, we get C = 2π(4.2) = 8.4π feet. Since we want the amount of trim needed, we can simply round the answer to the nearest whole number. Therefore, approximately 27 feet of trim would be needed to go around the window.
The average height of students at UH from an SRS of 12 students gave a standard deviation of 3.1 feet. Construct a 95% confidence interval for the standard deviation of the height of students at UH. Assume normality for the data.
Answer: [tex](2.20,\ 5.26)[/tex]
Step-by-step explanation:
The confidence interval for the standard deviation is given by :-
[tex]\left ( \sqrt{\dfrac{(n-1)s^2}{\chi^2_{n-1,\alpha/2}}},\ \sqrt{\dfrac{(n-1)s^2}{\chi^2_{n-1,1-\alpha/2}}} \ \right )[/tex]
Given : n= 12 ; s= 3.1
Significance level : [tex]\alpha=1-0.95=0.05[/tex]
Using Chi-square distribution table ,
[tex]\chi^2_{11,0.025}}=21.92\\\\\chi^2_{11,0.975}}=3.82[/tex]
Now, the 95% confidence interval for the standard deviation of the height of students at UH is given by :-
[tex]\left ( \sqrt{\dfrac{(11)(3.1)^2}{21.92}},\ \sqrt{\dfrac{(11)(3.1)^2}{3.82}} \ \right )\\\\=\left ( 2.19602743525, 5.26049188471\right )\approx(2.20,\ 5.26)[/tex]
To construct a 95% confidence interval for the standard deviation of the height of students at UH, use the chi-square distribution.
Explanation:To construct a 95% confidence interval for the standard deviation of the height of students at UH, we can use the chi-square distribution. Since the population standard deviation is unknown, we need to use the sample standard deviation as an estimate. The formula to calculate the confidence interval is:
Lower limit = ((n-1) * s^2) / X^2
Upper limit = ((n-1) * s^2) / X^2
Where n is the sample size, s is the sample standard deviation, and X^2 is the chi-square critical value corresponding to the desired confidence level and degrees of freedom (n-1).
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Determine the sample size needed to construct a 90% confidence interval to estimate the average GPA for the student population at a college with a margin of error equal to 0.5. Assume the standard deviation of the GPA for the student population is 1.5.
Answer: 24
Step-by-step explanation:
The formula to calculate the sample size is given by :-
[tex]n=(\frac{z_{\alpha/2}\ \sigma}{E})^2[/tex]
Given : Margin of error : [tex]E=0.5[/tex]
The significance level : [tex]\alpha=1-0.90=0.1[/tex]
Critical value : [tex]z_{0.05}=\pm1.645[/tex]
Standard deviation : [tex]\sigma=1.5[/tex]
Now, the required sample size will be :-
[tex]n=(\frac{1.645\times1.5}{0.5})^2=24.354225\approx24[/tex]
Hence, the required sample size = 24
Suppose that you invest S1,100 in stock. Four years later, your investment yields $1,775. What is the rate of return of your investment? The rate of return is %. (Round to one decimal place.)
Answer:
The rate of return is 61.3%.
Step-by-step explanation:
Rate of return is given by:
[tex]\frac{current price - original price}{original price}\times100[/tex]
= [tex]\frac{1775-1100}{1100} \times100[/tex]
= [tex]\frac{675}{1100} \times100[/tex]
= 61.36% ≈ 61.3%
Hence, the rate of return is 61.3%.
Divide. Reduce the answer to lowest terms.
3/8 divided by 1/7
[tex]2 \frac{5}{8}[/tex], or [tex]\frac{21}{8}[/tex]
Explanation:Use the Keep Change Flip method. Change the sign to multiplication and flip the second fraction. [tex]\frac{3}{8}*\frac{7}{1}[/tex]
Multiply the numerators and the denominators separatley. [tex]\frac{3*7}{8*1}=\frac{21}{8}[/tex]
This cannot be simplified further, but it can be converted to a mixed number (as opposed to an improper fraction). [tex]\frac{21}{8}=2 \frac{5}{8}[/tex]
Answer:
D. 2 5/8
Step-by-step explanation:
3/8÷1/7=?
Dividing two fractions is the same as multiplying the first fraction by the reciprocal (inverse) of the second fraction.
Take the reciprocal of the second fraction by flipping the numerator and denominator and changing the operation to multiplication. Then the equation becomes
3/8×7/1=?
For fraction multiplication, multiply the numerators and then multiply the denominators to get
3×7/8×1= 21/8
This fraction cannot be reduced.
The fraction
2/18
is the same as
21÷8
Convert to a mixed number using
long division for 21 ÷ 8 = 2R5, so
2/18= 2 5/8
Therefore:
3/8÷1/7= 2 5/8
In 1995, 70% of all children in the U.S. were living with both parents. If 25 children were selected at random in the U.S., what is the probability that at most 10 of them will be living with both of their parents? Round your answer to 4 decimal places.
Answer: 0.0018
Step-by-step explanation:
Binomial distribution formula :-
[tex]P(x)=^nC_xp^x(q)^{n-x}[/tex], here P(x) is the probability of getting success at x trial , n is the total number of trails, p is the probability of getting success in each trail.
Given : The probability that a child in the U.S. was living with both parents : p=0.70 ; q=1-0.70=0.30
If 25 children were selected at random in the U.S.,then the probability that at most 10 of them will be living with both of their parents will be :-
[tex]P(x\leq10)=P(0)+P(1)+P(2)+P(3)+P(4)+P(5)+P(6)+P(7)+P(8)+P(9)+P(10)\\\\=^{25}C_{0}(0.7)^0(0.3)^{25}+^{25}C_{1}(0.7)^1(0.3)^{24}+^{25}C_{2}(0.7)^2(0.3)^{23}+^{25}C_{3}(0.7)^3(0.3)^{22}+^{25}C_{4}(0.7)^4(0.3)^{21}+^{25}C_{5}(0.7)^5(0.3)^{20}+^{25}C_{6}(0.7)^6(0.3)^{19}+^{25}C_{7}(0.7)^7(0.3)^{`18}+^{25}C_{8}(0.7)^8(0.3)^{17}+^{25}C_{9}(0.7)^9(0.3)^{16}+^{25}C_{10}(0.7)^{10}(0.3)^{15}\\\\=(0.7)^0(0.3)^{25}+25(0.7)^1(0.3)^{24}+300(0.7)^2(0.3)^{23}+2300(0.7)^3(0.3)^{22}+ 12650(0.7)^4(0.3)^{21}+53130(0.7)^5(0.3)^{20}+177100(0.7)^6(0.3)^{19}+480700(0.7)^7(0.3)^{`18}+1081575(0.7)^8(0.3)^{17}+2042975(0.7)^9(0.3)^{16}+3268760(0.7)^{10}(0.3)^{15}\\\\=0.00177840487034\approx0.0018[/tex]
Hence, the probability that at most 10 of them will be living with both of their parents is 0.0018.
In 2005, there were 18,400 students at college A, with a projected enrollment increase of 500 students per year In the same year, there were 37,650 students at college B, with a projected enrollment dec line of 1250 students per year According to these projections, when will the colleges have the same enrollment? What will be the enrolliment in each college at that time? In the yearthe enrolment at both colleges will be the same The total enrolment at each college will bestudents
Answer:
all parts has been answered
Step-by-step explanation:
Let us assume
after T years both colleges have same enrollment
Enrollment at college A after T years = 18400+500*T
Enrollment at college B after T years = 37650-1250*T
Both the colleges will have same enrollment after T years
Hence
18400+500*T=37650-1250*T
1750*T=19250
T=11 years
Present year would be =2005+11=2016
In the year 2016, the enrollment of both the colleges will be same.
Total enrollment at each college = 18400+11*500=37650-1250*11=23900
The total enrollment at each college will be 23900 students
Four points question: Answer the following A- When selecting cards from a deck without replacing, the number of ways to draw the 3 card is (which of the following)? 52 52-1 52-2 B- When calculating the number of permutations of all letters in a word, the denominator of the calculation is which of the following? n! 0! C- How many ways can you draw a card from a normal deck and roll a number on a normal die? D- If you pick cards from a normal deck of cards, one at a time and replace the card and reshuffle the deck between draws, how many ways can you select 3 cards?
Answer:
A : 52-2
B : 0!
C : 312
D : 140608
Step-by-step explanation:
Part A:
It is given that that cards are selecting from a deck without replacing. So,
The number of ways to draw first card = 52
Now, one card is draw. The remaining cards are 51.
The number of ways to drawn second card = 52 - 1 =51
Now, one more card is drawn. The remaining cards are 50.
The number of ways to drawn third card = 52 - 2 =50
Therefore the number of ways to draw the 3 card is 52-2.
Part B:
Let a word has n letters and we need to find the number of permutations of all letters in a word, then the permutation formula is
[tex]^nP_n=\frac{n!}{(n-n)!}=\frac{n!}{0!}[/tex]
The denominator of the calculation is 0!.
Part C:
Total number of cards is a normal deck = 52
Total number sides in a die = 6
Total number of ways to draw a card from a normal deck and roll a number on a normal die = 52 × 6 = 312.
Therefore the total number of ways to draw a card from a normal deck and roll a number on a normal die is 312.
Part D:
It is given that we pick cards from a normal deck of cards, one at a time and replace the card and reshuffle the deck between draws.
Total number of ways to select each card = 52
Total number of ways to select 3 cards = 52³ = 140608
Therefore the total number of ways to select 3 cards is 140608.
18. Polar Bear Frozen Foods manufactures frozen French fries for sale to grocery store chains. The final package weight is thought to be a uniformly distributed random variable. Assume X, the weight of French fries, has a uniform distribution between 50 ounces and 68 ounces. What is the mean weight for a package? What is the standard deviation for the weight of a package? Round your answers to four decimal places, if necessary.
Answer:
μ = 59, σ = 5.1962
Step-by-step explanation:
For a uniform distribution, where a is the minimum and b is the maximum, the mean (or average) is:
μ = (a + b) / 2
And the standard deviation is:
σ = (b − a) / √12
Here, a = 50 and b = 68.
The mean is:
μ = (50 + 68) / 2
μ = 59
And the standard deviation is:
σ = (68 − 50) / √12
σ = 5.1962
The mean weight of a package of French fries produced by Polar Bear Frozen Foods is 59 ounces. The standard deviation of the weight is approximately 5.1962 ounces.
Explanation:The question pertains to understanding the mean and standard deviation of a uniformly distributed variable, specifically the weight of French fries produced by Polar Bear Frozen Foods. A uniform distribution is a type of probability distribution that has constant probability.
The formula for the mean (or average) of a uniform distribution is (a + b) / 2 where 'a' is the minimum value and 'b' is the maximum value. Given in the question a = 50 ounces and b = 68 ounces, we obtain the mean as (50+68)/2 = 59 ounces.
The formula for the standard deviation of a uniform distribution is sqrt[(b-a)^2 /12 ]. Thus, substituting values we get sqrt[(68-50)^2 / 12] = sqrt[324/12] = sqrt[27] = 5.1962 ounces (rounded to four decimal places).
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(3 points + 1 point BONuS) Many people grab a granola bar for breakfast or for a snack to make it through the afternoon slump at work. A Kashi GoLean Crisp Chocolate Caramel bar weights 45 grams. The mean amount of protein in each bar is 7.8 grams. Suppose the distribution of protein in a bar is normally distributed with a standard deviation of 0.2 grams and a random Kashi bar is selected. (0.5 pts.) a) What is the probability that the amount of protein is between 7.65 and 8.2 grams?
Answer: 0.7506
Step-by-step explanation:
Given :Mean : [tex]\mu=\text{ 7.8 grams}[/tex]
Standard deviation : [tex]\sigma =\text{ 0.2 grams}[/tex]
The formula for z -score :
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x= 7.65 ,
[tex]z=\dfrac{7.65-7.8}{0.2}=-0.75[/tex]
For x= 8.2 ,
[tex]z=\dfrac{8.2-7.8}{0.2}=2[/tex]
The p-value = [tex]P(-0.75<z<2)=P(z<2)-P(z<-0.75)[/tex]
[tex]=0.9772498-0.2266274=0.750622\approx0.7506[/tex]
Hence, the probability that the amount of protein is between 7.65 and 8.2 grams=0.7506.
Final answer:
To calculate the probability of a Kashi GoLean Crisp Chocolate Caramel bar having a protein content between 7.65 and 8.2 grams, we use the Z-score formula. The probability is found to be approximately 75.06%.
Explanation:
To find the probability that a randomly selected Kashi GoLean Crisp Chocolate Caramel bar has a protein content between 7.65 and 8.2 grams, we use the Z-score formula for a normal distribution, where Z = (X - μ) / σ. Here, μ (mu) is the mean, and σ (sigma) is the standard deviation. The mean (μ) is 7.8 grams, and the standard deviation (σ) is 0.2 grams.
To find the Z-scores for 7.65 grams and 8.2 grams:
Z for 7.65 = (7.65 - 7.8) / 0.2 = -0.75Z for 8.2 = (8.2 - 7.8) / 0.2 = 2Next, we consult the standard normal distribution table to find the probabilities corresponding to these Z-scores. For Z = -0.75, the probability is approximately 0.2266, and for Z = 2, it's approximately 0.9772.
The probability that the protein is between 7.65 and 8.2 grams is the difference between these two probabilities: 0.9772 - 0.2266 = 0.7506 or 75.06%.
Which of the following is equal to 6^5 - 6^4 / 5 = ?
a) 1/5 b) 6/5 c) 6^3 d) 6^4 / 5 e) 6^4
Answer:
[tex]6^4[/tex]
Step-by-step explanation:
We'd solve the exponents first:
[tex]6^5 = 7776[/tex]
[tex]6^4 = 1296[/tex]
Subtract:
7776 - 1296 = 6480
Divide:
[tex]6480\div5 = 1296[/tex]
We already know [tex]6^4 = 1296[/tex]
Our answer is [tex]6^4[/tex]
Answer:
6^4
Step-by-step explanation:
6^5 - 6^4 / 5 = ?
Factor out a 6^4
6^4(6-1)
---------------
5
Simplify
6^4(5)
---------------
5
Cancel the 5's
6^4
Use Laplace transforms to solve the initial value problem, then give the value of x(?).
x'' + 4x = 0; x(0) = 5, x'(0) = 0
Answer:
x = 5 cos 2t
Step-by-step explanation:
given equation
x'' + 4x = 0 ; x(0) = 5, x'(0) = 0
L{ x'' + 4 x } = 0
L{x''} + 4 L{x} = 0
s² . L(x) - s . x(0) - x'(0) + 4 L{x} = 0
( s² + 4 ). L(x) - 5 s = 0
L(x) = [tex]\dfrac{5s}{s^2 +4}[/tex]
[tex]L(\dfrac{s}{s^2 +a^2})[/tex] = cos at
so,
x = 5 [tex]L^{-1}(\dfrac{s}{s^2 +2^2})[/tex]
x = 5 cos 2t
Find a possible formula for a fourth degree polynomial g that has a double zero at -2, g(4) = 0, g(3) = 0, and g(0) = 12. g(x) =
The possible formula for a fourth degree polynomial g is:
[tex]g(x)=\dfrac{1}{4}(x^4-3x^3-12x^2+20x+48)[/tex]
Step-by-step explanation:We know that if a polynomial has zeros as a,b,c and d then the possible polynomial form is given by:
[tex]f(x)=m(x-a)(x-b)(x-c)(x-d)[/tex]
Here the polynomial g has a double zero at -2, g(4) = 0, g(3) = 0.
This means that the polynomial g(x) is given by:
[tex]g(x)=m(x-(-2))^2(x-4)(x-3)\\\\i.e.\\\\g(x)=m(x+2)^2(x-4)(x-3)\\\\i.e.\\\\g(x)=m(x^2+2^2+2\times 2\times x)(x(x-3)-4(x-3))\\\\i.e.\\\\g(x)=m(x^2+4+4x)(x^2-3x-4x+12)\\\\i.e.\\\\g(x)=m(x^2+4+4x)(x^2-7x+12)\\\\i.e.\\\\g(x)=m[x^2(x^2-7x+12)+4(x^2-7x+12)+4x(x^2-7x+12)]\\\\i.e.\\\\g(x)=m[x^4-7x^3+12x^2+4x^2-28x+48+4x^3-28x^2+48x]\\\\i.e.\\\\g(x)=m[x^4-7x^3+4x^3+12x^2+4x^2-28x^2-28x+48x+48]\\\\i.e.\\\\g(x)=m[x^4-3x^3-12x^2+20x+48][/tex]
Also,
[tex]g(0)=12[/tex]
i.e.
[tex]48m=12\\\\i.e.\\\\m=\dfrac{12}{48}\\\\i.e.\\\\m=\dfrac{1}{4}[/tex]
Hence, the polynomial g(x) is given by:
[tex]g(x)=\dfrac{1}{4}(x^4-3x^3-12x^2+20x+48)[/tex]
The formula for the fourth degree polynomial g(x) can be determined using the known roots and points. The polynomial will have form g(x) = a(x+2)² × (x-3) × (x-4). Value of 'a' is found by setting x=0 and solving for a.
Explanation:The subject of the question is determining the formula for a fourth degree polynomial, g(x), based on provided conditions. From the question we know that g(x) must have a double root at -2 (-2, -2); roots 3 and 4 (3, 0) and (4, 0); and that g(0) = 12.
With this information, we know that a fourth degree polynomial will look something like this: g(x) = ax4 + bx3 + cx2 + d = 0. But, per the conditions, it looks like g(x) = a(x+2)2 × (x-3) × (x-4), because it has roots at x = -2 (twice, or 'double'), x = 3, and x = 4.
To find the value of 'a', we use the information that at x = 0, g(x) should equal to 12. The function can then be processed as follows:
g(0) = a(0+2)2 × (0-3) × (0-4) = 12.
This will give us 'a', and the desired polynomial formula.g(x).
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In a sample of 800 U.S. adults, 218 dine out at a restaurant more than once a week. If two U.S adults are selected at random from the population of all U.S. adults without replacement. Assuming the sample is representative of all U.S. adults. (a) Find the probability that both adults dine out more than once per week\
Answer: 0.074
Step-by-step explanation:
Given : The total number of U.S. adults in the sample = 800
The number of U.S. adults dine out at a restaurant more than once a week = 216
The probability for an adult dine out more than once per week :-
[tex]\dfrac{218}{800}[/tex]
If another person is selected without replacement ,then
Total adults left = 799
Total adults left who dine out at a restaurant more than once a week = 217
The probability for the second person dine out more than once per week :-
[tex]\dfrac{217}{799}[/tex]
Now, the probability that both adults dine out more than once per week :-
[tex]\dfrac{218}{800}\times\dfrac{217}{799}=0.074[/tex]
Final answer:
To calculate the probability that both selected adults dine out more than once per week, multiply the probability of the first adult dining out more than once (218/800) by the probability of the second adult dining out more than once after the first has been selected (217/799).
Explanation:
The probability that both adults dine out more than once per week can be found using the formula for the probability of successive events without replacement. With 218 out of 800 US adults dining out more than once per week, the probability of the first adult dining out more than once is 218/800. If one adult who dines out more than once a week has been chosen, there are now 217 such adults left and 799 total adults. The probability of the second adult dining out more than once is 217/799. The joint probability of both events happening is calculated by multiplying these two probabilities together:
P(both dine out) = (218/800)
times (217/799)
Simplify this to find the requested probability.
Calculate the average density in kilograms [kg] per cubic meter [m3] for a White Dwarf with a mass of
1 solar mass and the size of planet Earth. Be sure to use the correct units. (b) Calculate your own weight on the surface of the white dwarf star.
Answer:
(a) 2 × 10^9 kg/m^3; (b) roughly the mass of the Statue of Liberty.
Step-by-step explanation:
(a) Density of white dwarf:
D = m/V
Data:
1 solar mass = 2 × 10^30 kg
1 Earth radius = 6.371 × 10^6 m
Calculations:
V = (4/3)πr^3 = (4/3)π × (6.371 × 10^6 m)^3 = 1.083 × 10^21 m^3
D = 2 × 10^30 kg/1.083 × 10^21 m^3 = 2 × 10^9 kg/m^3
2. Weight on a white dwarf
The formula for weight is
w = kMm/r^2
where
k = a proportionality constant
M = mass of planet
m = your mass
w(on dwarf)/w(on Earth) = [kM(dwarf)m/r^2] /[kM(Earth)m/r^2
k, m, and r are the same on both planets, so
w(on dwarf)/w(on Earth) = M(dwarf)/M(Earth)
w(on dwarf) = w(on Earth) × [M(dwarf)/M(Earth)]
Data:
M(Earth) = 6.0 × 10^24 kg
Calculation:
w(on dwarf) = w(on Earth) × (2 × 10^30 kg /6.0 × 10^24 kg)
= 3.3 × 10^5 × w(on Earth)
Thus, if your weight on Earth is 60 kg, your weight on the white dwarf will be
3.3 × 10^5 × 60 kg = 2 × 10^7 kg
That's roughly as heavy as the Statue of Liberty is on Earth.
Color blindness is an inherited characteristic that is more common in males than in females. If M represents male and C represents red-green color blindness, using the relative frequencies of the incidences of males and red-green color blindness as probabilities results in the values below. P(C)=0.042, P ( M intersect C)=0.022, P ( M union C)=0.412What is the probability that a randomly selected person is not color blind?
Answer:
The probability that a randomly selected person is not color blind is 0.958
Step-by-step explanation:
Given,
C represents red-green color blindness,
Also, the probability that a randomly selected person is color blind,
P(C) = 0.042,
Thus, probability that a randomly selected person is not color blind,
P(C') = 1 - P(C) = 1 - 0.042 = 0.958
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 7x6, y = 7x, x ≥ 0; about the x-axis
Answer:
The volume is [tex]\frac{490\pi}{39}[/tex] cubic units.
Step-by-step explanation:
The given curve is
[tex]y=7x^6[/tex]
The given line is
[tex]y=7x[/tex]
Equate both the functions to find the intersection point of both line and curve.
[tex]7x^6=7x[/tex]
[tex]7x^6-7x=0[/tex]
[tex]7x^6-7x=0[/tex]
[tex]7x(x^5-1)=0[/tex]
[tex]7x=0\rightarrow x=0[/tex]
[tex]x^5-1=0\rightarrow x=1[/tex]
According to washer method:
[tex]V=\pi \int_{a}^{b}[f(x)^2-g(x)^2]dx[/tex]
Using washer method, where a=0 and b=1, we get
[tex]V=\pi \int_{0}^{1}[(7x)^2-(7x^6)^2]dx[/tex]
[tex]V=\pi \int_{0}^{1}[49x^2-49x^{12}]dx[/tex]
[tex]V=49\pi \int_{0}^{1}[x^2-x^{12}]dx[/tex]
[tex]V=49\pi [\frac{x^3}{3}-\frac{x^{13}}{13}]_0^1[/tex]
[tex]V=49\pi [\frac{1^3}{3}-\frac{1^{13}}{13}-(0-0)][/tex]
[tex]V=49\pi [\frac{1}{3}-\frac{1}{13}][/tex]
[tex]V=49\pi (\frac{13-3}{39})[/tex]
[tex]V=49\pi (\frac{10}{39})[/tex]
[tex]V=\frac{490\pi}{39}[/tex]
Therefore the volume is [tex]\frac{490\pi}{39}[/tex] cubic units.
The volume of the solid obtained by rotating the region bounded by the curves y = 7x6 and y = 7x, for x ≥ 0, around the x-axis is determined by setting up and evaluating a volume integral using the method of cylindrical shells. The intersection points of the curves, that serve as the limits of the integral, are found by equating the two functions, giving x = 0 and x = 1.
Explanation:To find the volume of the solid obtained by rotating the area between the curves y = 7x6 and y = 7x around the x-axis, we use the method of cylindrical shells.
Firstly, we find the intersection points of the two curves by setting them equal to each other: 7x6 = 7x, which gives us x = 0 and x = 1.
Next, we set the volume integral and evaluate it over the range [0,1]. The general formula is V = 2π ∫ from a to b [x * (f(x) - g(x)) dx]. Here, f(x) = 7x6 and g(x) = 7x, therefore,
V = 2π ∫ from 0 to 1 [x * (7x6 - 7x) dx] = 2π ∫ from 0 to 1 [7x7 - 7x2 dx],
which can be integrated term-by-term. The antiderivatives of x7 and x2 are (1/8)x8 and (1/3)x3, respectively. By implementing these with the Fundamental Theorem of Calculus, we then substitute the limits of integration, subtract, and simplify to reach the final volume of the rotated solid.
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The scores on the entrance exam at a well-known, exclusive law school are normally distributed with a mean score of 200 and a standard deviation equal to 50. At what value should the lowest passing score be set if the school wishes only 2.5 percent of those taking the test to pass? (Round your answer to nearest whole number.)
Answer:
the lowest passing score would be x = 298
Step-by-step explanation:
School wishes that only 2.5 percent of students taking test pass
We are given
mean= 200,
standard deviation = 50
We need to find x
The area under the curve can be found by:
2.5 % = 0.025
So, 1- 0.025 = 0.975
We need to find the value of z for which the answer is 0.975
Looking at the z-score table, the value of z is: 1.96
Now, using the formula:
z = x - mean/standard deviation
1.96 = x - 200/50
=> 1.96 * 50 = x-200
98 = x - 200
=> x = 200+98
x = 298
So, the lowest passing score would be x = 298
The lowest passing score should be set at 102 to ensure that only 2.5 percent of the test takers pass.
Understand the Problem Context:7) State the prime factorization of 30.
Answer:
[tex]30=2\: *3\:*5[/tex]
Step-by-step explanation:
We analyze between which prime numbers we can divide the number 30. The smallest prime number by which we divide is 2. Then:
[tex]\frac{30}{2}=15[/tex]
We now look for the smallest prime number that divides the 15. Since 15 is not a multiple of 2, we make the division with the number 3 that is divisor of 15.
[tex]\frac{15}{3}=5[/tex]
We now look for a number that divides to 5, but since 5 is a prime number, the only divisor other than 1 is 5. Then:
[tex]\frac{5}{5}=1[/tex]
This ends the decomposition of 30 and we find 3 prime factors:
2,3 and 5.
A cellphone service provider charges $5.00 per month and $0.20 per minute per call. If a customer's current bill is $55, how many minutes did the customer use? (Round any intermediate calculations and your final answer to the nearest whole minute.) 300 minutes 275 minutes 250 minutes 225 minutes
Answer:
250 minutes
Step-by-step explanation:
Given,
Cell phone charges for a month = $ 5.00,
Additional charges per minute = $0.20,
Let the cell phone is used for x minutes for a month such that the total bill is $ 55,
⇒ Cell phone charge for the month + additional charges for x minutes = $ 55
⇒ 5.00 + 0.20x = 55
Subtracting 5 on both sides,
0.20x = 50
x = 250,
Hence, the cell phone is used for 250 minutes.
Third option is correct.
A patient needs atropine 0.2 mg IM STAT. It's available in 0.4 mg/mL. What's the correct dosage for this patient?
A. 0.5 mL B. 0.1 mL C. 1 mL D. 5 mL
Answer:
The correct dosage is 0.5 ml
Step-by-step explanation:
Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.
We can solve this question by using the simple Rule of Three property. The Property is the following
[tex]\frac{a}{x} = \frac{b}{c}[/tex] ⇒ [tex]x = \frac{a*c}{b}[/tex]
Now we can use the property above using the values given to us to find the correct dosage for the patient.
[tex]\frac{0.2mg}{x} = \frac{0.4mg}{1ml}[/tex]
[tex]x = \frac{0.2mg*1ml}{0.4mg}[/tex]
[tex]x = \frac{0.2ml}{0.4}[/tex]
[tex]x = 0.5ml[/tex]
So Now we can see that the correct dosage is 0.5 ml
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
Suppose you have the formula =$D$5*E5 in cell F5. When you copy the formula into cell F6 what will the new formula be?
a. =$D$5*E5
b. =D5*E6
c. =$D$5*E6
d. =$D$6*E6
Answer:
C. =$D$5*E6
Step-by-step explanation:
In excel Columns are A, B, C, D, .....
And, rows are, 1, 2, 3, ....
Also, the intersection of column and row is called cell,
eg : A1 is a cell,
When we copy a function formula from a cell A1 to B2,
Then in the formula we replace A1 by B2 or vice versa,
But the row or column which comes after the dollar sign is anchored or absolute.
That is, when we copy an excel formula with $ sign they will copy cells referred in that formula relative to the position where they are being copied to.
Thus, if we have the formula =$D$5*E5 in cell F5,
Then, $D$5*E6 must be the new formula into cell 6.
Option 'C' is correct.
Find an equation for the line that passes through the points (-2, 1) and (6, -5)
Answer:
y=-3/4x-1/2
Step-by-step explanation:
Using the slope equation,
m=y₂-y₁
____
x₂-x₁
you get the slope:
m=-5-1/6+2
m=-6/8
m=-3/4
We find that the intercept is -1/2.
The equation is y=-3/4x-1/2
The answer is:
The equation of the line that passes through the points (-2,1) and (6,-5) is:
[tex]y=-\frac{3}{4}x-\frac{1}{2}[/tex]
Why?To solve the problem, we can use the following formula:
We have that:
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}*(x-x_1)[/tex]
So, using the given points (-2,1) and (6,-5), we have:
[tex]y-1=\frac{-5-1}{6-(-2)}*(x-(-2))[/tex]
[tex]y-1=\frac{-6}{6+2}*(x+2)[/tex]
[tex]y-1=\frac{-6}{8}*(x+2)[/tex]
[tex]y-1=-\frac{3}{4}*(x+2)[/tex]
[tex]y=-\frac{3}{4}*(x+2)+1[/tex]
[tex]y=-\frac{3}{4}*(x)-\frac{3}{4}**(2)+1[/tex]
[tex]y=-\frac{3}{4}(x)-\frac{6}{4})+1[/tex]
[tex]y=-\frac{3}{4}(x)-\frac{3}{2})+1[/tex]
[tex]y=-\frac{3}{4}x-\frac{1}{2}[/tex]
Hence, we have that the equation of the line that passes through the points (-2,1) and (6,-5) is:
[tex]y=-\frac{3}{4}x-\frac{1}{2}[/tex]
Have a nice day!
Select all the levels of measurement for which data can be qualitative.A.Nominal.B.IntervalC.RatioD.Ordinal
Select all the levels of measurement for which data can be qualitative.A.Nominal.B.IntervalC.RatioD.Ordinal
ratio
Divide
(x^2 - 13x +40) divided by (x- 6)
Answer:
The quotient is: x-7
The remainder is: -2
Step-by-step explanation:
We need to divide (x^2 - 13x +40) divided by (x- 6)
The quotient is: x-7
The remainder is: -2
The division is shown in the figure attached