Final answer:
To find the number of adults in the family, we need to solve the system of equations. By multiplying the first equation by 2 and subtracting it from the second equation, we can eliminate x and solve for y. Substituting the value of y back into the first equation, we can solve for x. The number of adults in the family is 3.
Explanation:
To find the number of adults in the family, we need to solve the system of equations:
Equation 1: 8x + 4y = 32
Equation 2: 16x + 12y = 72
We can solve this system by first multiplying Equation 1 by 2 to make the coefficients of x in both equations the same. This gives us:
Equation 1 (multiplied by 2): 16x + 8y = 64
Next, we can subtract Equation 1 (multiplied by 2) from Equation 2 to eliminate x:
Equation 2 - Equation 1 (multiplied by 2): (16x + 12y) - (16x + 8y) = 72 - 64
Simplifying the equation, we get:
4y = 8
Dividing both sides by 4, we find:
y = 2
So, there are 2 children in the family. Substituting this value back into Equation 1, we can solve for x:
8x + 4(2) = 32
8x + 8 = 32
8x = 24
Dividing both sides by 8, we find:
x = 3
Therefore, there are 3 adults in the family.
Write the polynomial as the product of linear factors.
h(x) = x^2 ? 2x + 10
Answer:
If [tex]h(x) = x^2 + 2x + 10[/tex] then [tex](x+1-3i)*(x+1+3i)[/tex].
If [tex]h(x) = x^2 - 2x + 10[/tex] then [tex](x-1-3i)*(x-1+3i)[/tex].
Step-by-step explanation:
In order to write the polynomial as the product of linear factors, we need to find the roots of the polynomial. A quadratic equation is defined as:
[tex]ax^2+bx+c[/tex]
Because the given polynomial expression is a quadratic equation, we can use the following equations for calculating the roots:
[tex]x1=(-b/2a)+\sqrt{b^2-4ac}/2a[/tex]
[tex]x1=(-b/2a)-\sqrt{b^2-4ac}/2a[/tex]
Since the second term sign is not given, then we can write the expression as:
[tex]x^2+s2x+10[/tex], in which a=1, b=s2 where 's' represents a sign (- or +), and c=10.
Using the equation for finding the roots we obtain:
[tex]x1=(-sb/2a)+\sqrt{b^2-4ac}/2a[/tex]
[tex]x1=(-s2/2*1)+\sqrt{2^2-4*1*10}/(2*1)[/tex]; notice that [tex](sb)^{2} = 2^{2}[/tex]
[tex]x1=(-s1)+\sqrt{-36}/2[/tex]
[tex]x1=-(s1)+6i/2[/tex]
[tex]x1=-(s1)+3i[/tex]
[tex]x2=(-sb/2a)-\sqrt{b^2-4ac}/2a[/tex]
[tex]x2=(-s2/2*1)-\sqrt{2^2-4*1*10}/(2*1)[/tex]; notice that (s2)²=2^2
[tex]x2=(-s1)-\sqrt{-36}/2[/tex]
[tex]x2=-(s1)-6i/2[/tex]
[tex]x2=-(s1)-3i[/tex]
If we consider 's' as possitive (+) the roots are:
[tex]x1=-1+3i[/tex] and [tex]x2=-1-3i[/tex]
Whereas if we consider 's' as negative (-) the roots are:
[tex]x1=1+3i[/tex] and [tex]x2=1-3i[/tex]
The above means that if the equation is [tex]h(x) = x^2 + 2x + 10[/tex], then we can express the polynomial as: [tex](x+1-3i)*(x+1+3i)[/tex].
But, if the equation is [tex]h(x) = x^2 - 2x + 10[/tex], then we can express the polynomial as: [tex](x-1-3i)*(x-1+3i)[/tex].
The waiting times between a subway departure schedule and the arrival of a passenger are uniformly distributed between 0 and 5 minutes. Find the probability that a randomly selected passenger has a waiting time greater than 1.25 minutes.
Answer: 0.75
Step-by-step explanation:
Given : Interval for uniform distribution : [0 minute, 5 minutes]
The probability density function will be :-
[tex]f(x)=\dfrac{1}{5-0}=\dfrac{1}{5}=0.2\ \ ,\ 0<x<5[/tex]
The probability that a given class period runs between 50.75 and 51.25 minutes is given by :-
[tex]P(x>1.25)=\int^{5}_{1.25}f(x)\ dx\\\\=(0.2)[x]^{5}_{1.25}\\\\=(0.2)(5-1.25)=0.75[/tex]
Hence, the probability that a randomly selected passenger has a waiting time greater than 1.25 minutes = 0.75
The probability that a randomly selected passenger has a waiting time greater than 1.25 minutes is 1.
Explanation:To find the probability that a randomly selected passenger has a waiting time greater than 1.25 minutes, we need to find the area under the probability density function (PDF) curve for values greater than 1.25. Since the waiting times between a subway departure schedule and the arrival of a passenger are uniformly distributed between 0 and 5 minutes, the PDF is a rectangle with height 1/5 and base 5. The area of the rectangle represents the probability.
The probability of a waiting time greater than 1.25 minutes is the ratio of the area of the rectangle representing waiting times greater than 1.25 minutes to the total area of the rectangle representing all waiting times.
To calculate this probability, we first need to find the area of the rectangle representing waiting times greater than 1.25 minutes. Since the base of the rectangle is 5 minutes and the height is 1/5, the area is given by:
Area = base * height = 5 * (1/5) = 1
The total area of the rectangle representing all waiting times is the area of the entire rectangle, which is also equal to:
Area = base * height = 5 * (1/5) = 1
Therefore, the probability of a randomly selected passenger having a waiting time greater than 1.25 minutes is:
Probability = Area of waiting times greater than 1.25 minutes / Total area = 1/1 = 1
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10 companies sponsor a tournament and decided to give M rupees collectively. Two companies dropped and remaining agreed on paying their share equally. What was the increase in share of each company ?
a) M-20/2 b) M/50 c) M/40 d) M/2 e) 2M
Answer:
c) M/40
Step-by-step explanation:
The share of each company was originally M/10. Now it is M/8. The increase is ...
M/8 -M/10 = M(1/8 -1/10) = M(10-8)/(10·8) = M/40
The increase in the share of each company after two companies dropped out is M/40.
Explanation:The total amount of money to be paid by the companies is M rupees. Initially, these M rupees were to be divided among 10 companies, but when two dropped out, they had to be divided among 8 companies instead. So, the share of each company initially was M/10 and after the drop out, the share of each company became M/8.
The increase in the amount that each company had to give is given by subtraction of the initial share from the final share, that is, (M/8) - (M/10). To solve this equation, we first must find a common denominator for the fractions, which is 40 in this case. Therefore, the equation becomes (5M/40) - (4M/40) = M/40.
So, the increase in share of each company was M/40. Therefore, the correct answer is (c) M/40.
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Dave is buying pizza and soda. Suppose that a slice of pizza andd a can of soda cost $3. Let E be the amount in dollars that Dave spends on pizza and soda. If Dave buys P slices of pizza and S cans of soda, which equation correctly describes the amount of money he spends? E=P+3S E=3P+S E=P+S E=3P+3S 3E=P+S 2. Now rearrange the equation you wrote above so that S is written in terms of E and P. Which of the following is correct? S=P+E S=(1/3)E-P S=E-3P S=3E-3P S=E-P
Answer:
E = 3P + 3S,
S=(1/3)E-P
Step-by-step explanation:
Given,
A slice of pizza and a can of soda cost $3,
That is, total cost of P slices of pizza and S cans of soda = ( 3P + 3S ) dollars,
According to the question,
Total cost of P slices of pizza and S cans of soda = E dollars,
⇒ 3P + 3S = E
Which is the required equation that correctly describes the amount of money spent,
For rearranging the equation in terms of E and P,
We need to isolate S,
Subtract 3P on both sides,
3S = E - 3P
Divide both sides by 3,
[tex]S=\frac{E-3P}{3}[/tex]
[tex]\implies S=(\frac{1}{3})E-P[/tex]
Which is the required equation.
Find the interest rate on a loan charging $855 simple interest on a principal of $3750 after 6 years.
Answer:
3.8%
Step-by-step explanation:
Simple interest formula is I=P*R*T
where:
I=interest
P=principal
R=rate
T=time.
So let's plug in our information we are given:
I=855
P=3750
T=6
R=?.
The equation becomes 855=3750*R*6.
Multiplication is commutative so we could write this as 3750*6*R=855.
After the multiplication of 3750 and 6 we obtain 22500*R=855.
Now we just divide both sides by 22500. This will give us:
R=855/22500 which when entered into the calculator as 855 division sign 22500 gives us 0.038 or 3.8%.
A student takes an exam containing 18 multiple choice questions. The probability of choosing a correct answer by knowledgeable guessing is 0.3. If the student makes knowledgeable guesses, what is the probability that he will get between 8 and 12 (both inclusive) questions right? Round your answer to four decimal places.
When you have to repeatedly take the same test, with constant probability of succeeding/failing, you have to use Bernoulli's distribution. It states that, if you take [tex]n[/tex] tests with "succeeding" probability [tex]p[/tex], and you want to "succeed" k of those n times, the probability is
[tex]\displaystyle P(n,k,p) = \binom{n}{k}p^k(1-p)^{n-k}[/tex]
In your case, you have n=18 (the number of tests), and p=0.3 (the probability of succeeding). We want to succeed between 8 and 12 times, which means choosing k=8,9,10,11, or 12. For example, the probability of succeeding 8 times is
[tex]\displaystyle P(18,8,0.3) = \binom{18}{8}(0.3)^8(0.7)^{10}[/tex]
you can plug the different values of k to get the probabilities of succeeding 9, 10, 11 and 12 times, and your final answer will be
[tex]P = P(18,8,0.3) + P(18,9,0.3) + P(18,10,0.3) + P(18,11,0.3) + P(18,12,0.3)[/tex]
Final answer:
In mathematics, calculate the probability of a student guessing between 8 and 12 questions right out of 18 when the probability of guessing correctly is 0.3.
Explanation:
To calculate the probability of getting between 8 and 12 questions right through knowledgeable guessing, we can use the binomial probability formula:
P(k successes in n trials) = nCk × [tex]p^k[/tex] × (1-[tex]p^{(n-k)[/tex])
where:
k = number of successes (correct answers) = 8 to 12
n = total number of questions = 18
p = probability of success (correct answer by guessing) = 0.3
(1-p) = probability of failure (incorrect answer) = 0.7
We need to calculate the probability for each possible number of correct answers between 8 and 12 and then sum them up.
Here's the calculation:
Probability of getting 8 correct:
P(8 successes in 18 trials) = 18C8 ×0.3⁸ × 0.7¹⁰ ≈ 0.076
Probability of getting 9 correct:
P(9 successes in 18 trials) = 18C9 × 0.3⁹ ×0.7⁹ ≈ 0.124
Probability of getting 10 correct:
P(10 successes in 18 trials) = 18C10 ×0.3¹⁰ ×0.7⁸ ≈ 0.122
Probability of getting 11 correct:
P(11 successes in 18 trials) = 18C11 × 0.3¹¹ × 0.7⁷ ≈ 0.072
Probability of getting 12 correct:
P(12 successes in 18 trials) = 18C12 × 0.3¹² × 0.7⁶ ≈ 0.021
Total probability:
P(8 to 12 correct) = 0.076 + 0.124 + 0.122 + 0.072 + 0.021 ≈ 0.415
Therefore, the probability that the student gets between 8 and 12 questions right by knowledgeable guessing is approximately 41.5%.
How many different four-letter passwords can be formed from the letters Upper A comma Upper B comma Upper C comma Upper D comma Upper E comma Upper F comma and Upper G if no repetition of letters is allowed?
Answer with explanation:
Number of Letters from which we have to make four-letter passwords
={A,B,C,D,E,F,G}=7
Since repetition of digit is not allowed.In Permutation order of arrangement is Important,while in combination order of arrangement is not Important.
Out of seven letters we have to select four letters, keeping in mind the order of Alphabets.So, we will use the concept of Permutation.
Number of ways of arrangement,that is to make four letter password from 7 Alphabets
[tex]_{4}^{7}\textrm{P}\\\\=\frac{7!}{(7-4)!}\\\\=\frac{7!}{3!}\\\\=4\times 5\times 6 \times 7=840[/tex]
So, total number of ways of making password from 7 alphabets=840 ways
Second Method⇒
Since repetition of alphabets is not allowed, first Alphabet can be chosen in 7 ways , second alphabet can be chosen in 6 ways, third alphabet can be chosen in 5 ways, fourth alphabet can be chosen from four alphabets.
So, total number of ways of making four letter password
=7×6×5×4
= 840 ways
We can form 840 unique four-letter passwords with no repeats from the letters A, B, C, D, E, F, G. This is a problem in combinatorics, calculating permutations of positions in the password.
Explanation:We are trying to determine the number of unique four-letter passwords that can be created using the letters A, B, C, D, E, F, G with no repetition of letters. This is a problem of permutations, a concept in combinatorics, a branch of mathematics that deals with counting.
In this case, for the first position of the password, we have 7 choices. Once we've picked this first letter, we then have 6 remaining letters to choose from for the second position. Similarly, there are 5 choices remaining for the third position and then 4 for the last position.
The total number of password combinations will then be a product of the number of choices for each of the positions. Mathematically, this can be represented as 7*6*5*4.
Consequently, there are 840 different four-letter passwords that can be created from the given letters without repeating any letter.
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An excavation has a volume of 203 cubic feet. The material weighs 1,200 pounds per cubic yard. What is the weight of the exca- vation? a. 2,700 pounds b. 9,000 pounds c. 81,120 pounds d. 243,600 pounds
Answer: Option 'd' is correct.
Step-by-step explanation:
Since we have given that
Volume of an excavation = 203 cubic feet
Weight of the material = 1200 pounds per cubic yard
So, the weight of the excavation would be
Volume of an excavation × Weight of the material
[tex]=203\times 1200\\\\=243,600\ pounds[/tex]
Hence, the weight of the excavation is 243,600 pounds.
Therefore, Option 'd' is correct.
Write a differential equation to represent each situation below. Do NOT solve them. a. A new technology is introduced into a community of 5000 people. If the rate at which the technology is adopted in the community is jointly proportional to the number of people who have adopted the technology and the number of people who have not adopted it, write a differential equation to represent the number of people, x(t), who have adopted the technology by time t. b. A tank with a capacity of 1000 gal originally contains 800 gal of water with 200 lbs of salt in the solution. Water containing 3.5 lbs of salt per gal is entering at a rate of 2 gal/min, and the mixture is allowed to flow out of the tank at a rate of 5 gal/min. Write a differential equation reflecting the information above, clearly stating the requested intermediate results below. Let A(t) represent the amount of salt (in pounds) in the tank after t minutes. Do NOT solve the DE! Show units with each factor in the three setup steps below, and simplify each expression, showing the resulting units as well. *Be sure to include the initial conditions for this DE in your final equation. R_in = Concentration of salt in the tank: c(t) = R_out = Differential equation: A(0) =
Answer:
a. x'(t) = kx(5000-x)
b. R_in = 7 lbs/min
R_out = 5c(t) lbs/min
c(t) = A(t)/(800 -3t) lbs/gal
A'(t) = R_in - R_out; A(0) = 200 lbs
Step-by-step explanation:
A. When proportionality is joint between x and y, the expression describing that is kxy, where k is the constant of proportionality. Here, the rate of change is jointly proportional to number who have adopted (x) and number of the community of 5000 who have not (5000-x). The corresponding equation is ...
x'(t) = kx·(5000-x)
We know the initial number of adopters must be greater than 0, or the equation's only solution would be x=0.
__
B. The rate of influx of salt is ...
R_in = (3.5 lbs/gal)×(2 gal/min) = 7 lbs/min
The number of gallons in the tank at time t is ...
v = 800 gal + (2 gal/min - 5 gal/min)(t min) = (800 -3t) gal
For amount of salt A(t) pounds, the concentration c(t) will be ...
c(t) = A(t)/v = A(t)/(800 -3t) . . . . lbs/gal
The outflow rate will be the product of outflow volume and concentration:
R_out = (c(t) lbs/gal)(5 gal/min) = 5c(t) lbs/min
And the differential equation for A(t) is ...
A'(t) = R_in -R_out . . . with initial condition A(0) = 200 lbs.
__
When you go to solve the equation for A'(t), it will need to be cast in terms of A(t):
A'(t) = 7 -5A(t)/(800 -3t)
The daily number in the State Lottery is a three-digit nuber from 000 to 999. To win in "straight play", a ticket must have the exact number. To win in a "box play" , a ticket can have the three digit of the daily number in any order. Find the probabilty of winning in :
a) Straight play
b) Box play
Answer:
a. [tex]P = \frac{1}{1000}[/tex]
b. [tex]P = \frac{3}{500}[/tex]
Step-by-step explanation:
To win the 'straight play' a ticket must have the exact number. Given that there are 1000 numbers between 000 and 999. The probability of winning in striaght play is:
[tex]P = \frac{1}{1000}[/tex]
To win in a box play, a ticket can have the three digit in daily order.
Given three numbers A, B and C. There are 6 ways to arrange the numbers:
ABC, ACB, BAC, BCA, CAB, CBA.
Therefore the probability is: [tex]P = \frac{3}{500}[/tex]
Find an equation for a circle satisfying the given conditions. (a) Center (-1, 4), passes through (3, 7) (b) The points (7, 13) and (-3, -11) are at the ends of a diameter.
Answer:
Step-by-step explanation:
In order to find the equations we need the circle's general equation:
[tex](x-h)^{2}+(y-k)^{2}=r^2[/tex] where:
(h,k) is the center and 'r' is the radius.
A. Because the center is (-1,4) then h=-1 and k=4.
Now we can find the radius as:
[tex]distance=\sqrt{(x2-x1)^{2}+(y2-y1)^{2}}[/tex]
[tex]distance=\sqrt{(3-(-1))^{2}+(7-4)^{2}}[/tex]
[tex]distance=5[/tex] so we have r=5
Then the equation is [tex](x+1)^{2}+(y-4)^{2}=25[/tex]
B. Because we have two points defining a diameter we can find the radius as follows:
[tex]diameter=\sqrt{(x2-x1)^{2}+(y2-y1)^{2}}[/tex]
[tex]diameter=\sqrt{(7-(-3))^{2}+(13-(-11))^{2}}[/tex]
[tex]diameter=26[/tex]
[tex]radius=26/2=13[/tex]
Now let's find the center of the circle as follows:
[tex]C=(\frac{x1+x2}{2} , \frac{y1+y2}{2})[/tex]
[tex]C=(\frac{7-3}{2} , \frac{13-11}{2})[/tex]
[tex]C=(2,1)[/tex]
Then the equation is [tex](x-2)^{2}+(y-1)^{2}=169[/tex]
Randy deposits $9,500 in an IRA. What will be the value of his investment in 3 years if the investment is earning 1.75% per year and is compounded continuously? Round to the nearest cent.
Answer:
The value of his investment in 3 years = $10007.53
Step-by-step explanation:
Compound interest
A = P[1 +R/n]^nt
Where A - amount
P - principle amount
R = rate of interest
t - number of years
n - number of times compounded yearly
To find value if investment
Here P = $9500, R = 1.75% = 0.0175 n = 1 and t = 3
A = P[1 +R/n]^nt
= 9500[1 + 0.0175/1]^(1 * 3)
= 10007.53
Answer:
$10,012.07
Step-by-step explanation:
Identify the values of each variable in the formula. Remember to express the percent as a decimal.
APrt=?=$9,500=0.0175=3 years
For compounding continuously, use the formula A=Pert.
Substitute the values into the formula and compute the amount to find
AA=9,500e0.0175⋅3=$10,012.07.
if ( 43.65 ) ( 8.79 ) / x = ( 0.4365 ) ( 87.9 ) then value of x is:
(a) .01 (b) 0.1 (c) 1 (d) 10 (e) 100
Answer:
(d)
Step-by-step explanation:
Multiple your numbers out first. This always makes it less hectic to look at. Multiply both sides by x to move the x to the right side. Then divide both sides by ((0.4365)(87.9)). You should get 10.
swimsuit buyer reduced a group of designer swimwear from $75.00 to $50.00 for a special sale. If 40 swimsuits sold at the reduced price and the remaining 25 swimsuits were returned to the original price after the sale, calculate the total markdowns, markdown cancellations, and net markdown achieved.
Answer:
The original price is = $75
The reduced price = $50
So, price reduced is = [tex]75-50=25[/tex] dollars
Total swimsuits are = [tex]40+25=65[/tex]
Total markdown = [tex]65\times25=1625[/tex] dollars
Now, 25 swimsuits were returned to the original price. Means 25 swimsuits were returned to $75, increasing $25 again.
So, markdown cancellation = [tex]25\times25=625[/tex] dollars
Net markdown = total markdown - markdown cancellation
= [tex]1625-625=1000[/tex] dollars
A television network commissioned a telephone poll of randomly sampled men. Of the 708 respondents who had children, 19% said "yes" to the question "Are you a stay-at-home dad?" To help market commercial time, the network wants an accurate estimate of the true percentage of stay-at-home dads. Construct a 90% confidence interval. left parenthesis nothing % comma nothing % right parenthesis
Answer: (16.6%, 21.4%)
Step-by-step explanation:
The confidence interval for proportion is given by :-
[tex]p\pm z_{\alpha/2}\sqrt{\dfrac{p(1-p)}{n}}[/tex]
Given : Sample size : n= 708
The proportion of respondents who had children = [tex]p=0.19[/tex]
Significance level : [tex]\alpha=1-0.90=0.1[/tex]
Critical value : [tex]z_{\alpha/2}=z_{0.05}=\pm1.645[/tex]
Now, the 90% confidence interval for proportion will be :-
[tex]0.19\pm (1.645)\sqrt{\dfrac{0.19(1-0.19)}{708}}\approx0.19\pm 0.024\\\\=(0.19-0.024,0.19+0.024)=(0.166,\ 0.214)=(16.6\%,\ 21.4\%)[/tex]
Hence, the 90% confidence interval for the proportion = (16.6%, 21.4%)
A survey of 400 randomly selected high school students determined that 68 play organized sports. (a) What is the probability that a randomly selected high school student plays organized sports? (b) Interpret this probability.
Answer: a) The probability that a randomly selected high school student plays organized sports = 0.17
b) The randomly selected high school student is unlikely to play organized sports.
Step-by-step explanation:
Given : A survey of 400 randomly selected high school students determined that 68 play organized sports.
Then , the probability that a randomly selected high school student plays organized sports is given by :-
[tex]\dfrac{\text{Favorable outcomes}}{\text{Total outcomes}}\\\\=\dfrac{68}{400}=0.17[/tex]
In percent , the probability that a randomly selected high school student plays organized sports is 17%.
since 17% lies in the interval of unlikely events (0%, 25%).
It means that the randomly selected high school student is unlikely to play organized sports.
The probability that a randomly selected high school student plays organized sports is 0.17, which means there is a 17% chance, or a 1 in 6 chance, that a randomly picked student plays organized sports.
Explanation:This question relates to the field of Statistics, particularly to the concept of Probability. In this problem, we are provided with a total number of High School students (400), and a number of students who play organized sports (68).
To find the probability, you would divide the number of students who play sports by the total number of students. That is, Probability = Number with desired characteristic (play sports) / Total number In this case, it would be 68 / 400 = 0.17
Interpreting the probability means describing what it represents in practical terms. Here, a probability of 0.17 means that if you were to randomly select a high school student, there is a 17% chance, or around 1 in 6 chance, that this student plays organized sports.
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Find the slope and the y -intercept of the line.
Write your answers in simplest form.
-7x + y = 1
Answer:
The slope is: 7
The y-intercept is: 1
Step-by-step explanation:
The equation of the line in Slope-Intercept form is:
[tex]y=mx+b[/tex]
Where "m" is the slope of the line and "b" is the y-intercept.
To find the slope and the y-intercept of the given line, we can write it in Slope-Intercept form. We can do this by solving for "y".
Then, this is:
[tex]-7x + y = 1\\\\y=7x+1[/tex]
Therefore, you can identify that the slope of this line is:
[tex]m=7[/tex]
And the y-intercept is:
[tex]b=1[/tex]
From a shipment of 65 transistors, 6 of which are defective, a sample of 5 transistors is selected at random.
(a) In how many different ways can the sample be selected?
________ways
(b) How many samples contain exactly 3 defective transistors?
________samples
(c) How many samples do not contain any defective transistors?
________ samples
Answer:
a) 8259888
b) 34220
c) 45057474
Step-by-step explanation:
Given,
The total number of transistor = 65,
In which, the defective transistor = 6,
So, the number of non defective transistor = 65 - 6 = 59,
Since, out of these transistor 5 are selected,
a) Thus, the number of ways = the total possible combination of 5 transistors = [tex]{65}C_ 5[/tex]
[tex]=\frac{65!}{(65-5)!5!}[/tex]
[tex]=8259888[/tex]
b) The number of samples that contains exactly 3 defective transistors = the possible combination of exactly 3 defective transistors = [tex]{6}C_3\times {59}C_2[/tex]
[tex]=\frac{6!}{(6-3)!3!}\times \frac{59!}{(59-2)!\times 2!}[/tex]
[tex]=20\times 1711[/tex]
[tex]=34220[/tex]
c) The number of sample without any defective transistor = The possible combination of 0 defective transistor = [tex]^6C_0\times ^{59}C_5[/tex]
[tex]=1\times 45057474[/tex]
[tex]=45057474[/tex]
(a) 8259888 ways, (b) 34220 samples and (c) 5006386 samples.
Let's solve each part:
(a) There are 65 total transistors, and we want to select 5 of them. So, the number of ways to select a sample is the number of combinations of 65 things taken 5 at a time. This can be calculated using the combination formula:
Number of combinations = [tex]C(n, k) = \frac{n!}{ k! \times (n-k)!)}[/tex]
where:
n is the total number of items (65 transistors)
k is the number of items to select (5 transistors)
Plugging in the values:
[tex]C(65, 5) = \frac{65!}{ 5! \times (65-5)!)} = 8259888[/tex]
(b) We want samples that contain exactly 3 defective transistors. We can achieve this by selecting 3 defective transistors from the 6 available and 2 non-defective transistors from the remaining 59 (65 total - 6 defective).
So, the number of ways to select this specific scenario is:
Number of ways to choose 3 defective transistors out of 6: C(6, 3)
Number of ways to choose 2 non-defective transistors out of 59: C(59, 2)
Using the combination formula again:
[tex]C(6, 3) \times C(59, 2) = (\frac{6!}{ (3! \times 3!)}) \times (\frac{59!} { (2! \times 57!)}) = 34220[/tex]
(c) Samples with no defective transistors simply means selecting all 5 transistors from the 59 non-defective ones.
Therefore:
[tex]C(59, 5) = \frac{59!}{ (5! \times 54!)} = 5006386[/tex]
Let X be a continuous random variable with a uniform distribution on the interval open square brackets 0 comma 10 close square brackets. Find straight P open parentheses straight X less than 1 space or space straight X greater than 8 close parentheses.
[tex]X[/tex] has PDF
[tex]f_X(x)=\begin{cases}\frac1{10}&\text{for }0\le x\le10\\0&\text{otherwise}\end{cases}[/tex]
and thus CDF
[tex]F_X(x)=\begin{cases}0&\text{for }x<0\\\frac x{10}&\text{for }0\le x\le10\\1&\text{for }x>10\end{cases}[/tex]
Because [tex]X[/tex] is continuous, we have
[tex]P(X<1\text{ or }X>8)=1-P(1\le X\le8)=1-(P(X\le8)-P(X\le1))[/tex]
[tex]P(X<1\text{ or }X>8)=1-P(X\le 8)+P(X\le1)[/tex]
[tex]P(X<1\text{ or }X>8)=1-F_X(8)+F_X(1)[/tex]
[tex]P(X<1\text{ or }X>8)=1-\dfrac8{10}+\dfrac1{10}=\boxed{\dfrac3{10}}[/tex]
Compute the entry (the number in the second row and second column) of the product matrix resulting from the following multiplication:
[1 2] [9 6]
[3 4] [5 7]
Answer:
The entry on the second row and second column of the product matrix is [tex]c_{22} = 40[/tex].
Step-by-step explanation:
Let's define as A and B the given matrixes:
[tex]A = \left[\begin{array}{cc}1&2\\3&4\end{array}\right][/tex]
[tex]B = \left[\begin{array}{cc}9&6\\5&7\end{array}\right][/tex]
The product matrix C entry in the first row and first column [tex]c_{1,1}[/tex] or [tex]c_{11}[/tex] can be computer multiplying first row of A by first column of B (see example attached).
The product matrix C entry in the first row and second column [tex]c_{1,2}[/tex] or [tex]c_{12}[/tex] can be computer multiplying first row of A by second column of B.
The product matrix C entry in the second row and first column [tex]c_{2,1}[/tex] or [tex]c_{21}[/tex] can be computer multiplying second row of A by first column of B.
The product matrix C entry in the second row and second column [tex]c_{2,2}[/tex] or [tex]c_{22}[/tex] can be computer multiplying second row of A by second column of B.
Then, let's compute [tex]c_{22}[/tex] by doing the dot product between [3 4] and [6 7]...
[tex]c_{22} = [3 4] . [6 7] = 3*4 + 4*7 = 12 + 28 = 40[/tex]
A doctor at a local hospital is interested in estimating the birth weight of infants. How large a sample must she select if she desires to be 99% confident that the true mean is within 3 ounces of the sample mean? The standard deviation of the birth weights is known to be 5 ounces.
Answer: 18
Step-by-step explanation:
Given : Margin of error = 3 ounces
Significance level : [tex]\alpha = 1-0.99=0.1[/tex]
Critical value : [tex]z_{\alpha/2}=2.576[/tex]
Standard deviation :[tex]\sigma=5\text{ ounces}[/tex]
The formula to calculate the sample size is given by :-
[tex]n=(\dfrac{z_{\alpha/2}\ \sigma}{E})^2[/tex]
[tex]\Rightarrow\ n=(\dfrac{2.576\times5}{3})^2\\\\\Rightarrow\ n=18.4327111111\approx18[/tex]
Hence, the minimum sample size should be 18.
To estimate the birth weight of infants, the doctor must select a sample size of at least 19 infants to be 99% confident that the true mean is within 3 ounces of the sample mean. This calculation is based on the given standard deviation of 5 ounces.
Explanation:To determine the sample size needed to estimate the birth weight of infants, we will use the formula for a margin of error:
Margin of Error = (Z-value) * (Standard Deviation / Square Root of Sample Size)
Since the doctor wants to be 99% confident that the true mean is within 3 ounces of the sample mean, the Z-value will be 2.576 (corresponding to a 99% confidence level). The given standard deviation is 5 ounces. Plugging these values into the formula, we get:
3 = 2.576 * (5 / Square Root of Sample Size)
Solving for the sample size:
Square Root of Sample Size = 2.576 * 5 / 3
Square Root of Sample Size ≈ 4.293
Sample Size ≈ 18.41
Since the sample size must be a whole number, the doctor must select a sample size of at least 19 infants to be 99% confident that the true mean is within 3 ounces of the sample mean.
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Solve for x,W 3x+3W-66 (1) 7 12x+15W-300 (2)
Answer:
x=10 and W=12
Step-by-step explanation:
Let's solve the equations. First we need to understand that the problem can be solved because we have two variables (x, W) and two equations.
Now, we have the following equations:
3x+3W-66 making the equation equal to 0:
3x+3W-66=0 which can be express as:
3x=-3W+66
x=(-3W+66)/3
x=-W+22 (equation 1)
The next equation is:
12x+15W-300 making the equation equal to 0 and then divided by 3:
(12x+15W-300)/3=0 which is:
4x+5W-100=0 (equation 2), using equation 1 we can write:
4(-W+22)+5W-100=0
-4W+88+5W-100=0
W-12=0
W=12
Using W=12 in equation 2 we have:
4x+5W-100=0
4x+5*(12)-100=0
4x+(60)-100=0
4x-40=0
4x=40
x=40/4
x=10
In conclusion the solution for the equations are: x=10 and W=12.
5. Find the general solution to y'''-y''+4y'-4y = 0
For any equation,
[tex]a_ny^(n)+\dots+a_1y'+a_0y=0[/tex]
assume solution of a form, [tex]e^{yt}[/tex]
Which leads to,
[tex](e^{yt})'''-(e^{yt})''+4(e^{yt})'-4e^{yt}=0[/tex]
Simplify to,
[tex]e^{yt}(y^3-y^2+4y-4)=0[/tex]
Then find solutions,
[tex]\underline{y_1=1}, \underline{y_2=2i}, \underline{y_3=-2i}[/tex]
For non repeated real root y, we have a form of,
[tex]y_1=c_1e^t[/tex]
Following up,
For two non repeated complex roots [tex]y_2\neq y_3[/tex] where,
[tex]y_2=a+bi[/tex]
and,
[tex]y_3=a-bi[/tex]
the general solution has a form of,
[tex]y=e^{at}(c_2\cos(bt)+c_3\sin(bt))[/tex]
Or in this case,
[tex]y=e^0(c_2\cos(2t)+c_3\sin(2t))[/tex]
Now we just refine and get,
[tex]\boxed{y=c_1e^t+c_2\cos(2t)+c_3\sin(2t)}[/tex]
Hope this helps.
r3t40
To find the general solution for the differential equation y''' - y'' + 4y' - 4y = 0, we solve the characteristic equation r³ - r² + 4r - 4 = 0, and find the roots r = 1 and r = ±2i. Thus, the general solution is y(x) = C1eˣ + C2cos(2x) + C3sin(2x).
The given differential equation is:
y''' - y'' + 4y' - 4y = 0
First, we find the characteristic equation corresponding to this differential equation. Replace derivatives with powers of r:This is the general solution to the differential equation y''' - y'' + 4y' - 4y = 0.
Consider a sample with data values of 27, 25, 20, 15, 30, 34, 28, and 25. Compute the range, interquartile range, variance, and standard deviation (to a maximum of 2 decimals, if decimals are necessary).
So we have a set of integers,
[tex]X=\{15,20,25,25,27,28,30,34\}[/tex]
And we need to compute some stats.
First the range,
The range of data set is difference between the maximum and the minimum of the data set.
So the minimum is 15 and maximum is 34 therefore the range is 34 - 15 = 19.
Second the interquartile range,
The interquartile range of data set is the difference of the first and third quartiles.
First quartile is the value separating the lower quarter and higher three - quarters of the set. The first quartile is computed by taking median of the lower half of the sorted set. This is 15, 20, 25, 25 and it's median is 22.5
Third quartile is therefore the median of 27, 28, 30, 34 which is 29.
Next up, variance.
The variance of set measures how much data is spread out. For data set [tex]x_1,\dots, x_n[/tex] with an average [tex]a[/tex],
[tex]\mathsf{var}(X)=\Sigma_{i=1}^{n}\dfrac{(x_i-a)^2}{n-1}[/tex]
So first compute the average value which is actually a mean,
[tex]a=\dfrac{1}{2}\Sigma_{i=1}^{n}a_i[/tex]
The sum [tex]\Sigma_{i=1}^{n}a_i[/tex] of numbers in set is 204.
Divide this by number of elements in the set (8).
[tex]a=\dfrac{204}{8}=25.5[/tex]
Then compute the variance,
[tex]\mathsf{var}(X)=\dfrac{\Sigma_{i=1}^{n}(x_i-a)^2}{n-1}\approx\boxed{34.57}[/tex]
And finally standard deviation,
Since have computed variance the standard deviation is nothing too hard. It's defined as a square root of variance,
[tex]\mathsf{sde}(X)=\sqrt{\dfrac{\Sigma_{i=1}^{n}(x_i-a)^2}{n-1}}=\sqrt{34.57}\approx\boxed{5.88}[/tex]
Hope this helps.
r3t40
The values of the statistical measures for the distribution are :
Range = 19Interquartile range = 6.5Variance = 34.57Standard deviation = 5.88Given the data :
27, 25, 20, 15, 30, 34, 28, 25.
Rearranging the values :
15, 20, 25, 25, 27, 28, 30, 34
A.)
The range = Maximum - Minimum
Maximum = 34 ; Minimum = 15
Range = 34 - 15 = 19
B.)
The interquartile range = Upper quartile - Lower quartile
Upper quartile (Q3) = 3/4(n+1)th term
n = number of values = 8
Q3 = 3/4(9) = 6.75 th term = (28+30)/2 = 29
Lower quartile (Q1) = 1/4(n+1)th term
Q1 = 1/4(9) = 2.25th term = (20+25)/2 = 22.5
Interquartile range = 29 - 22.5 = 6.5
C.)
Variance :
Σ[(X - mean)²] / (n-1)
ΣX/ n
Mean = 204 / 8 = 25.5
[(15-25.5)² + (20-25.5)² + (25-25.5)² + (25+25.5)² + (27-25.5)² + (28-25.5)² + (30-25.5)² + (34 - 25.5)²] ÷ (8 - 1)
Variance = 242 / 7
Variance = 34.57
Standard deviation = √variance
Standard deviation = √34.57
Standard deviation = 5.879
Standard deviation = 5.88 (2 decimal place).
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As men and women age, their risk of death increases. There are also health issues that can greatly impair older citizens' abilities to participate in activities and travel. Assume you are planning a 60th high school reunion and are wondering how many of the older adults would be able to attend. If you know that 22 women and 18 men are still alive, and are told that exactly 8 of them will attend, what is the probability that you would have exactly four men and exactly four women attend?
Answer:
The total number of men and women is 42.
So the number of ways of selecting 8 people from 42 is C(42,8) = 118030185.
The number of ways of selecting 4 men from 19 and 4 women from 23 is C(19,4)*C(23,4) = 3876*8855 = 34321980
Therefore, the required probability is 34321980 / 118030185 = 0.2907
Step-by-step explanation:
To determine the probability of exactly four men and four women attending a 60th high school reunion given certain conditions, we use the combination formula. The probability is found by dividing the product of the combinations of 4 men from 18 and 4 women from 22 by the total combinations of 8 people from 40.
Explanation:The question is asking us to find the probability that exactly four men and exactly four women would attend the high school reunion. Considering that there are 40 (22 women + 18 men) people still alive, we are told that exactly 8 will attend the reunion.
We utilise the formula for combinations, to calculate this problem.
So, the possible combinations of 4 men attending from 18 is calculated as C(18,4).
And the possible combinations of 4 women attending from 22 is calculated as C(22,4).
Finally, the total combinations of 8 people attending from 40 is calculated as C(40,8).
Probability is calculated as the desired outcomes over total possibilities. Therefore, the desired probability will be [C(18,4) * C(22,4)] / C(40,8).
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Find a polynomial f(x) of degree 4 that has the following zeros.
0, -4, (multiplicity ), 2
Leave your answer in factored form.
Answer:
[tex]f(x)=(x-2)(x)(x+4)^2[/tex]
Step-by-step explanation:
Since there is a multiplicity at x=-4, that would meant that that part of the factor would have to have an even degree. This means that it would have to be 2.
This would give you [tex]f(x)=(x-2)(x)(x+4)^2[/tex]
Answer:
[tex]f (x) = x (x + 4)^2(x-2)[/tex]
Step-by-step explanation:
The zeros of the polynomial are all the values of x for which the function [tex]f (x) = 0[/tex]
In this case we know that the zeros are:
[tex]x = -4,\ x+4 =0[/tex]
[tex]x = -4,\ x+4 =0[/tex]
[tex]x = 0[/tex]
[tex]x = 2[/tex], [tex]x - 2 = 0[/tex]
Now we can write the polynomial as a product of its factors
[tex]f (x) = x (x + 4)(x+4) (x-2)[/tex]
[tex]f (x) = x (x + 4)^2(x-2)[/tex]
Note that the polynomial is of degree 4 because the greatest exponent of the variable x that results from multiplying the factors of f(x) is 4
The diameters of red delicious apples of an orchard have a normal distribution with a mean of 3 inches and a standard deviation of 0.5 inch. One apple will be randomly chosen. What is the probability of picking an apple with diameter greater than 3.76 inches?
To find the probability of picking an apple with a diameter greater than 3.76 inches from a normally distributed population with a mean of 3 inches and a standard deviation of 0.5 inches, we calculate a z-score and then find the corresponding probability using a z-table or calculator.
Explanation:The question is about finding the probability in a situation that follows a normal distribution, specifically, the likelihood of picking an apple with a diameter greater than 3.76 inches given a mean of 3 inches and a standard deviation of 0.5 inches.
To solve this, we must first calculate the z-score, which is the number of standard deviations a data point (in this case, the apple size of 3.76 inches) is from the mean. The z-score is calculated as follows: Z = (X - μ) / σ, where X is the data point, μ is the mean, and σ is the standard deviation. Substituting the given values, we get Z = (3.76 - 3) / 0.5, which results in a z-score of 1.52.
Next, we will refer to the z-table to find the probability corresponding to this z-score. However, most z-tables give the probability from the mean to our z-score (the left-hand side). But we want the probability that the apple's diameter is greater than 3.76 inches, i.e., the right-hand side of the distribution curve. As a result, we have to subtract the value we get from the z-table from 1 (because the cumulative probability under the entire normal curve is 1).
If a z-table is not readily available, there are many online calculators available to find this probability.
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If a set of sample measurements has a mean of 100, a normal distribution, a standard deviation of 2, and control limits of 94 and 106, what percentage of the samples are expected to be between 94 and 106? Explain your answer.
Answer: 99.73%
Step-by-step explanation:
Given : Mean : [tex]\mu=100[/tex]
Standard deviation : [tex]\sigma=2[/tex]
Let X be the random variable that represents the data values.
Formula for Z-score : [tex]z=\dfrac{X-\mu}{\sigma}[/tex]
For x=94, we have
[tex]z=\dfrac{94-100}{2}=-3[/tex]
For x=106, we have
[tex]z=\dfrac{106-100}{2}=3[/tex]
The probability that the samples are between 94 and 106:-
[tex]P(-3<x<3)=1-2P(z<-3)\\\\=1-2(0.0013499)=0.9973002\approx0.9973=99.73\%[/tex]
Hence, the percent of the samples are expected to be between 94 and 106 = 99.73%
Approximately 95% of the samples are expected to be between 94 and 106.
Explanation:The percentage of samples that are expected to be between 94 and 106 can be determined using the empirical rule, also known as the 68-95-99.7 rule. According to this rule, approximately 68% of the samples will fall within one standard deviation of the mean, and approximately 95% of the samples will fall within two standard deviations of the mean.
In this case, the control limits are 94 and 106, which are one standard deviation below and above the mean, respectively. Therefore, 95% of the samples are expected to be between 94 and 106.
Note that this rule applies to data that follow a bell-shaped, symmetric distribution, which is assumed to be the case for this set of sample measurements.
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find a solution using the power series method
y'+ty=0, y(0)=1
Answer: The required solution is
[tex]y=(t-\dfrac{t^3}{2}+~~.~~.~~.).[/tex]
Step-by-step explanation: We are given to find the solution of the following differential equation using power series method :
[tex]y^\prime+ty=0,~~y(0)=1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
Let [tex]y=\sum_{n=0}^{\infty}a_nq^n[/tex] be the solution of the given equation.
Then, we have
[tex]y^\prime=\sum_{n=1}^{\infty}a_nnt^{n-1}.[/tex]
From equation (i), we get
[tex]\sum_{n=1}^{\infty}a_nnt^{n-1}+t\sum_{n=0}^{\infty}a_nt^n=0\\\\\\\Rightarrow \sum_{n=1}^{\infty}a_nnt^{n-1}+t\sum_{n=-1}^{\infty}a_nt^{n+1}=0.[/tex]
Comparing the coefficients of [tex]t^n,~t^{n+1},~,~~.~~.~~.[/tex] from both sides of the above, we get
[tex]2a_2+a_0\\\\\Rightarrow a_2=-\dfrac{a_0}{2},\\\\\\3a_3+a_1=0\\\\\Rightarrow a_3=-\dfrac{a_1}{3},\\\\\\\vdots~~~\vdots~~~\vdots[/tex]
Therefore, we get
[tex]y=a_0(1-\dfrac{1}{2}t^2+~~.~~.~~.)+a_1(t-\dfrac{t^3}{2}+~~.~~.~~.).[/tex]
The condition y(0) = 1 gives
[tex]a_0=0.[/tex]
So, the required solution is
[tex]y=t-\dfrac{t^3}{2}+~~.~~.~~.[/tex]
Answer with explanation:
The diiferential equation of first order is,
y'+t y=0
[tex]y=\sum_{n=0}^{\infty}a_{n}x^n\\\\y'=\sum_{n=1}^{\infty}na_{n}x^{n-1}\\\\y'+ty=\sum_{n=0}^{\infty}na_{n-1}x^{n-1}+t\sum_{n=0}^{\infty}a_{n}x^n\\\\=\sum_{n=0}^{\infty}(a_{n-1}\frac{n}{x}+a_{n}t)x^n\\\\\rightarrow y'+t y=0\\\\\rightarrow a_{n-1}\frac{n}{x}+a_{n}t=0\\\\a_{n-1}=\frac{-xta_{n}}{n}[/tex]
For, n=1
[tex]a_{0}=\frac{-xta_{1}}{1}\\\\a_{0}=-txa_{1}\\\\a_{1}=\frac{-a_{0}}{tx}\\\\\text{for}, n=2\\\\a_{1}=\frac{-xta_{2}}{2}\\\\a_{2}=\frac{-2a_{1}}{xt}\\\\=\frac{2a_{0}}{x^2t^2}\\\\\text{for}, n=3\\\\a_{2}=\frac{-xta_{3}}{3}\\\\a_{3}=\frac{-6a_{0}}{x^3t^3}\\\\a_{4}=\frac{24a_{0}}{x^4t^4}\\\\a_{n}=(-1)^n\frac{n!a_{0}}{x^nt^n}[/tex]
So,the series can be written as
[tex]y_{n}=\sum_{n=1}^{\infty}(-1)^n\frac{n!a_{0}}{x^nt^n}[/tex]
Let A be a 5 × 4 matrix with a pivot in each column. Is A invertible? Why or why not?
Answer:
Matrix A is not invertible.
Step-by-step explanation:
Invertible matrix are those matrix which are square matrix or non singular matrix.
For any matrix to be invertible, matrix should be non singular i.e. det(x)[tex]\neq[/tex] 0.
But for the question given above we cannot find determinant of matrix A as it is not square matrix. so inverse of given matrix does not exist. so it is not possible to have non trival solutions.