Step-by-step answer:
Given:
class time uniformly distributed between 48 and 53 minutes (range 5 minutes)
Find probability that the class period runs between 51.5 and 51.75 minutes.
Solution:
Uniformly distributed means that the probability that the class time is the same for any minute (out of 5), i.e. 20%, or probability density is 0.2 / minutes, no matter which minute.
The interval between 51.5 and 51.75 is 0.25 minutes, so the probability of class period within that range is 0.25 minutes * 0.2 / minute = 0.05.
Final answer:
The probability that a class period runs between 51.5 and 51.75 minutes, when class lengths are uniformly distributed between 48.0 and 53.0 minutes, is 0.05 or 5%.
Explanation:
To find the probability that a given class period runs between 51.5 and 51.75 minutes when class lengths are uniformly distributed between 48.0 and 53.0 minutes, we utilize the properties of the uniform distribution. The probability is the area under the uniform probability density function (PDF) between the two specified lengths. For a uniform distribution, this area corresponds to the length of the interval divided by the length of the entire interval in which the variable is uniformly distributed.
Here, we're looking for the probability between 51.5 minutes and 51.75 minutes. The total length of the available interval is 53.0 - 48.0 = 5 minutes. The length of the desired interval is 51.75 - 51.5 = 0.25 minutes. So, we divide the length of the desired interval by the total interval to get the probability.
Probability = (51.75 - 51.5) / (53.0 - 48.0) = 0.25 / 5 = 0.05
Therefore, the probability that a class period runs between 51.5 and 51.75 minutes is 0.05 or 5%.
If a 42-piece set of stainless steel flatware costs $113.40 at a certain store, what is the cost (in dollars) per piece?
Answer:$2.7 dollars per piece
Step-by-step explanation: Divide 113.40 by 42
$2.7 per piece is the answer.
$113.40 / 42 piece set of stainless steel = $2.7
What is problem-solving?Problem-solving is the act of defining a problem; figuring out the purpose of the trouble; identifying, prioritizing, and selecting alternatives for an answer; and imposing an answer.
What are problem-solving and examples?Problem-solving starts with identifying the issue. As an example, a trainer may need to parent out a way to improve a scholar's overall performance on a writing talent test. To do that, the instructor will overview the writing exams looking for regions for improvement.
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Solve the inequality and graph the solution on a number line.
–3(5y – 4) ≥ 17
please show work!
Answer:
Step-by-step explanation:
In order to solve the inequality, follow the simple steps:
–3(5y – 4) ≥ 17 .
Dividing both sides with -3:
5y - 4 ≤ -17/3 (the sign of the inequality becomes opposite whenever a negative number is either multiplied or divided on both the sides of the inequality).
Adding 4 on both sides:
5y ≤ -5/3
Dividing 5 on both sides:
y ≤ -1/3.
This shows that all the values of y less than and equal to -1/3 satisfy the inequality.
The number line has been attached. Since it involves ≤ sign, therefore, a filled circle will be used to plot the inequality. Less than means that all the values on the left hand side of the number line will be included. This is denoted with an arrow (see the diagram)!!!
Pierre sold 340 tickets for a concert. Balcony tickets cost $5 while tickets for the lower floor cost $10. If Pierre sold $2,700 worth of tickets, how many lower floor tickets did he sell?
Answer:
He sold 200 lower floor tickets.
Step-by-step explanation:
Consider the provided information.
Pierre sold 340 tickets for a concert.
Let Balcony tickets represents by x and lower floor ticket represents by y.
As we know he sold 340 tickets in total, i.e.
x + y = 340 ......(1)
Balcony tickets cost $5 while tickets for the lower floor cost $10.
If he sold x balcony tickets, so the money he got after selling x balcony tickets is 5x. Similarly, If he sold y lower floor tickets, so the money he got after selling y lower floor tickets is 10y.
Pierre sold $2,700 worth of tickets, i.e.
5x + 10y = 2700 ......(2)
Now use elimination method to solve the system of equations:
Multiply the equation (1) with 5 and subtract them as shown below:
5x + 5y = 1700
5x + 10y = 2700
_____________
-5y = -1000
y = 200
Thus, he sold 200 lower floor tickets.
1. Assume that R and S are symmetric relations on a set A. Prove that Rns is symmetric.
Answer with explanation:
Suppose, A={(a,b),(b,a), (c,d),(d,c),(p,q),(q,p),(a,a),(b,b)}
A Relation M is Symmetric , if (p,q)∈M , then (q,p)∈M.
⇒It is given that, R and S are symmetric Relation on a Set A.
⇒If R is symmetric, then if (a,b)∈R, means,(b,a)∈R.So, R={(a,b),(b,a)}.
⇒If S is Symmetric, then if (c,d)∈S, means,(d,c)∈S.So, S={(c,d),(d,c)}.
⇒R ∩ S ={(a,b),(b,a),(c,d),(d,a)}
⇒If you will look at the elements of Set , R∩S, there is (a,b)∈ R∩S,so as (b,a)∈ R∩S.Also, (c,d)∈ R∩S,so as (d,a)∈ R∩S.
Which shows Relation in the set , R∩S is symmetric.
Find the general solution to 2y ′′ − y ′ − y = 0.
Answer: y(x) = [tex]C_{1} e^{x} + C_{2} e^{\frac{-x}{2} }[/tex]
Step-by-step explanation:
2y ′′ − y ′ − y = 0
The characteristic equation is:
[tex]2r^{2} - r - 1 = 0[/tex]
[tex]2r^{2} - 2r + r - 1 = 0[/tex]
2r(r-1) + 1(r-1) = 0
(r-1)(2r+1) = 0
[tex]r_{1} = 1 , r_{2} = \frac{-1}{2}[/tex]
∴ there are two distinct roots
so the general equation is as follows:
y(x) = [tex]C_{1} e^{r_{1}x } + C_{2} e^{r_{2}x }[/tex]
y(x) = [tex]C_{1} e^{x} + C_{2} e^{\frac{-x}{2} }[/tex]
If the area under the standard normal curve to the left of zequalsminus1.72 is 0.0427, then what is the area under the standard normal curve to the right of zequals1.72?
Answer: 0.0427
Step-by-step explanation:
Given : The area under the standard normal curve to the left of z = -1.72 is 0.0427
We know that the normal curve is a bell shaped curve that is symmetric such that half of the data falls to the left of the mean (Mean lies at the middle of the curve) and half of data falls to the right.
Now, z=-1.72 lies on the left side and z=1.72 lies right side.
Since, normal curve is symmetric and the magnitude of the values if same , then the area under the standard normal curve to the left of z = -1.72 is equals to the area to the right of z = 1.72 is 0.0427
Therefore, the area under the standard normal curve to the right of z = 1.72 is 0.0427
The area under the standard normal curve to the right of z=1.72 is 0.9573 or 95.73%, which can be calculated by subtracting the area to the left of z from 1 (1 - 0.0427 = 0.9573).
Explanation:The area under the standard normal curve to the left of z=-1.72 is 0.0427. This essentially means that 4.27% of all observations fall under this score. Conversely, the area to the right of the z-score represents the proportion of observations that are greater than z. In a normal distribution, the sum of the areas to the left and to the right of a z-score must always equal to 1 (0.0427 + x = 1), because all possible outcomes are accounted for by a normal distribution. Hence, we can find out the area to the right by simply subtracting the area to the left from 1. So, the area under the standard normal curve to the right of z=1.72 is 1 - 0.0427 = 0.9573 or 95.73%.
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discrete math:
find the rule for determining when a number is divisible by 11.
Answer:
Step-by-step explanation:
Divisibility rule when a number is divisible by 11:
Take the alternating sum of digits in a number read from left to right .If that is divisible by 11 then the the given number is divisible by 11.
It is a divisibility rule of 11.
Let 1342 is a number .The number is divisible by 11 or not
Alternating sum of digits =1-3+4-2=0
The alternating sum of digits is divisible by 11 .
Therefore, the number 1342 is divisible by 11.
Let a number 2728
Alternating sum of digits = 2-7+2-8=-11.
The alternating sum of digits is divisible by 11 .
Therefore, the number 2728 is divisible by 11.
The rule for determining divisibility by 11 involves alternately adding and subtracting a number's digits; if the result is 0 or divisible by 11, the original number is too. Examples illustrate the application of the divisibility test.
Rule for Divisibility by 11
The rule for determining when a number is divisible by 11 is quite straightforward. To apply the rule, you alternate between adding and subtracting the digits of the number in question from left to right. If the resulting sum (which may be negative) is 0 or divisible by 11 itself, then the original number is divisible by 11. For example, consider the number 2728. Apply the rule like this: 2 - 7 + 2 - 8 = -11. Since -11 is divisible by 11, 2728 is also divisible by 11.
Examples
Another example: Take the number 5831. Using the rule: 5 - 8 + 3 - 1 = -1, which is not divisible by 11, hence 5831 is not divisible by 11.
This divisibility test is a fantastic example of how certain mathematical rules are used to quickly determine properties of numbers without needing to perform long divisions.
Working together, two secretaries can stuff the envelopes for a political fund-raising letter in 3 hours. Working alone, it takes the slower worker 8 hours longer to do the job than the faster worker. How long does it take each to do the job alone?
Answer:
Faster worker takes 4 hours and slower worker takes 12 hours.
Step-by-step explanation:
Let x be the time ( in hours ) taken by faster worker,
So, according to the question,
Time taken by slower worker = (x+8) hours,
Thus, the one day work of faster worker = [tex]\frac{1}{x}[/tex]
Also, the one day work of slower worker = [tex]\frac{1}{x+8}[/tex]
So, the total one day work when they work together = [tex]\frac{1}{x}+\frac{1}{x+8}[/tex]
Given,
They take 3 hours in working together,
So, their combined one day work = [tex]\frac{1}{3}[/tex]
[tex]\implies \frac{1}{x}+\frac{1}{x+8}=\frac{1}{3}[/tex]
[tex]\frac{x+8+x}{x^2+8x}=\frac{1}{3}[/tex] ( Adding fractions )
[tex]3(2x+8)=x^2+8x[/tex] ( Cross multiplication )
[tex]6x+24=x^2+8x[/tex] ( Distributive property )
[tex]x^2+2x-24=0[/tex] ( Subtraction property of equality )
By quadratic formula,
[tex]x=\frac{-2\pm \sqrt{100}}{2}[/tex]
[tex]x=\frac{-2\pm 10}{2}[/tex]
[tex]\implies x=4\text{ or }x=-6[/tex]
Since, hours can not negative,
Hence, time taken by faster worker = x hours = 4 hours,
And, the time taken by slower worker = x + 8 = 12 hours.
To solve this work rate problem, we set up an equation with combined work rates and find that the faster worker takes 3 hours alone, while the slower worker takes 11 hours alone.
Explanation:The question states that two secretaries can stuff envelopes together in 3 hours. The slower worker takes 8 hours more than the faster worker to complete the job alone. To find how long it takes each secretary to complete the job alone, we can set up an equation using the reciprocal of their work rates.
Let x be the time it takes for the faster worker to stuff the envelopes alone. Then, the slower worker will take x + 8 hours. The work rate of the faster worker is 1/x and the slower worker's rate is 1/(x + 8). Working together, their combined work rate is 1/3 per hour (since they complete the task in 3 hours).
The combined work rate equation will be:
1/x + 1/(x + 8) = 1/3
To solve this equation:
Multiply every term by the common denominator, which is 3x(x + 8).This gives us 3(x + 8) + 3x = x(x + 8).Simplify and solve the resulting quadratic equation.This results in x² + 8x - 3x - 24 = 0, which simplifies to x² + 5x - 24 = 0.Factor the quadratic equation to find the values of x.The factors of 24 that add up to 5 are 8 and -3, so (x + 8)(x - 3) = 0.Therefore, x = 3 or x = -8. Since time cannot be negative, we disregard x = -8.The faster worker takes 3 hours and the slower worker takes 3 + 8 = 11 hours.As a result, the faster worker takes 3 hours to complete the job alone, and the slower worker takes 11 hours to complete the job alone.
In a certain study comma the chance of encountering a car crash on the roadstudy, the chance of encountering a car crash on the road is stated as 66%. Express the indicated degree of likelihood as a probability value between 0 and 1 inclusive.
Answer:
The probability is 0.66.
Step-by-step explanation:
Given,
The chance of encountering a car crash on the road is stated as 66%,
That is, out of 100% cases the percentage of the number of car crash cases is 66%,
⇒ Total outcomes = 100%, favourable outcome = 66 %
So, the probability of occurrence a car crash = [tex]\frac{66\%}{100\%}[/tex]
[tex]=\frac{66/100}{100/100}[/tex]
[tex]=\frac{66}{100}[/tex]
[tex]=0.66[/tex]
Where, 0 < 0.66 < 1.
Hence, the indicated degree of likelihood as a probability value between 0 and 1 inclusive is 0.66.
Find the area of the region trapped between LaTeX: y=1-2x^2 y = 1 − 2 x 2 and LaTeX: y=\left|x\right| y = | x | , shown above. The answer is LaTeX: \frac{A}{12} A 12 . Below, enter only the whole number LaTeX: A A .
The area is given by the integral,
[tex]\displaystyle\int_{-1/2}^{1/2}(1-2x^2-|x|)\,\mathrm dx[/tex]
The integrand is even, so we can simplify the integral somewhat as
[tex]\displaystyle2\int_0^{1/2}(1-2x^2-|x|)\,\mathrm dx[/tex]
When [tex]x\ge0[/tex], we have [tex]|x|=x[/tex], so this is also the same as
[tex]\displaystyle2\int_0^{1/2}(1-2x^2-x)\,\mathrm dx[/tex]
which has a value of
[tex]2\left(x-\dfrac23x^3-\dfrac12x^2\right)\bigg|_0^{1/2}=2\left(\dfrac12-\dfrac1{12}-\dfrac18\right)=\boxed{\dfrac7{12}}[/tex]
so that A = 7.
On their first date, Kelly asks Mike to guess the date of her birth, not including the year. Complete parts a through c below. a. What is the probability that Mike will guess correctly? (Ignore leap years.)
Answer:
The probability that mike will guess correctly is 0.0027397 or [tex]\frac{1}{365}[/tex].
Step-by-step explanation:
Consider the provided information.
The number of days in a year is 365 (Ignore leap years).
[tex]Probability = \frac{favorable\ outcomes}{possible\ outcomes}[/tex]
Here, favorable outcomes is 1 and total number of outcomes are 365.
Substitute these value in above formula.
[tex]Probability = \frac{1}{365}[/tex]
[tex]Probability = 0.0027397[/tex]
Thus, the probability that mike will guess correctly is 0.0027397 or [tex]\frac{1}{365}[/tex].
Probability of an event represents the chances of occurrence of that event.
The probability that Mike will guess Kelly's birth date correctly (ignoring leap years) is [tex]\dfrac{1}{365} \approx 0.00027[/tex]
How to calculate the probability of an event?Suppose that there are finite elementary events in the sample space of the considered experiment, and all are equally likely.
Then, suppose we want to find the probability of an event E.
Then, its probability is given as
[tex]P(E) = \dfrac{\text{Number of favorable cases}}{\text{Number of total cases}}[/tex]
Where favorable cases are those elementary events who belong to E, and total cases are the size of the sample space.
Since we know that in a year (non leap year), there are 365 days, and Kelly's birthday can be on any one of those date, so the total number of days to chose from is 365 and since birthday of Kelly is going to be on single day of whole year, so favorable case is only single.
Thus,
if E = Selecting Kelly's birth date correctly,
Then
[tex]P(E) = \dfrac{1}{365} \approx 0.00027[/tex]
Thus,
The probability that Mike will guess Kelly's birth date correctly (ignoring leap years) is [tex]\dfrac{1}{365} \approx 0.00027[/tex]
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Arianna kicks a soccer ball off the ground and into the air with an initial velocity of 42 feet per second. Assume the starting height of the ball is 0 feet. Approximately what maximum height does the soccer ball reach? 1.3 ft 2.6 ft 26.0 ft 27.6 ft
Answer:
Option D.
Step-by-step explanation:
Arianna kicks a soccer ball into the air with an initial velocity = 42 feet per second
If the starting height of the ball is 0 feet then we have to find the approximate maximum height the soccer ball attached.
Since we know vertical motion is represented by
v² = u² - 2gh
Where v = final velocity of the object
u = initial velocity of the object
g = gravitational force
h = h = maximum height achieved by the object
Here v = 0, u = 42 feet per second and g = 32 feet per second²
Now we plug in these values in the formula
0 = (42)² - 2(32)(h)
1764 = 64h
[tex]h=\frac{1764}{64}[/tex]
h = 27.56 feet
≈ 27.60 feet
Therefore, maximum height achieved by the soccer ball is 27.6 feet.
Option D is the answer.
Answer:
Option D is 2.76 ft
Step-by-step explanation:
Its correct
consider the function represented by the table
the ordered pair given in the bottom row can be written using function notation as,
a) f(9)=5
b) f(5)=9
c) f(5,9)=14
d) f(9,5)=14
Answer:
f(9) =5
Step-by-step explanation:
We have an input x and an output f(x)
The input in the last row is 9 and the output is 5
The input x=9 and the output f(9) =5
f(9) =5
The correct option is (a) because [tex]f(9)=5[/tex].
Important information:
The given table represents the ordered pairs.Function notation:The function notation of a point [tex](x,f(x))[/tex] or [tex](x,y)[/tex] is [tex]f(x)=y[/tex].
Using the definition of function notation. The function notation of given ordered pairs [tex](2,6),(7,3)[/tex] and [tex](9,5)[/tex] are:
[tex]f(2)=6[/tex]
[tex]f(7)=3[/tex]
[tex]f(9)=5[/tex]
Thus, option (a) is correct and the other options are incorrect.
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Determine whether f(x) = –5x^2 – 10x + 6 has a maximum or a minimum value.
Find that value and explain how you know
First,
We are dealing with parabola since the equation has a form of,
[tex]y=ax^2+bx+c[/tex]
Here the vertex of an up - down facing parabola has a form of,
[tex]x_v=-\dfrac{b}{2a}[/tex]
The parameters we have are,
[tex]a=-5,b=-10, c=6[/tex]
Plug them in vertex formula,
[tex]x_v=-\dfrac{-10}{2(-5)}=-1[/tex]
Plug in the [tex]x_v[/tex] into the equation,
[tex]y_v=-5(-1)^2-10(-1)+6=11[/tex]
We now got a point parabola vertex with coordinates,
[tex](x_v, y_v)\Longrightarrow(-1,11)[/tex]
From here we emerge two rules:
If [tex]a<0[/tex] then vertex is max valueIf [tex]a>0[/tex] then vertex is min valueSo our vertex is minimum value since,
[tex]a=-5\Longleftrightarrow a<0[/tex]
Hope this helps.
r3t40
Find the general solution to y'" - y" + 2y = 0
Answer: The required general solution is
[tex]y(x)=Ae^{-x}+e^x(B\cos x+C\sin x).[/tex]
Step-by-step explanation: We are given to find the general solution of the following differential equation :
[tex]y^{\prime\prime\prime}-y^{\prime\prime}+2y=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
Let y = y(x) and [tex]y=e^{mx}[/tex] be an auxiliary solution of equation (i).
Then, we have
[tex]y^\prime=me^{mx},~~~y^{\prime\prime}=m^2e^{mx},~~~y^{\prime\prime\prime}=m^3e^{mx}.[/tex]
Substituting these values in equation (i), we have
[tex]m^3e^{mx}-m^2e^{mx}+2e^{mx}=0\\\\\Rightarrow (m^3-m^2+2)e^{mx}=0\\\\\Rightarrow m^3-m^2+2=0~~~~~~~~~~~~~~~~~~~~~~~~~~[\textup{since }e^{mx}\neq0]\\\\\Rightarrow m^2(m+1)-2m(m+1)+2(m+1)=0\\\\\Rightarrow (m+1)(m^2-2m+2)=0\\\\\Rightarrow m+1=0~~~~~\Rightarrow m=-1[/tex]
and
[tex]m^2-2m+2=0\\\\\Rightarrow (m^2-2m+1)+1=0\\\\\Rightarrow (m-1)^2=-1\\\\\Rightarrow m-1=\pm\sqrt{-1}\\\\\Rightarrow m=1\pm i~~~~~~~~~~~~~~~~~~~~~~~~~~~[\textup{where }i^2=-1][/tex]
So, we get
[tex]m_1=-1,~~m_2=1+i,~~m_3=1-i.[/tex]
Therefore, the general solution of the given equation is given by
[tex]y(x)=Ae^{m_1x}+e^{1\times x}(B\cos 1x+C\sin 1x)}\\\\\Rightarrow y(x)=Ae^{-x}+e^x(B\cos x+C\sin x).[/tex]
Thus, the required general solution is
[tex]y(x)=Ae^{-x}+e^x(B\cos x+C\sin x).[/tex]
Which of the following is equivalent to the formula =C2+C3+C4+C7?
=SUM(C2:C7)
=SUM(C2,C4:C7)
=SUM(C2, C3, C4:C7)
=SUM(C2:C4, C7)
Answer:
=SUM(C2:C4, C7)
Step-by-step explanation:
In excel 'C' represent the cell ( intersection of row and column ),
Also, '=sum( )' represents the function of addition in the excel,
Also, when we operate all consecutive cells from [tex]Cn[/tex] to [tex]Cm[/tex]
Then under the function formula we write [tex]Cn:Cm[/tex]
And, when we operate a cell [tex]Cn[/tex] with non consecutive cell [tex]Cl[/tex]
Then under the function formula we write [tex]Cn, Cl[/tex]
Hence, C2+C3+C4+C7 can be written as,
=SUM(C2:C4, C7)
PLEASE HELP THIS SHOULD BE MY LAST ONE
Answer:
g(-4) = -4
g(-2) = 1
g(1) = -4
Step-by-step explanation:
The value of the given function is -4 for all values of x other than -2 and 1 if x=-2
So,
For x=-4 the value of function will be -4.
g(-4) = -4
For x=-2
The value of function is -2.
g(-2) = 1
And for x=1, the value will be -4.
g(1) = -4 ..
if abc=1 prove that 1/(1+a+b^-1) + 1/(1+b+c^-1) +1/(1+c+a^-1) =1
Answer with explanation:
It is given that, abc=1
[tex]\rightarrow \frac{1}{1+a+b^{-1}}+\frac{1}{1+b+c^{-1}}+\frac{1}{1+c+a^{-1}}\\\\\rightarrow \frac{b}{b+ab+1}+\frac{c}{c+bc+1}+\frac{a}{a+ac+1}\\\\abc=1\\\\\rightarrow \frac{b}{b+ab+abc}+\frac{c}{c+bc+abc}+\frac{a}{a+ac+abc}\\\\\rightarrow \frac{1}{1+a+ac}+\frac{1}{1+b+ab}+\frac{1}{1+c+bc}\\\\\rightarrow \frac{1}{abc+a+ac}+\frac{1}{1+b+ab}+\frac{1}{1+c+bc}\\\\\rightarrow \frac{1+a}{a(bc+1+c)}+\frac{c}{c+bc+1}\\\\\rightarrow\frac{1+a+ac}{a(bc+1+c)}\\\\\rightarrow\frac{1+a+ac}{abc+a+ac)}\\\\\rightarrow\frac{1+a+ac}{1+a+ac)}\\\\=1[/tex]
Hence proved.
Answer:
ABC=1
Step-by-step explanation:
Madison is picking out some movies to rent, and she has narrowed down her selections to 7 foreign films, 3 horror films, 6 action movies, and 4 dramas. How many different combinations of 9 movies can she rent if she wants all 4 dramas?
Final answer:
Madison can choose from 4368 different combinations of 9 movies, including all 4 dramas and 5 out of the 16 non-drama movies.
Explanation:
Madison is picking out movies to rent and wants all 4 dramas out of her 9 selections. With 7 foreign films, 3 horror films, 6 action movies, and 4 dramas, the total number of non-drama movies is 16 (7+3+6). Since she must select all 4 dramas, she needs to choose the remaining 5 movies from the 16 non-drama movies available.
To find how many different combinations of 9 movies she can rent, we use the combination formula C(n, r) = n! / [r!(n-r)!], where n is the total number of items to choose from, r is the number of items to choose, and ! denotes factorial, representing the product of all positive integers up to that number.
The question asks us to calculate C(16, 5) because Madison is choosing 5 movies out of the 16 non-drama movies. This calculation yields:
C(16, 5) = 16! / [5!(16-5)!] = 4368
Therefore, there are 4368 different combinations of 9 movies Madison can rent if she wants all 4 dramas.
A grocery bag can be classified as either paper or plastic. Suppose that 97% of grocery bags are classified as plastic. (a) Two grocery bags are chosen at random. What is the probability that both grocery bags are plastic? (b) Five grocery bags are chosen at random. What is the probability that all five grocery bags are plastic? (c) What is the probability that at least one of five randomly selected grocery bags is paper? Would it be unusual that at least one of five randomly selected grocery bags is paper?
Answer with explanation:
Given : The probability of grocery bags are classified as plastic = 0.97
(a) f two bags are chosen at random.
Then , the probability that both grocery bags are plastic is given by :-
[tex]^2C_2(0.97)^2(1-0.97)^0=0.9409[/tex]
(b) If five grocery bags are chosen at random.
Then , the probability that all five grocery bags are plastic is given by :-
[tex]^5C_5(0.97)^5(1-0.97)^0\approx0.8587[/tex]
(c) The probability of getting paper = 1-0.97=0.03
The probability that at least one of five randomly selected grocery bags is paper :-
[tex]P(x\geq1)=1-P(0)\\\\1-^5C_0(0.03)^0(0.97)^5\\\\=1-(0.03)^0(0.97)^5=0.1412659\approx0.14>0.05[/tex]
Thus , it would not be unusual that at least one of five randomly selected grocery bags is paper.
Probability is used to determine the chances of an event
The probability that both grocery bags are plastic is 0.9409The probability that all five grocery bags are plastic is 0.8587The probability that at least one grocery bags is paper is 0.1413It would not be unusual that at least one of five randomly selected grocery bags is paper.[tex]p = 97\%[/tex] -- chances that a bag is plastic
(a) Both plastics selected are plastic
This is calculated as:
[tex]P(2) = p^2[/tex]
So, we have:
[tex]P(2) = (97\%)^2[/tex]
[tex]P(2) = 0.9409[/tex]
Hence, the probability that both grocery bags are plastic is 0.9409
(b) All five are plastic
This is calculated as:
[tex]P(5) = p^5[/tex]
So, we have:
[tex]P(5) = (97\%)^5[/tex]
[tex]P(5) = 0.8587[/tex]
Hence, the probability that all five grocery bags are plastic is 0.8587
(c) At least one of the five is paper
The probability that none of the five is paper is the same as the probability that all five is plastic.
So:
[tex]P(None) = 0.8587[/tex]
Using complement rule,
[tex]P(At\ least\ 1) = 1 - P(None)[/tex]
So, we have
[tex]P(At\ least\ 1) = 1 - 0.8587[/tex]
[tex]P(At\ least\ 1) = 0.1413[/tex]
Hence, the probability that at least one grocery bags is paper is 0.1413
The above probability is greater than 0.05
Hence, it would not be unusual that at least one of five randomly selected grocery bags is paper.
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20. Use Barrow's a, e method to determine the slope of the tangent line to the curve x^3 + y^3=C^3
Answer: Slope would be,
[tex]-\frac{x^2}{y^2}[/tex]
Step-by-step explanation:
Here, the given curve,
[tex]x^3 + y^3=C^3[/tex]
[tex]\implies x^3 + y^3 - C^3=0[/tex]
In Barrow's method,
Steps are as follows,
Step 1 : put, x = x - e, y = y - a
[tex](x-e)^3 + (y-a)^3 - C^3=0[/tex]
[tex]x^3-3x^2e+3xe^2-e^3+y^3-3y^2a+3ya^2-a^3+C^3=0[/tex]
Step 2 : Reject terms which do not contain a or e,
[tex]-3x^2e+3xe^2-e^3-3y^2a+3ya^2-a^3=0[/tex]
Step 3 : Reject all terms in which a or e have exponent greater than 1,
[tex]-3x^2e-3y^2a=0[/tex]
Step 4 : Find the ratio of a : e,
[tex]-3y^2a=3x^2e[/tex]
[tex]\implies \frac{a}{e}=-\frac{x^2}{y^2}[/tex]
Hence, the slope of the given curve is [tex]-\frac{x^2}{y^2}[/tex]
A client has an order for 1000 mL 0.45 Saline to be infused over 8 hours. How many mL/hr should the IV pump be programmed for?
Answer:
125 mL/h
Step-by-step explanation:
Intravenous pumps are used when there is a need of constant pumping of fluids into the body. Injecting every hour with the required volume causes damage.
Volume of 0.45 saline = 1000 mL
Time to be infused = 8 hours
The Intravenous pump should be programmed in the following way
[tex]\frac{Volume}{time}=\frac{1000}{8}\\\Rightarrow flow\ rate=125\ mL/hr[/tex]
Hence, the IV pump should be programmed for 125 mL/hr.
Graduation is 3 years away and you want to have $850 available for a trip. If your bank is offering a 3-year CD (certificate of deposit) paying 2% simple interest, how much do you need to put in this CD to have the money for your trip?
Answer:
We need to put $801.88 amount in the bank.
Step-by-step explanation:
given that
t=3 yr
Need amount $850 after 3 yr so P= $850
Interest rate=2%
We know that for simple interest
[tex]P=A\left(1+\dfrac{rt}{100}\right)[/tex]
Where r is the Interest rate,t is the time,A is the present amount and P is principle amount after t time.
Here given that P= $850
So now putting the values
[tex]850=A\left(1+\dfrac{2\times 3}{100}\right)[/tex]
So A=$801.88
We need to put $801.88 amount in the bank.
Thorium 234 is a radioactive substance that decays at a rate proportional to the amount present. 1 gram of this material is reduced to 0.8 grams in one week
(a) Find an expression that expresses the amount of Thorium 234 present at any time
(b) Find half life of Thorium 234
(c) Find amount of Thorium 234 present after 10 weeks
Answer: a) [tex]t=\frac{2.303}{k}\log\frac{a}{a-x}[/tex]
b) 3.15 weeks.
c) 0.11 grams
Step-by-step explanation:
a) radioactive decay follows first order kinetics and thus Expression for rate law for first order kinetics is given by:
[tex]k=\frac{2.303}{t}\log\frac{a}{a-x}[/tex]
where,
k = rate constant = ?
t = time taken for decomposition = 1 week
a = initial amount of the reactant = 1 g
a - x = amount left after decay process = 0.8 g
Now put all the given values in above equation, we get
[tex]k=\frac{2.303}{1}\log\frac{1}{0.8}[/tex]
[tex]k=0.22weeks^{-1}[/tex]
2) To calculate the half life, we use the formula :
[tex]t_{\frac{1}{2}=\frac{0.693}{k}[/tex]
[tex]t_{\frac{1}{2}=\frac{0.693}{0.22}=3.15weeks[/tex]
Thus half life of Thorium 234 is 3.15 weeks.
3) amount of Thorium 234 present after 10 weeks:
[tex]10=\frac{2.303}{0.22}\log\frac{1}{a-x}[/tex]
[tex](a-x)=0.11g[/tex]
Thus amount of Thorium 234 present after 10 weeks is 0.11 grams
Suppose that a movie theater snack bar turns over its inventory of candy 3.3 times per month. (Round your answer to 2 decimal places.) If the snack bar has an average of 340 boxes of candy in inventory, what is its average daily sales rate for candy
Answer:
The average daily sales rate for candy is 37.4.
Step-by-step explanation:
We know that,
[tex]\text{Inventory turnover }= \frac{\text{Total sale}}{\text{Average inventories}}[/tex]
[tex]\implies \text{Total sale}=\text{Inventory turnover }\times \text{Average inventories}[/tex]
Given,
Inventory turnover of candies = 3.3,
Average inventories = 340
So, sale of candies = 3.3 × 340 = 1122
Now,
[tex]\text{Average daily sales rate}=\frac{\text{total sale}}{\text{Number of days}}[/tex]
Since, 1 month = 30 days ( approx ),
Hence, the average daily sales rate for candy = [tex]\frac{1122}{30}[/tex]=37.4
In a study of the relationship between geographical mobility (number of times a person has changed residences) and number of friends, Pearson's r2 is reported as .40. Which of the following would be a correct interpretation? Mobility explains 40% of the variation in number of friends There is a strong positive relationship between number of friends and mobility As mobility increases, number of friends decreases Mobility explains 16% of the variation in number of friends
Answer:
Option A
Step-by-step explanation:
Given that in a study of the relationship between geographical mobility (number of times a person has changed residences) and number of friends, Pearson's r^2 is reported as .40.
r square, being the coefficient of determination explains the variablity of one variable due to the variability of the other. Here Mobility explains 40% variation in the number of friends is right answer.
Option B is wrong because r = ±6324, so cannot say positive or negative.
Similarly option c is wrong because we are unsure whether negative correlation. Option d is wrong since 16% is not right .
Twenty percent of all telephones of a certain type are submitted for service while under warranty. Of these, 60% can be repaired, whereas the other 40% must be replaced with new units. If a company purchases ten of these telephones, what is the probability that exactly two will end up being replaced under warranty?
The probability that exactly two telephones will end up being replaced under warranty is found to be 0.147.
How to find probability using binomial distribution?The binomial distribution is based on the binomial theorem which can be written as (a + b)ⁿ = ⁿC₀aⁿb⁰ + ⁿC₁aⁿ⁻¹b¹ + ⁿC₂aⁿ⁻²b²+ ....+ ⁿCₙa₀bⁿ.
In order to use in the probability, there should be events independent of each other and the sum of there probabilities is 1.
Suppose the total number of telephones be x.
Then, the telephones submitted under warranty is 20% × x = 0.2x.
The number of telephones to be repaired is 60% × 0.2x = 0.12x
And, those to be replaced are given as 40% × 0.2x = 0.08x.
Now, the probability of exactly two units being replaced out of 10 is given by binomial distribution as,
P(Exactly two units being replaced) = ¹⁰C₂(0.08)²(0.92)⁸
⇒ P(Exactly two units being replaced) = 0.147
Hence, the probability for the given case is obtained as 0.147.
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To solve this problem, we can use the concept of conditional probability. Given that 20% of telephones are submitted for service while under warranty and of these, 60% can be repaired and 40% must be replaced with new units, we can calculate the probability that exactly two telephones will be replaced under warranty.
Explanation:To solve this problem, we can use the concept of conditional probability. Let's break down the information given:
20% of all telephones are submitted for service while under warranty.Of these, 60% can be repaired.40% must be replaced with new units.Now, let's calculate the probability that exactly two telephones will be replaced:
First, calculate the probability of a telephone being replaced: 20% (probability of being submitted) * 40% (probability of needing replacement) = 8%.Now, calculate the probability of a telephone being repaired: 20% (probability of being submitted) * 60% (probability of being repaired) = 12%.Since we need exactly two telephones to be replaced, we need two telephones to be replaced and eight telephones to be repaired. The probability of two telephones being replaced and eight telephones being repaired can be calculated as 10C2 (number of ways to choose 2 out of 10) * (8%)^2 * (12%)^8 = 45 * 0.08^2 * 0.12^8 ≈ 0.0572 (rounding to four decimal places).Therefore, the probability that exactly two telephones will end up being replaced under warranty is approximately 0.0572, or 5.72%.
Question 11 (2.5 points)
The perimeter of a rectangular poster is 108 in. The length is 12 in, greater than the
width. Find the length,
a) 21
b) 42
c) 33
d) 48
Answer:
42 in
Step-by-step explanation:
108 in = 2l + 2w
l = 12 in
108 = 24 + 2w
84 = 2w
w = 42
The width is 21 inches, leading to a length of 33 inches, which corresponds to option c.
To find the length of the rectangular poster, we will use the formula for the perimeter of a rectangle, which is P = 2l + 2w, where P represents the perimeter, l represents the length, and w represents the width. According to the question, we have a perimeter P of 108 inches and the length is 12 inches greater than the width. If we let w represent the width, then the length can be represented as w + 12. Plugging into the perimeter formula we get: 108 = 2(w + 12) + 2w.
Now, let's solve for w:
108 = 2w + 24 + 2w
108 = 4w + 24
108 - 24 = 4w
84 = 4w
w = 84 / 4
w = 21 inches
Now that we have the width, we can find the length by adding 12 inches:
Length = w + 12
Length = 21 + 12
Length = 33 inches
Therefore, the length of the poster is 33 inches, which corresponds to option c.
Among 8846 cases of heart pacemaker malfunctions, 375 were found to be caused by firmware, which is software programmed into the device. If the firmware is tested in 3 different pacemakers randomly selected from this batch of 8846 and the entire batch is accepted if there are no failures, what is the probability that the firmware in the entire batch will be accepted? Is this procedure likely to result in the entire batch being accepted?
Step-by-step explanation:
Probability, P = [tex]\frac{no. of favourable outcomes}{Total no. of possible outcomes}[/tex]
Out of 8846 heart pacemaker malfunctions cases, caused by firmware cases are 375
then, no. of cases not caused by firmware are:
8846 - 375 = 8471
Probability for three different pacemakers respectively is given by:
P(1) = [tex]\frac{8471}{8846}[/tex]
we select the next malfunctioned pacemaker from the remaining i.e., out of 8471, excluding the chosen malfunctioned pacemaker
P(2|1) = [tex]\frac{8470}{8845}[/tex]
Therefore, the events are not independent of each other
Now, if the selection is without replacement, then
P(3|1 & 2) = [tex]\frac{8469}{8844}[/tex]
Now, by general multiplication rule(as the events are not independent):
P(none of the 3 are caused by malfunction) =
[tex]\frac{8471}{8846}\times\frac{8470}{8845}\times\frac{8469}{8844}[/tex]
= 0.9019
P(none of the 3 are caused by malfunction) = 90.19%
The probability is high, therefore the whole batch will be accepted.
Probability of an event represents the chances of that event to occur.
The probability of the entire batch to get selected is 0.879Yes, this procedure is likely to result in the entire batch being accepted.What is the multiplication rule of probability for independent events?Suppose there are n mutually independent events.
The probability of their simultaneously occurrence is given as
[tex]P(A_1 \cap A_2 \cap ... \cap A_n) = P(A_1) \times P(A_2) \times ... \times P(A_n)[/tex]
(This is true only if all those events are mutually independent).
The probability of getting a failure(malfunction) in a randomly selected pacemaker for the given case is
[tex]\dfrac{375}{8846} \approx 0.042[/tex]
The probability of a pacemaker working fine = 1- its failure probability = 1 - 0.042 = 0.958
Since each of the 3 selected pacemakers are independent for their failure or success of each other, thus, if we have:
[tex]A_1[/tex] = Event of properly working of first pacemaker
[tex]A_2[/tex] = Event of properly working of second pacemaker
[tex]A_3[/tex] = Event of properly working of third pacemaker
Then, as all three are independent for their working from each other, thus,
[tex]P(A_1 \cap A_2 \cap A_3) = P(A_1) \times P(A_2)\times P(A_3) = (0.958)^3 \approx 0.879\\\\\begin{aligned}{P(\text{batch getting selected}) &= P(\text{All three pacemaker working})\\& = P(A_1 \cap A_2 \cap A_3) \\&\approx 0.879\\\end{aligned}[/tex]
Thus, as this probability is high, thus, this procedure is likely to result in the entire batch being accepted.
Thus,
The probability of the entire batch to get selected is 0.879Yes, this procedure is likely to result in the entire batch being accepted.Learn more about multiplication rule of probability here:
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hi, I would greatly appreciate if someone left the work step by step to solve one of these questions so I can understand it? if it's possible
Answer:
see below
Step-by-step explanation:
a
y = 2x -3
Standard form is Ax + By =C where A is a positive integer and B and C are integers
Subtract 2x from each side
y-2x = 2x-2x -3
-2x+y = -3
We want A to be positive
Multiply each side by -1
2x -y = 3
This is in standard form
b
y = 2/3 x -7
Subtract 2/3x from each side
y-2/3x = 2/3x-2/3x -7
-2/3x+y = -7
We want A to be positive integer
Multiply each side by -3
-3*(-2/3x+y) = -7*-3
2x -3y = 21
c
y = -3x +1/2
Add 3x to each side
3x +y = -3x+3x +1/2
3x+y = 1/2
Multiply each side by 2
2(3x+y) = 1/2*2
6x+2y = 1