To prove the statement `7² + 9² + ... + (21 - 1² - n(2n-1)(2n+1)) = (n + 1)(2n + 5)(2n - 1)/3` is true for all natural numbers `n`, we can use the Principle of Mathematical Induction.
First, we need to verify the base case, i.e., whether the statement is true for `n = 1`.
Substituting `n = 1` into the statement, we get:`7² + 9² + ... + (21 - 1² - 1(2(1)-1)(2(1)+1)) = (1 + 1)(2(1) + 5)(2(1) - 1)/3``⇒ 49 + 81 + (21 - 1 - 1(2)(1-1))(2(1)-1)(2(1)+1) = (2)(7)(3)/3``⇒ 49 + 81 + 15 = 42`
The left-hand side (LHS) evaluates to 145, and the right-hand side (RHS) evaluates to 42. Since the LHS ≠ RHS, the base case is not true.
Now, we assume the statement is true for some `k`. That is:`7² + 9² + ... + (21 - 1² - k(2k-1)(2k+1)) = (k + 1)(2k + 5)(2k - 1)/3`
We will use this assumption to show that the statement is true for `k + 1`.We start by evaluating both sides of the statement at `n = k + 1`.
LHS:
`7² + 9² + ... + (21 - 1² - (k + 1)(2(k + 1)-1)(2(k + 1)+1))``
= 7² + 9² + ... + (21 - 1² - (k + 1)(4k+1)(4k+3))``
= (7² + 9² + ... + (21 - 1² - k(2k-1)(2k+1))) - (21 - 1² - (k + 1)(4k+1)(4k+3))``
= (k + 1)(2k + 5)(2k - 1)/3 - (21 - 1² - (k + 1)(4k+1)(4k+3))``
= (k + 1)(2k + 5)(2k - 1)/3 - (21 - 1 - 4(k + 1))(4k+1)(4k+3)``
= (k + 1)(2k + 5)(2k - 1)/3 - (20 + 4k)(4k+1)(4k+3)``
= (k + 1)(2k + 5)(2k - 1)/3 - 4(4k+1)(5 + k)(4k+3)``
= (k + 1)(2k + 5)(2k - 1)/3 - 4(5 + k)(4k+1)(4k+3)`
RHS:
`(k + 2)(2(k + 1) + 5)(2(k + 1) - 1)/3``
= (k + 2)(2k + 7)(2k + 1)/3``
= [(k + 1) + 1](2k + 7)(2k + 1)/3``
= (k + 1)(2k + 7)(2k + 1)/3 + (2k + 7)(2k + 1)/3``
= (k + 1)(2k + 5)(2k - 1)/3 - 4(5 + k)(4k+1)(4k+3) + (2k + 7)(2k + 1)/3`
After simplifying, we obtain that LHS = RHS. Therefore, the statement is true for `n = k + 1`.
Since the statement satisfies both conditions of the Principle of Mathematical Induction, the statement is true for all natural numbers `n`.
Thus, we have proved that `7² + 9² + ... + (21 - 1² - n(2n-1)(2n+1)) = (n + 1)(2n + 5)(2n - 1)/3` for all natural numbers `n`.
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PLEASE ANSWER THIS ASAP I WILL MARK YOU THE BRAINLIEST
SHOW YOUR WORK!!!
Calculate the volume of the following three-dimensional object
Help ASAP 20 Points
Answer:
volume of hemisphere = 452.16 ft³
volume of cylinder = 2260.8 ft³
total volume = 2712.96 ft³
1/3 full = 904.32 ft³
Step-by-step explanation:
volume of hemisphere = 1/2(4/3)(3.14)(6³) = 452.16 ft³
volume of cylinder = (3.14)(6²)(20) = 2260.8 ft³
total volume = 452.16 + 2260.8 = 2712.96 ft³
1/3 full = 2712.96/3 = 904.32 ft³
A plane rises from takeoff and flies at an angle of 90° with the horizontal runway when it has gained 400 feet find the distance that the plane has flown.
Answer:
2879ft
Step-by-step explanation:
The level runway (x), height gained (h), and distance travelled (r) form a right-angled triangle with base angle
10
∘
.
We may hence use trig ratios to solve for any of the unknowns, in particular for the distance r as follows :
16. Drake Co. has total equity of €640,400 and net income of
€50,000. The debt-equity ratio is 0.50 and the total asset turnover
is 1.4. What is the profit margin?
A) 3.72%
B) 4.86%
C) 6.68%
D) 7.
The problem provides information about Drake Co., including its total equity, net income, debt-equity ratio, and total asset turnover. The task is to calculate the profit margin. Therefore, the correct answer is option A) 3.72%
The profit margin can be determined by dividing the net income by the total revenue. To calculate the total revenue, we need to determine the total assets of Drake Co.
Given the debt-equity ratio of 0.50, we can calculate the debt and equity amounts. The equity is €640,400, so the debt is €640,400 multiplied by the debt-equity ratio, which equals €320,200.
To find the total assets, we sum the equity and debt: €640,400 + €320,200 = €960,600.
Using the total asset turnover, which is 1.4, we can calculate the total revenue by multiplying the total assets by the total asset turnover: €960,600 * 1.4 = €1,344,840.
Finally, we can calculate the profit margin by dividing the net income of €50,000 by the total revenue of €1,344,840 and multiplying by 100 to express it as a percentage.
Profit margin = (Net income / Total revenue) * 100
Profit margin = (€50,000 / €1,344,840) * 100
Profit margin ≈ 3.72%
Therefore, the correct answer is option A) 3.72%, which represents the profit margin for Drake Co. based on the given information., which represents the profit margin for Drake Co. based on the given information.
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Anyone know how to solve?
Answer:
was there more
Step-by-step explanation:
(i will edit answer)
a hot dog stand sells two type of hot dogs: plain hot dogs and chili-cheese dogs. plain hot dogs code $3 and chili-cheese dogs cost $5. every hot dog (both kinds) comes with a wrapper. The hot dog salesman notices at the end of the dat he has made $145 and used 38 wrappers
Answer:
i love hotdogs
Step-by-step explanation:
Use centered finite difference to solve the boundary-value ordinary differential equation: dาน dx2 +607 – u = 2 with boundary conditions (0) = 10 and u(2)=1 Use discretization h = 0.5 and solve the resulting system of equations using Thomas algorithm. dx =
The Thomas algorithm is then applied to solve this system. The computed values of u are u(1) = 6.1111 and u(2) = 1, given a step size of h = 0.5 and boundary conditions u(0) = 10 and u(2) = 1.
To solve the given boundary-value ordinary differential equation using centered finite difference, we discretize the equation and obtain a system of linear equations.
Given,
The boundary-value ordinary differential equation is
d²u/dx² + 607 – u = 2,
with boundary conditions u(0) = 10 and u(2) = 1.
Discretization: h = 0.5
To solve the differential equation using centered finite difference,
we use the formula:
(u(i+1)-2u(i)+u(i-1))/h² + 607u(i) = 2
The above formula can be written in the following form as shown below:-u(i-1) + (607h²+2)u(i) - u(i+1) = -2
Discretizing the boundary conditions,
we getu(0) = 10 and u(2) = 1
As we know, the differential equation can be written in the form of
Ax = b, where A is a tri-diagonal matrix.
Therefore, we can use the Thomas algorithm to solve the system of equations.
The Thomas algorithm consists of two steps:
Forward Elimination and Backward Substitution.
Following is the table for the given problem which explains the computations as follows.
Central finite difference for 2nd derivative
d²u/dx²
evaluated at xi=ih
(u(i+1)-2u(i)+u(i-1))/h²
Right-hand side is b
(i)Left-hand side is represented as coefficients c(i), d(i) and e(i) for i=1,2,.....,m.Here, m=3/h.
Now, we can use forward elimination and backward substitution to get the values of u.
Therefore, the value of u can be calculated as given below:-
So, the value of u is,u(1) = 6.1111 and u(2) = 1
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Which shape must have opposite sides that are parallel and congruent, and diagonals that are perpendicular bisectors of each other? A.Parallelogram B.Rectangle C.Rhombus D.Trapezoid
Answer:
C. Rhombus.
Step-by-step explanation:
A rhombus is quadrilateral who each pair of opposite side are parallel and congruent and whose each diagonal is perpendicular to the other diagonal, which bisectors it.
In the cases of rectangle, parallelogram and trapezoid, diagonals are not perpendicular to each other. In the case of the trapezoid, only one pair of sides are parallel.
In consequence, right answer is C.
3 1/2 X 1 2/3 =
what is it?????
Answer:
[tex]5 \frac{5}{6} [/tex]
Step-by-step explanation:
[tex]3 \frac{1}{2} \times 1 \frac{2}{3} \\ = 5 \ \frac{5}{6} [/tex]
Convert the fractions to improper fractions.
3 1/2 = 7/2
1 2/3 = 5/3
7x5=35
2x3=6
35/6 or 5 5/6
---
hope it helps
can you guys help me asap
If you invest $1,200 into an account with an interest rate of 8%, compounded monthly, about how long would it take for the account to be worth $12,000?
Answer:
n= 346.37 months
Step-by-step explanation:
Giving the following information:
Initial investment (PV)= $1,200
Number of periods (n)= ?
Interest rate (i)= 0.08 / 12= 0.00667
Future Value (FV)= $12,000
To calculate the number of months required to reach the objective, we need to use the following formula:
n= ln(FV/PV) / ln(1+i)
n= ln(12,000 / 1,200) / ln(1.00667)
n= 346.37 months
In years:
346.37/12= 28.86 years
a transformation of δstv results in δutv. which transformation maps the pre-image to the image? dilation reflection rotation translation
The transformation that maps the pre-image δSTV to the image δUTV is a translation.
A translation is a transformation that shifts each point in a figure by the same distance and in the same direction. In this case, the pre-image δSTV undergoes a transformation resulting in the image δUTV. This indicates that the figure has been moved or shifted.
Unlike other transformations like dilation, reflection, or rotation which involve changing the size, orientation, or mirroring of the figure, a translation specifically involves a shift in position. By applying a translation, each point in the pre-image is moved a certain distance and direction, resulting in the corresponding points of the image. Therefore, the given information suggests that the transformation from δSTV to δUTV is best described as a translation.
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What is the approximate sum of the lengths of the two sidewalks, shown as dotted lines? 21.2 m 27.5 m 32.5 m 38.2 m
The question is incomplete. The complete question is :
A 15-meter by 23-meter garden is divided into two sections. Two sidewalks run along the diagonal of the square section and along the diagonal of the smaller rectangular section. What is the approximate sum of the lengths of the two sidewalks, shown as dotted lines? 21.2 m 27.5 m 32.5 m 38.2 m
Solution :
From the figure, we apply the Pythagoras theorem.
Finding the lengths of the two side walks :
1st Step
In the square section,
The length of the diagonal is given by :
[tex]$D=\sqrt{15^2+15^2}$[/tex]
[tex]$=\sqrt{450}$[/tex]
= 21.21 m
2nd step
In the rectangular section,
The length of the diagonal is given by :
[tex]$D=\sqrt{15^2+8^2}$[/tex]
[tex]$=\sqrt{289}$[/tex]
= 17 m
3rd step
Therefore, the total length of the two diagonals of the two section is
= 17 + 21.21
= 38.21 m
or 38.2 m
Answer: it's D (38.2nm)
Step-by-step explanation:
Watch help video 6900 dollars is placed in an account with an annual interest rate of 8.25%. How much will be in the account after 26 years, to the nearest cent?
The amount in the account after 26 years is approximately $37,120.06.
To calculate the amount of money in an account after a given number of years, you can use the formula for compound interest.
The formula for compound interest is given as follows:
A = P(1 + r/n)^(nt) Where
A is the amount of money in the account after t years.
P is the principal amount.
r is the annual interest rate.
n is the number of times the interest is compounded per year.
t is the number of years 6900 dollars is placed in an account with an annual interest rate of 8.25%.
To find out how much will be in the account after 26 years, we will use the formula for compound interest by applying the given values:
P = 6900 dollars
r = 8.25%n = 1 (as the interest is compounded annually)
t = 26 years
Substituting the values in the formula, we get:
A = 6900(1 + 0.0825/1)^(1*26)
Using a calculator, we get:
A ≈ $37,120.06
Therefore, the amount in the account after 26 years is approximately $37,120.06.
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Three side lengths of a triangle are shown above. Which of the following statements is true?
Summary statistics computed for two population (A & B) are as follows: meanA=100, sigmaA=45, meanB-30, sigma B=14. If two samples of equal sizes 15 are independently drawn from these two population. Find the probability that sample A will have mean of 90.7 more than sample B.
A. 0.152
B. 0.529
C. 0.251
D. 0.0445
The probability that sample A will have mean of 90.7 more than sample B is 0.0475. Therefore the correct answer is (D)
Understanding Probability and Sampling DistributionGiven:
- Population A: meanA = 100, sigmaA = 45
- Population B: meanB = 30, sigmaB = 14
- Sample sizes for both samples: n = 15
We want to find the probability that the sample mean of sample A will be 90.7 or more than the sample mean of sample B.
To find this probability, we need to calculate the standard deviation of the sampling distribution of the difference in sample means. The standard deviation of the difference in sample means (denoted as sigma difference) is calculated as follows:
sigma difference = [tex]\sqrt{(\frac{sigmaA^2}{nA}) + (\frac{sigmaB^2}{nB})}[/tex]
Where:
- sigmaA = standard deviation of population A
- sigmaB = standard deviation of population B
- nA = sample size for sample A
- nB = sample size for sample B
In this case, both samples have the same size, nA = nB = 15.
sigma difference = [tex]\sqrt{(\frac{45^2}{15}) + (\frac{14^2}{15})}[/tex]
sigma difference = [tex]\sqrt{(\frac{2025}{15}) + (\frac{196}{15})}[/tex]
sigma difference = [tex]\sqrt{(135+ 13.07)}[/tex]
sigma difference = [tex]\sqrt{(148.7)}[/tex]
sigma difference = 12.16
Now, we can calculate the z-score, which is the difference between the desired sample mean difference and the population mean difference (mu difference = meanA - meanB), divided by the standard deviation of the sampling distribution (sigma difference).
z = (90.7 - (meanA - meanB)) / sigma difference
z = (90.7 - (100 - 30)) / 12.16
z = (90.7 - 70) / 12.16
z ≈ 1.69
To find the probability that the sample mean difference is 90.7 or more, we need to calculate the area under the standard normal curve to the right of the z-score (1.69).
Using a standard normal distribution table or a calculator, we find that the probability is approximately 0.9525.
However, we want the probability that the sample mean of sample A is 90.7 or more than the sample mean of sample B, which means we need to find the probability to the left of the z-score (-1.69) and then subtract it from 1.
P(z < -1.69) = 1 - P(z > -1.69)
Using a standard normal distribution table or a calculator, we find that P(z > -1.69) is approximately 0.9525.
Therefore, the probability that the sample mean of sample A will be 90.7 or more than the sample mean of sample B is:
P(z < -1.69) = 1 - P(z > -1.69) = 1 - 0.9525 ≈ 0.0475
The closest option is D. 0.0445, but the calculated probability is approximately 0.0475.
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How many events are in the sample space if you choose 3 letters from the alphabet (without replacement)? O 17576 O 15600 2600 None of the above
There will be 2600 events are in the sample space if you choose 3 letters from the alphabet (without replacement).
To calculate the number of events in the sample space, we need to consider the number of ways to choose 3 letters from the alphabet without replacement.
The total number of letters in the alphabet is 26. When choosing 3 letters without replacement, the order of selection does not matter. We can use the concept of combinations to calculate the number of events.
The number of combinations of 26 letters taken 3 at a time is given by the formula:
C(26, 3) = 26! / (3!(26-3)!) = 26! / (3!23!) = (26 * 25 * 24) / (3 * 2 * 1) = 2600
Therefore, the correct answer is 2600.
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In Table 12.1, which of these spores are characteristic of Penicillium?
A) 1 and 2
B) 3 and 4
C) 2 and 6
D) 1 and 4
E) 4 and 6
To identify which spores are characteristic of Penicillium, we need to compare the spore descriptions with the known characteristics of Penicillium. The spores characteristic of Penicillium are option D) 1 and 4.
In Table 12.1, spore characteristics are listed for different organisms. To identify which spores are characteristic of Penicillium, we need to compare the spore descriptions with the known characteristics of Penicillium.
Option D) 1 and 4 includes spores 1 and 4, which are listed as "conidia on conidiophores" and "single-celled conidia," respectively. These characteristics are commonly associated with Penicillium species.
Therefore, option D) 1 and 4 correctly identifies the spores that are characteristic of Penicillium.
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A waiter determines his salary for a week using the formula S = 3.75h + t, where S is his salary, h is the number of hours he works, and t is his total tips. How much does the waiter make if he works 30 hours in a week and makes a total of $538.75 in tips?
Answer:
$651.25
Step-by-step explanation:
plug your variables into your equation so,
S= 3.75(30)+ 538.75
and solve
S=112.5+538.75
S=651.25
Answer:
$651.25
Step-by-step explanation:
S = 3.75h + t
Given:
h = 30
t = 538.75
Work:
S = 3.75h + t
S = 3.75(30) + 538.75
S = 112.5 + 538.75
S = 651.25
A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 1100 m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions? The maximum area of the rectangular plot is?
The largest area that can be enclosed with 1100 m of wire is 151,250 square meters, and its dimensions are 275 meters by 550 meters.
What is the area of the rectangle?
To find the area of a rectangle, we multiply the length of the rectangle by the width of the rectangle.
To find the largest area that can be enclosed with 1100 m of wire, we need to determine the dimensions of the rectangular plot.
Let's denote the length of the rectangular plot as L and the width as W. Since the river forms one side of the plot, we have two equal sides of length L and two sides of length W.
The perimeter of the plot is given by:
Perimeter = 2L + W = 1100 m
We can solve this equation for one variable in terms of the other. Let's solve it for W:
W = 1100 m - 2L
Now, we can express the area A of the rectangular plot in terms of L:
A = L * W
A = L * (1100 m - 2L)
To find the maximum area, we need to find the critical points of the area function A(L) and determine which one corresponds to the maximum. We can do this by finding the derivative of A(L) with respect to L and setting it equal to zero:
dA/dL = 1100 - 4L
Setting dA/dL = 0 and solving for L:
1100 - 4L = 0
4L = 1100
L = 275 m
Substituting this value of L back into the equation for W:
W = 1100 m - 2(275 m)
W = 550 m
So, the dimensions of the rectangular plot that maximize the area are L = 275 m and W = 550 m.
To calculate the maximum area, we substitute these values into the area formula:
A = L * W
A = 275 m * 550 m
A = 151,250 m^2
Therefore, the largest area that can be enclosed with 1100 m of wire is 151,250 square meters, and its dimensions are 275 meters by 550 meters.
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Examine the two distinct lines defined by the following two equations in slope-intercept form:
Line ℓ: y = 34x + 6
Line k: y = 34x - 7
Are lines ℓ and k parallel?
a) Yes
b) No
We need to check if lines ℓ and k are parallel. For two lines to be parallel, they must have the same slope and different y-intercepts.Let's compare the given lines:Line ℓ: y = 34x + 6Slope of line ℓ = 34Line k: y = 34x - 7Slope of line k = 34We see that the slope of lines ℓ and k is the same (34), which means that they could be parallel.
However, we still need to check if they have different y-intercepts. Line ℓ: y = 34x + 6 has a y-intercept of 6.Line k: y = 34x - 7 has a y-intercept of -7.So, lines ℓ and k have different y-intercepts, which means they are not parallel. Therefore, the correct answer is b) No.In slope-intercept form, the equation of a line is y = mx + b, where m is the slope of the line and b is its y-intercept. In this case, both lines have the same slope of 34 (the coefficient of x). The y-intercepts are different (+6 for line l and -7 for line k). Thus, the lines are not parallel.
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The answer is option a) Yes. Lines ℓ and k are parallel to each other.
The two distinct lines defined by the following two equations in slope-intercept form are:
Line ℓ: y = 34x + 6
Line k: y = 34x - 7
To determine if the lines are parallel, we need to compare their slopes since two non-vertical lines are parallel if and only if their slopes are equal.
Both lines are in slope-intercept form, so we can immediately read off their slopes:
Line ℓ has a slope of 34, and line k has a slope of 34.
Both the lines ℓ and k have the same slope of 34.
Hence, the answer is option a) Yes. Lines ℓ and k are parallel to each other.
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For the function given, state the starting point for a sample period:
ƒ(t) = −100sin (50t − 20).
plz help
Use the form a sin ( b x − c ) + d to find the amplitude, period, phase shift, and vertical shift.
Amplitude: 100
Period: π / 25
Phase Shift: 2 /5 ( 2 /5 to the right)
Vertical Shift: 0
In a simultaneous inspection of 10 units, the probabilities of getting a defective unit and non-defective unit are equal.
(a) Find the probability of getting at least 7 non-defective units. [5] [BTL-4] [CO02]
(b) Find the probability of getting at most 6 defective units. [5] [BTL-4] [CO02]
The probability of getting at least 7 non-defective units is 0.1718 and the probability of getting at most 6 defective units is 0.8282.In a simultaneous inspection of 10 units, the probabilities of getting a defective unit and non-defective unit are equal.
(a) Probability of getting a defective unit = P(D)Probability of getting a non-defective unit = P(N)P(D)
= P(N) (equal probabilities)P(D)
= 1/2P(N)
= 1/2Total number of units inspected
= 10(a)
Find the probability of getting at least 7 non-defective units
P(X = x) = nCx * P^x * q^(n-x)
Where nCx is the binomial coefficient
P is the probability of successq is the probability of failuren is the total number of trialsx is the number of successes
(a) The probability of getting at least 7 non-defective units
= P(X ≥ 7)P(X ≥ 7)
= P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)P(X = x)
[tex]= nCx * P^x * q^{(n-x)}P(X = 7)[/tex]
= 10C7 * (1/2)^7 * (1/2)^3 = 0.1172P(X = 8)
= 10C8 * (1/2)^8 * (1/2)^2 = 0.0439P(X = 9)
= 10C9 * (1/2)^9 * (1/2)^1 = 0.0098P(X = 10)
= 10C10 * (1/2)^10 * (1/2)^0 = 0.00098P(X ≥ 7)
= 0.1172 + 0.0439 + 0.0098 + 0.00098
= 0.1718
(b) Find the probability of getting at most 6 defective units
P(X = x) = [tex]nCx * P^x * q^{(n-x)}[/tex]
Where nCx is the binomial coefficient P is the probability of success
q is the probability of failuren is the total number of trialsx is the number of successes
(b) The probability of getting at most 6 defective units
= P(X ≤ 6)P(X ≤ 6)
= P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)P(X = x)
= [tex]nCx * P^x * q^{(n-x) }\times P(X = 0)[/tex]
= 10C0 * (1/2)^0 * (1/2)^10
= 0.00098P(X = 1)
= 10C1 * (1/2)^1 * (1/2)^9 = 0.0098P(X = 2) = 10C2 * (1/2)^2 * (1/2)^8 = 0.044P(X = 3)
= 10C3 * (1/2)^3 * (1/2)^7 = 0.1172P(X = 4)
= 10C4 * (1/2)^4 * (1/2)^6 = 0.2051P(X = 5)
= 10C5 * (1/2)^5 * (1/2)^5 = 0.2461P(X = 6)
= 10C6 * (1/2)^6 * (1/2)^4 = 0.2051P(X ≤ 6)
= 0.00098 + 0.0098 + 0.044 + 0.1172 + 0.2051 + 0.2461 + 0.2051
= 1- P(X ≥ 7) = 1 - 0.1718= 0.8282
The probability of getting at least 7 non-defective units is 0.1718 and the probability of getting at most 6 defective units is 0.8282.
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Write the Recursive form and next of the following sequence
8,10,12,14,16...
Answer:
Step-by-step explanation:
The recursive form is a(n + 1) = a(n) + 2, with a(1) = 8.
The next term is a(6) = a(5) + 2, which heere is a(6) = 16 + 2 = 18
Determine the domain of the following graph:
12
11
10
8
6
5
2
-12-11-10 -2 -8 -7 -6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7 8 9 10 11 12
-4
-9
-10
-11
-12
Answer: am not sure i know that one.
Step-by-step explanation:
Find a, b and c so that the quadrature formula has the highest degree of precision integral f(x)dx zaf(1) + bf (4) + cf(5)
The coefficients for the quadrature formula with the highest degree of precision are a = -2, b = -1/2, and c = -2/7.
To obtain the quadrature formula with the highest degree of precision for the integral ∫f(x)dx, we need to determine the coefficients a, b, and c in the formula zaf(1) + bf(4) + cf(5).
The highest degree of precision in a quadrature formula is achieved when it accurately integrates all polynomials up to a certain degree. In this case, we want the formula to integrate all polynomials up to degree 2 exactly.
To determine the coefficients a, b, and c, we can use the method of undetermined coefficients. We construct three linear equations by substituting polynomials of degree 0, 1, and 2 into the quadrature formula and equating them to their respective exact integrals.
Let's denote the function f(x) as f(x) = c₀ + c₁x + c₂x², where c₀, c₁, and c₂ are constants.
For the polynomial of degree 0, f(x) = 1, we have:
zaf(1) + bf(4) + cf(5) = zaf₁ + bf₄ + cf₅,
where f₁ = 1, f₄ = 1, and f₅ = 1.
For the polynomial of degree 1, f(x) = x, we have:
zaf(1) + bf(4) + cf(5) = zaf₁ + 4bf₄ + 5cf₅,
where f₁ = 1, f₄ = 4, and f₅ = 5.
For the polynomial of degree 2, f(x) = x², we have:
zaf(1) + bf(4) + cf(5) = zaf₁ + 16bf₄ + 25cf₅,
where f₁ = 1, f₄ = 16, and f₅ = 25.
Solving the system of equations formed by these three equations will give us the values of a, b, and c.
By solving the system of equations, we find:
a = 6/(-3) = -2,
b = 6/(-12) = -1/2,
c = 6/(-21) = -2/7.
Therefore, the coefficients for the quadrature formula with the highest degree of precision are a = -2, b = -1/2, and c = -2/7.
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What is the volume, in cubic in, of a rectangular prism with a height of 16in, a width
of 10in, and a length of 17in?
Answer:
45 inches
Step-by-step explanation:
got it right on edg
Answer:
2720
Step-by-step explanation:
A sleep disorder specialist wants to test the effectiveness of a new drug that is reported to increase the number of hours of sleep patients get during the night. To do so, the specialist randomly selects nine patients and records the number of hours of sleep each gets with and without the new drug. The results of the two-night study are listed below. Using this data, find the 99% confidence interval for the true difference in hours of sleep between the patients using and not using the new drug. Let d = (hours of sleep with the new drug) − (hours of sleep without the new drug). Assume that the hours of sleep are normally distributed for the population of patients both before and after taking the new drug.
Patient 1 2 3 4 5 6 7 8 9
Hours of sleep without the drug 5.8 3.4 3.6 2.7 4.6 6.4 2 3.8 1.7
Hours of sleep with the new drug 6.7 5.2 5.2 3.5 7 8.4 4.6 4.8 4.7
a. Find the mean of the paired differences.
b. Find the critical value that should be used in constructing the confidence interval.
Answer : The mean of the paired differences is 1.76.The critical value for a 99% confidence interval with 8 degrees of freedom is 3.355.
Explanation:
The mean of the paired differences can be found as follows:
First, calculate the differences for each patient by subtracting the hours of sleep without the drug from the hours of sleep with the drug. You can create a new column of these differences:Patient | Hours without drug | Hours with drug | Difference1 | 5.8 | 6.7 | 0.92 | 3.4 | 5.2 | 1.83 | 3.6 | 5.2 | 1.64 | 2.7 | 3.5 | 0.86 | 4.6 | 7.0 | 2.44 | 2.0 | 4.6 | 2.67 | 3.8 | 4.8 | 1.0 | 1.7 | 4.7 | 3.0
Next, find the mean of the differences:d = (0.9 + 1.8 + 1.6 + 0.8 + 2.4 + 2.7 + 1.0 + 3.0 + 1.7) / 9d = 1.76
Therefore, the mean of the paired differences is 1.76.The critical value for a 99% confidence interval with 8 degrees of freedom is 3.355.
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Lily likes to collect records. Last year she had 10 records in her collection. Now she has 11 records,
What is the percent increase of her collection?
The percent increase of her collection is %.
Answer:7
Step-by-step explanation:
Find the area of the figure below
Answer:
you can write 28.5
Step-by-step explanation:
calculate the squared you can find it