A toy manufacturer wants to know how many new toys children buy each year. A sample of 601 children was taken to study their purchasing habits. Construct the 95% confidence interval for the mean number of toys purchased each year if the sample mean was found to be 6.7. Assume that the population standard deviation is 1.5. Round your answers to one decimal place.

Answer:

35

Step-by-step explanation:

Given that A,B, C are three non empty sets.

[tex]n(A) =n(B) =n(C) =17\\n(A \bigcap C) = 7\\n(A \bigcap B) = 5\\n(B \bigcap C) = 6\\n(A \bigcap B \bigcap C) = 7[/tex]

Use the addition theory for finding no of elements in union of two or more sets

We have addition theorem as

[tex]n(AUBUC) = n(A)+n(B)+n(C)-n(A \bigcap B)-n(B  \bigcap C)-n(A  \bigcap C)+n(A  \bigcap B \bigcap C)[/tex]

Now substitute for each entry from the given information

[tex]n(AUBUC) = 17+17+17-5-6-7+2\\= 53-18\\=35[/tex]

Answer:

See definitions and relation below

Step-by-step explanation:

Given points x and y of a certain set S in a metric space, a path from x to y is a continuous map  f:[a,b]-->S of some closed interval [a,b] in the real line into S, such that

f(a)=x and f(b)=y

In this case, we can also say that the points x and y are joined by a path or arc.

A set S in metric space is said to be path connected or arcwise connected if every pair of points x, y of S  can be joined by a path.

The relation between arcwise connectedness and connectedness of a set is that every arcwise connected set is also connected, but the converse does not hold; not every connected space is also path connected.

As an example, consider the unit square [0,1]X[0,1] with the dictionary order topology.

It can be proved that this space is connected but not path connected.

Arcwise connectedness is defined as the presence of a continuous path between any two points in a set in a metric space. There is a relation between arcwise connectedness and connectedness, where any path connected set is also connected, but the converse is not necessarily true.

A set in a metric space is said to be arcwise connected or path connected if there exists a continuous curve or path that connects any two points in the set.

The relation between arcwise connectedness and connectedness of a set is that any arcwise connected set is also connected, but the converse is not necessarily true. In other words, every path connected set is connected, but not every connected set is path connected.

For example, consider a set consisting of two separate points in a metric space. This set is connected because we cannot find two disjoint open sets that cover the set, but it is not arcwise connected because there is no continuous path connecting the two points.

Answer:

Price of the drill kit should be set as $4180.

Step-by-step explanation:

Daily supply of the most popular cordless drill kit is represented by the equation

S(q) = 100q² + 100q + 20

where S(q) = price of the kits at which q units are supplied

q = number of drill kits supplied

Now we have to calculate the price of the drill kits if company plans to supply 16 kits a day.

S(16) = 10(16)² + 100(16) + 20

       = 10×256 + 1600 + 20

       = 2560 + 1600 + 20

       = $4180

Therefore, cost of the drill set should be set as $4180.

To determine the price for 16 drill kits, substitute q = 16 into the supply equation S(q) = 10q^2 + 100q + 20, resulting in a price of $4180.

To find the price at which the home improvement company should set the drill kit if they plan to supply 16 a day, we need to plug the quantity (q) into the given supply equation S(q) = 10q2 + 100q + 20.

Substituting q = 16, we get:
S(16) = 10(16)2 + 100(16) + 20
= 10(256) + 1600 + 20
= 2560 + 1600 + 20
= 4180.

So, the company should set the price of the cordless drill kit at $4180 if they plan to supply 16 units a day.

Answer:

x = 4

y = 1

z= -3

Step-by-step explanation:

Given equations are

-2x + 3y - 4z = 7

5x - y + 2z = 13

3x + 2y - z = 17

We can write the above equations in matrix augmented form as

[tex]\left[\begin{array}{ccc}-2&3&-4:7\\5&-1&2:13\\3&2&-1:17\end{array}\right][/tex]

[tex]R_1=>\dfrac{R_1}{-2}[/tex]

[tex]=\ \left[\begin{array}{ccc}1&\dfrac{-3}{2}&2:\dfrac{-7}{2}\\5&-1&2:13\\3&2&-1:17\end{array}\right][/tex]

[tex]R_2=>R_2-5R_1\ and\ R_3=>\ R_3-3R_1[/tex]

[tex]=\ \left[\begin{array}{ccc}1&\dfrac{-3}{2}&2:\dfrac{-7}{2}\\0&-1+\dfrac{15}{2}&-8:13+\dfrac{35}{2}\\0&0&-7:17+\dfrac{21}{2}\end{array}\right][/tex]

[tex]=\ \left[\begin{array}{ccc}1&\dfrac{-3}{2}&2:\dfrac{-7}{2}\\\\0&\dfrac{13}{2}&-8:\dfrac{61}{2}\\\\0&\dfrac{13}{2}&-7:\dfrac{55}{2}+\dfrac{21}{2}\end{array}\right][/tex]

[tex]R_2=>\ \dfrac{2}{13}R_2[/tex]

[tex]=\ \left[\begin{array}{ccc}1&\dfrac{-3}{2}&2:\dfrac{-7}{2}\\\\0&1&\dfrac{-16}{13}:\dfrac{61}{13}\\\\0&\dfrac{13}{2}&-7:\dfrac{55}{2}\end{array}\right][/tex]

[tex]R_3=>R_3-\dfrac{13}{2}R_2[/tex]

[tex]=\ \left[\begin{array}{ccc}1&\dfrac{-3}{2}&2:\dfrac{-7}{2}\\\\0&1&\dfrac{-16}{13}:\dfrac{61}{13}\\\\0&0&1:-3\end{array}\right][/tex]

So, from the above augmented matrix, we can write

[tex]x+\dfrac{-3}{2}y+2z=\dfrac{-7}{2}.......(1)[/tex]

[tex]y+\dfrac{-16}{13}z=\dfrac{61}{13}......(2)[/tex]

z= -3.....(3)

From eq(2) and (3)

[tex]y+\dfrac{-16}{13}(-3)=\dfrac{61}{13}[/tex]

=> y = 1

Now, by putting the value of y and z in equation (1), we will get

[tex]x+\dfrac{-3}{2}(1)+2(-3)=\dfrac{-7}{2}[/tex]

=> x = 4

Hence, the value of

x = 4

y = 1

z= -3

Answer:  Perimeter = 66 cm and area =[tex]66\ m^2[/tex]

Step-by-step explanation:

The perimeter of rectangle is given by :-

[tex]P=2(l+w)[/tex], where l is length and w is width of the rectangle.

Given : A rectangle has a length of 5.50 m and a width of 12.0 m .

Then, the perimeter of rectangle :

[tex]P=2(12+5.50)\\\\\Rightarrow\ P=2(17.50)=35\ m[/tex]

Also, area of rectangle is given by :-

[tex]A=l\times w=5.50\times12=66 \ m^2[/tex]

Area of rectangle = [tex]66\ m^2[/tex]

Answer:

It isn't the correct dosage, you should get 8 tablets.

Step-by-step explanation:

First, if the doctors orders 120 mg twice a day, it means that you need 240 mg of medicine. That is calculated as:

120 mg * 2 = 240 mg

Then if each tablet has 30 mg, the number of tablets that you should get is calculated as:

[tex]\frac{240mg}{30mg} = 8[/tex]

So, 240 mg of medicine are equivalents to 8 tablets.

8 tablets are different of 6 tablets, so the dosage given by the nurse is incorrect and you should get 8 tablets every day.

Answer:

[tex](\sqrt{3} + \frac{1}{\sqrt[4]{6}})^{15}[/tex]

Binomial expansion formula,

[tex](a+b)^n=\sum_{r=0}^{n} ^nC_r (a)^{n-r} (b)^r[/tex]

Where,

[tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]

[tex]\implies (\sqrt{3} + \frac{1}{2})^{15}=\sum_{r=0}^{15} ^{15}C_r (\sqrt{3})^{15-r} (\frac{1}{\sqrt[4]{6}})^r[/tex]

[tex]=(\sqrt{3})^{15}+15(\sqrt{3})^{14}(\frac{1}{\sqrt[4]{6}})^1+105(\sqrt{3})^{13}(\frac{1}{\sqrt[4]{6}})^2+455(\sqrt{3})^{12}(\frac{1}{\sqrt[4]{6}})^3+1365(\sqrt{3})^{11}(\frac{1}{\sqrt[4]{6}})^4+3003(\sqrt{3})^{10}(\frac{1}{\sqrt[4]{6}})^5+5005(\sqrt{3})^{9}(\frac{1}{\sqrt[4]{6}})^6+6435(\sqrt{3})^{8}(\frac{1}{\sqrt[4]{6}})^7+6435(\sqrt{3})^{7}(\frac{1}{\sqrt[4]{6}})^8+5005(\sqrt{3})^{6}(\frac{1}{\sqrt[4]{6}})^9+3003(\sqrt{3})^{5}(\frac{1}{\sqrt[4]{6}})^{10}+1365(\sqrt{3})^{4}(\frac{1}{\sqrt[4]{6}})^{11}+455(\sqrt{3})^{3}(\frac{1}{\sqrt[4]{6}})^{12}+105(\sqrt{3})^{2}(\frac{1}{\sqrt[4]{6}})^{13}+15(\sqrt{3})^{1}(\frac{1}{\sqrt[4]{6}})^{14}+(\frac{1}{\sqrt[4]{6}})^{15}[/tex]

∵ both [tex]\sqrt{3}[/tex] and [tex]\frac{1}{\sqrt[4]{6}}[/tex] are irrational numbers,

And, if the power of √3 is even, it converted to a rational number,

If its power is odd it remained as irrational number,

But, the product of a rational number and irrational number is irrational,

Thus, all terms in the above expansion are irrational. ( which can not expressed in the form of p/q, where, p and q are integers s.t. q ≠ 0 )

Answer:

a=b

Step-by-step explanation:

An antisymmetric relation () satisfies the following property:

If (a, b) is in R and (b, a) is in R, then a = b.

This means that if a|b and b|a then a = b

If a|b then, b can be written as b = an for an integer n

If b|a then a can be written as a= bm for an integer m

Now we have b = (a)n = (bm)n

b = bmn

1 =mn

But since m and n are integers, the only two integers that satisfy this property would be m = n = 1

Therefore, b = an = a (1) = a ⇒b = a

Answer:

The product represent the number [tex]3\times 0.3=0.9[/tex]

Step-by-step explanation:

To find : The decimal 0.3 represents What type of number best describes 0.9, which is 3.0.3?

Solution :

0.3 represents [tex]0.3=\frac{3}{10}[/tex]

0.9 represents [tex]0.9=\frac{9}{10}[/tex]

If we multiply 0.3 by 3 we get 0.9

As, [tex]3\times 0.3=3\times \frac{3}{10}[/tex]

[tex]3\times 0.3=\frac{9}{10}[/tex]

[tex]3\times 0.3=0.9[/tex]

Therefore, The product represent the number [tex]3\times 0.3=0.9[/tex]

Answer:

0.6604 m

Step-by-step explanation:

The convertion from inches to meters is 1 inch= 0.024 meters, so:

26 inches = 26 inch* 0.024 meters/inch = 0.6604 meters

Good luck!

Answer:

(a) 0.9838 (b) 0.6553 (c) 13.05198

Step-by-step explanation:

We have that the daily viewing time is a random variable normally distributed with mean and standard deviation

[tex]\mu[/tex] = 8.35 hours  and

[tex]\sigma[/tex] = 2.5 hours

respectively. If we call the random variable X, the density function of this random variable is given by

f(x) = [tex]\frac{1}{\sqrt{2\pi}2.5}\exp[-\frac{(x-8.35)^{2}}{2(2.5)^{2}}][/tex], and we can calculate the next probabilities using a computer or a table from a book.

(a) P(X>3)=[tex]\int\limits^{\infty}_3 {f(x)} \, dx[/tex]=0.9838

in the R statistical programming language we use the instruction pnorm(3, mean = 8.35, sd = 2.5, lower.tail = FALSE)

(b) P([tex]5\leq X\leq 10[/tex]) = [tex]\int\limits^{10}_5 {f(x)} \, dx[/tex] = 0.6553

in the R statistical programming language we use the instruction

pnorm(10, mean = 8.35, sd = 2.5) - pnorm(5, mean = 8.35, sd = 2.5)

(c) You should find a value [tex]x_{0}[/tex] such that

[tex]P(X\geq x_{0}) = 0.03[/tex], this value is  [tex]x_{0}[/tex]=13.05198

The instruction qnorm(0.03, mean = 8.35, sd = 2.5, lower.tail = FALSE) give us 13.05198 in the R statistical programming language.

Answer:

a. P=0.98

b. P=0.66

c. The top 3% of all TV viewing households watch 12.95 hours or more.

Step-by-step explanation:

We have a normal distribution with these parameters:

[tex]\mu=8.35\\\\\sigma=2.50[/tex]

a. What is the probability that a household views television more than 3 hours a day?

To calculate this, first we calculate the z-value for X=3 and then calculate the probability according to the standard normal distribution.

[tex]z=(X-\mu)/\sigma=(3-8.25)/2.50=-2.1\\\\P(X>3)=P(z>-2.1)=0.98214[/tex]

b. What is the probability that a household spends 5 – 10 hours watching television more a day?

[tex]z_1=(X_1-\mu)/\sigma=(5-8.25)/2.50=-1.3\\\\z_2=(X_2-\mu)/\sigma=(10-8.25)/2.50=0.7\\\\P(5<X<10)=P(-1.3<z<0.7)\\\\P(-1.3<z<0.7)=P(z>-1.3)-P(z>0.7)=0.9032-0.2412=0.662[/tex]

c. How many hours of television viewing must a household have in order to be in the top 3% of all television viewing households?

To calculate this we have to know the z-value for [tex]P(z>z_1)=0.03[/tex].

This z-value, according to the standard normal distribution is z=1.88.

Then, we can calculate the number of hours X as:

[tex]X=\mu+z*\sigma=8.25+1.88*2.5=8.25+4.7=12.95[/tex]

The top 3% of all TV viewing households watch 12.95 hours or more.

Answer:

Alice's weight is 100 lbs.

Step-by-step explanation:

Let's denote Jane's weight by J, and Alice's weight by A.

The exercise says that Jane is 20 lbs heavier than Alice. So that if you add 20 lbs to Alice's weight, you get Jane's weight. In equation form:

[tex]20+A=J[/tex]

It also mentions that Jane's weight is 120% that of Alice. So that if you multiply Alice's weight by 1.2, you get Jane's weight. In equation form:

[tex]1.2 \cdot A=J[/tex]

Plugging this second equation onto the first equation, we get:

[tex]20+A=1.2A[/tex]

And now solving for A:

[tex]20=1.2A-A[/tex]

[tex]20=0.2A[/tex]

[tex]\frac{20}{0.2}=A[/tex]

[tex]100=A[/tex]

Therefore Alice's weight is 100 lbs.

Answer:

The business should reject the supplier's claim as mean length is not equal to claimed value of 26.70 inches.      

Step-by-step explanation:

We are given the following in the question:

Population mean, μ = 26.70 inches

Sample mean, [tex]\bar{x}[/tex] = 26.77 inches

Sample size, n = 48

Alpha, α = 0.05

Population standard deviation, σ = 0.20 inches

First, we design the null and the alternate hypothesis

[tex]H_{0}: \mu = 26.70\text{ inches}\\H_A: \mu \neq 26.70\text{ inches}[/tex]

We use Two-tailed z test to perform this hypothesis.

Formula:

[tex]z_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}} }[/tex]

Putting all the values, we have

[tex]z_{stat} = \displaystyle\frac{26.77 - 26.70}{\frac{0.20}{\sqrt{48}} } = 2.425[/tex]

Now, [tex]z_{critical} \text{ at 0.05 level of significance } = 1.96[/tex]

Since,

[tex]z_{stat} > z_{critical}[/tex]

We reject the null hypothesis and accept the alternate hypothesis. Thus, the business should reject the supplier's claim as mean length is not equal to claimed value of 26.70 inches.

Answer: There are 20 ways and 1680 ways respectively.

Step-by-step explanation:

Since we have given that

Total number of blocks = 6

Number of red blocks = 3

Number of green blocks = 3

So, Number of patterns she can make by placing them all in a line is given by

[tex]\dfrac{6!}{3!\times 3!}\\\\=20[/tex]

If there are 3 white blocks

so, total number of white blocks becomes 9

So, Number of total pattern she can make by placing all nine blocks in a line is given by

[tex]\dfrac{9!}{3!\times 3!\times 3!}\\\\=1680\ ways[/tex]

Hence, there are 20 ways and 1680 ways respectively.

The child can create 20 different patterns if she uses just the 6 blocks (3 red, 3 green), and she can create 14,040 different patterns if she uses all 9 blocks (3 red, 3 green, 3 white). This is calculated using a branch of mathematics known as combinatorics.

In this math problem, we are dealing with a concept known as permutations in combination, which is part of combinatorics branch of Mathematics. When placing the blocks in a line, the order in which you arrange them matters, which makes this a permutation problem.

For the first case where she has 6 blocks, 3 red and 3 green, the number of different patterns she can create is calculated by the equation 6! / (3! * 3!). Here, the '!' character means factorial, which is the product of all positive integers up to that number. So, 6! = 6 * 5 * 4 * 3 * 2 * 1 and similarly 3! = 3 * 2 * 1. Plugging these in, the equation becomes 720 / (6*6) = 20 patterns.

For the second case, if she is given 3 white blocks, she then has a total of 9 blocks (3 red, 3 green, 3 white). The number of different patterns she can create is calculated similarly, but this time the equation is 9! / (3! * 3! * 3!). Plugging in the factorials, we have 362,880 / (6 * 6 * 6) = 14,040 patterns.

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Answer: The purchase price of the property is $112,500.

Step-by-step explanation:

Let the original value of property be 'x'.

Discount rate = 3%

Amount of discount = $2700

According to question, it becomes,

[tex]\dfrac{3}{100}\times x=2700\\\\x=\dfrac{2700\times 100}{3}\\\\x=\$90000[/tex]

Rate of down payment = 20%

So, Remaining rate of payment = 100-20 = 80%

So, Purchase price of the property would be

[tex]\dfrac{80}{100}\times x= 90000\\\\0.8\times x=90000\\\\x=\dfrac{90000}{0.8}\\\\x=\$112,500[/tex]

Hence, the purchase price of the property is $112,500.

Answer:

  (a)  C(x) = 5550 +8.25x

  (b)  R(x) = 36x

  (c)  P(x) = 27.75x -5550; 200 hours to break even

Step-by-step explanation:

(a) Howen's costs include fixed costs and a cost per hour. Then her total cost will be the sum of the fixed cost (5550) and the product of hours (x) and the cost per hour (8.25):

  C(x) = 5550 +8.25x

__

(b) Howen plans to charge a given amount (36) per hour, so her revenue will be the product of that amount and the number of hours she works:

  R(x) = 36x

__

(c) Her profit function is the difference between revenue and cost:

  P(x) = R(x) -C(x)

  P(x) = 36x -(5550 +8.25x)

  P(x) = 27.75x -5550

Howen's break-even point is the number of hours required to make profit be zero:

  0 = 27.75x -5550

  0 = x - 200 . . . . . . . . . divide by 27.75

  200 = x . . . . . . . . . . . . add 200

She needs to work 200 hours to break even.

Bea's cost function for operating her snow plow is C(x) = 5550.00 + 8.25x. Her revenue function for the amount she earns is R(x) = 36.00x. The profit function, which is the revenue minus the cost, simplifies to P(x) = 27.75x - 5550. To break even, she needs to work approximately 200 hours.

The cost function C(x) for operating the snow plow for x hours includes the initial cost of the snow plow plus the hourly operating cost. This can be written as C(x) = 5550.00 + 8.25x.

The revenue function R(x), representing the amount of revenue gained from x hours of use, can be given as R(x) = 36.00x as she charges $36 for each hour.

The profit function P(x), representing the amount of profit, is the revenue function minus the cost function.so, P(x) = R(x) - C(x) which simplifies to P(x) = 36x - (5550 + 8.25x). Simplify that to get P(x) = 27.75x - 5550.

To find when she breaks even, we set the profit function equal to zero and solve for x:
0 = 27.75x - 5550
Adding 5550 to both sides gives: 27.75x = 5550
Dividing both sides by 27.75 gives: x ≈ 200. Therefore, she needs to work approximately 200 hours to break even.

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Answer:

The given curve c(t) is a is a flow line of given velocity vector field F(x, y, z).

Step-by-step explanation:

We are given the following information in the question:

[tex]c(t) = (t^2, 2t-6, 3\sqrt{t}), t > 0\\\\ F(x, y, z) =(y+6, 2, \frac{9}{2z} )[/tex]

Now, we evaluate the following:

[tex]c'(t) = \frac{d(c(t))}{dt} = (2t, 2, \frac{3}{2\sqrt{t}} )[/tex]

Now, we have to evaluate:

[tex]F(c(t)) = (2t-6+6, 2, \frac{9}{6\sqrt{t}} ) = (2t, 2, \frac{3}{2\sqrt{t}} )[/tex]

When F(c(t)) = c'(t), then c(t) is a flow line of given velocity vector field F(x, y, z).

Since, [tex]F(c(t)) = c'(t)[/tex], we can say that c(t) is a flow line of given velocity vector field F(x, y, z).

The derivative of c(t) is calculated by differentiating each component with respect to t, resulting in c'(t)=(2t, 2, 3/2*t^(-1/2)). The velocity field F(c(t)) is found by substituting the equation of c(t) into F(x, y, z), resulting in F(c(t))=(2t+6, 2, 3/2*t^(-1/2)). As the results are equivalent, it's confirmed that c(t) is a flow line of the velocity vector field F(x, y, z).

In order to show that the curve c(t) is a flow line of the velocity vector field F(x, y, z), we first need to find c'(t), the derivative of c(t), and F(c(t)), the velocity field evaluated at the points along the curve c(t).

First, let's find c'(t). c(t) is given by (t^2, 2t − 6, 3sqrt(t)), so its derivative c'(t) is given by differentiating each component with respect to t: c'(t)=(2t, 2, 3/2*t^(-1/2)).

Next, let's find F(c(t)). F(x,y,z) is given by (y+6, 2, 9/2*z) so F(c(t)) is evaluated by substituting the equation of c(t) into F(x, y, z). Thus F(c(t))=(2t+6,2,3/2*t^(-1/2)).

Since F(c(t)) and c'(t) are equivalent, c(t) is indeed a flow line of the velocity vector field F(x, y, z).

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Step-by-step explanation:

To prove it we just use the definition of similar matrices and properties of determinants:

If [tex] A,B[/tex] are similar matrices, then there is an invertible matrix [tex]C[/tex], such that [tex] A=C^{-1}BC}[/tex] (that's the definition of matrices being similar). And so we compute the determinant of such matrix to get:

[tex]det(A)=det(C^{-1}BC)=det(C^{-1})det(B)det(C)[/tex]

[tex]=\frac{1}{det(C)}det(B)det(C)=det(B)[/tex]

(Determinant of a product of matrices is the product of their determinants, and the determinant of [tex]C^{-1}[/tex] is just [tex]\frac{1}{det(C)}[/tex])

Final answer:

The negation, contrapositive, converse, and inverse of a statement relating the divisibility of a number by 6, 2, and 3 are constructed by logically altering the original condition and consequent. These reflect different ways to express the relationship between the divisibility properties.

Explanation:

The original statement is: "If n is divisible by 6, then n is divisible by 2 and n is divisible by 3." Let's define the following propositions:

P: n is divisible by 6.

Q: n is divisible by 2.

R: n is divisible by 3.

The original statement can be written in logical form as P → (Q ∧ R).

Negation

The negation of the original statement is: "It is not the case that if n is divisible by 6, then n is divisible by 2 and n is divisible by 3." In logical form: ¬(P → (Q ∧ R)).

Contrapositive

The contrapositive of the original statement is: "If n is not divisible by 2 or n is not divisible by 3, then n is not divisible by 6." In logical form: (¬Q ∨ ¬R) → ¬P.

Converse

The converse of the original statement is: "If n is divisible by 2 and n is divisible by 3, then n is divisible by 6." In logical form: (Q ∧ R) → P.

Inverse

The inverse of the original statement is: "If n is not divisible by 6, then n is not divisible by 2 or n is not divisible by 3." In logical form: ¬P → (¬Q ∨ ¬R).

Answer:

Conjecture: next term is 45

Step-by-step explanation:

One very common sequence is one where the difference between one term and the previous one follows a recognizable pattern. Let's inspect the difference from one term to the previous one in the sequence:

Difference from 9 to 5:      9 - 5 = 4

Difference from 15 to 9:    15 - 9 = 6

Difference from 23 to 15: 23 -15 = 8

Difference from 33 to 23: 33-23= 10

At this point the pattern is clear, the differences are just even numbers increasing 2 by 2. We would expect next difference to be 12, and so the next term on the sequence should be 33 + 12 = 45.

[tex]\mbox{First, we compute the derivative of $y$ at $x_0=1$. So, we get}\\$$ y' = 3x^2 - 6x \, , \, y'(1) = -3 $$[/tex].

Therefore, the linear approximation at the point (1,-3) is

[tex]$$ y = -3 - 3(x -1) \ . $$[/tex]

To find the linear approximation of the non-linear system at the point (1, -3), first find the derivative of the function to get the slope of the tangent line at that point. Then, plug the slope and the point into the linearization formula. For the plotting part in Matlab, it should be a separate discussion as this platform does not support programming languages.

The subject of this question is a non-linear system given by the equation y=x^3 - 3x^2 - 1. The student is asked to find the linear approximation at the point (1, -3). The linear approximation of a function at a given point is the tangent line to the function at the given point, and it's also the best linear approximation of the function near that point.

Before we begin, let's define some terms. Linear approximation is a process of approximating the values of a nonlinear function using a line near a point. To find the linear approximation, we use the formula for the linearization of a function, L(x) = f(a) + f'(a)(x - a), where 'a' is the x-value of the point of tangency, f(a) is the y-value, and f'(a) is the slope of the tangent line at point 'a'. Tangent line is a straight line that just touches a curve at a given point. The tangent line is the best linear approximation to the curve at that point.

First, we need to find the derivative of the function, f'(x), which is 3x^2 - 6x. Then, evaluate f'(1) to find the slope of the tangent line. Plug these values into the linearization formula to get L(x) =  -3 + (3 - 6)(x - 1). Now, you can plot the original function and the linearization on the same graph.

Please note, for the Matlab portion of the question, it should be a separate discussion as this website is designed to walk through problems in a step-by-step manner and doesn't support running such programming languages directly. However, there are many online resources that can provide specific Matlab example codes for plotting functions and their linear approximations.

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To find the demand forecast for period t using the four-period weighted average, multiply each demand observation by its corresponding weight and sum the results.

To find the demand forecast for period t using the four-period weighted average, we multiply each demand observation by its corresponding weight and sum the results. In this case, we have:

Adding these weighted demands together gives us the demand forecast for period t:

Demand forecast for period t = 38 + 82 + 117 + 160 = 397.

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Answer:

The confidence interval is -5.3444 to 6.453 .

Step-by-step explanation:

We are given that  In a survey of 458 likely voters, 254 said that they would vote "yes" on the referendum.

So, n = 458

x = 254

We will use sample proportion over here

[tex]\widehat{p}=\frac{x}{n}[/tex]

[tex]\widehat{p}=\frac{254}{458}[/tex]

[tex]\widehat{p}=0.5545[/tex]

Confidence level = 95% = 0.95

Level of significance = 1-0.95 = 0.05

z value at 0.05 significance level = 1.96

Formula of confidence interval : [tex]\widehat{p}-x\times \sqrt{\frac{\widehat{p} \times (1-\widehat{p})}{n}[/tex] to [tex]\widehat{p}+x\times \sqrt{\frac{\widehat{p} \times (1-\widehat{p})}{n}[/tex]

Confidence interval : [tex]0.5545-254\times \sqrt{\frac{0.5545\times (1-0.5545)}{458}}[/tex] to [tex]0.5545+254\times \sqrt{\frac{0.5545\times (1-0.5545)}{458}}[/tex]

Confidence interval : [tex]-5.3444[/tex] to [tex]6.453[/tex]

Hence The confidence interval is -5.3444 to 6.453 .

Answer:

Step-by-step explanation:

We have given,              

x=254          

n=458          

Estimate for sample proportion

Level of significance is =1-0.95=0.05      

Z critical value(using Z table)=1.96      

Confidence interval formula is              

 =(0.5091,0.6001)              

Lower limit for confidence interval=0.5091

Upper limit for confidence interval=0.6001

Answer:

10 cans

Step-by-step explanation:

Number of servings to be prepared = 32

Weight of each pound = 3 oz

Yield factor per can = 0.8

Now,

Total weight of the cans = 32 × 3 = 96 oz

Actual weight required with yield factor 0.8 = [tex]\frac{96\ oz}{0.8}[/tex] =  120 oz

Therefore,

The number of 12 oz cans required = [tex]\frac{120}{12}[/tex]  = 10 cans

The Chef should order 10 cans.

To determine how many 12 oz. cans the Chef should order, we need to calculate the total amount of cooked beans required.

Since each serving is 3 oz. and there are 32 servings, the total amount needed is

3 oz. * 32 = 96 oz..

The yield factor per can is 0.8, so each can provides 0.8 * 12 oz. = 9.6 oz.

Therefore, the Chef should order 96 oz. / 9.6 oz. per can = 10 cans.

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Answer:

a) There is a 61,41% of none of the samples containing high levels of contamination.

b)There is a 32.52% probability that exactly one sample contains high levels of contamination.

c) There is a 38.59% probability that at least one contains high levels of contamination

Step-by-step explanation:

The probabilities are independent from each other. It means that the probability of selecting a lab specimen being contaminated is always 15%, no matter how many contaminated lab specimen have been chosen.

a) There are 3 independent samples. For each sample, the probability of it not being contaminated is 85%. So, the probability that none of the sample are contaminated is

[tex]P = (0.85)^3 = 0.6141 = 61,41%[/tex]

There is a 61,41% of none of the samples containing high levels of contamination.

b) There are 3 independent samples. For each sample, the probability of it being contaminated is 15% and not contaminated 85%.

So the probability the exactly one sample contains high levels of contamination is:

[tex]P = (0.85)^2(0.15) = 0.1084 = 10,84%[/tex]

There can be 3 orderings of the sample in these conditions.(C-NC-NC, NC-C-NC, NC-NC,C), so the probability that exactly one contains high levels of contamination is

P = 3*0.1084 = 0.3252 = 32.52%.

There is a 32.52% probability that exactly one sample contains high levels of contamination.

c) The sum of the probabilities is always 100%.

In relation to the existence of a contaminated sample, either:

-None of the samples are contaminated.

-At least one of the samples are contaminated.

So, the probability of at least one of the samples being contaminated is 100% - the probability that none of the samples are contaminated, that we have already found in a).

So, it is

100% - 61.41% = 38.59%

There is a 38.59% probability that at least one contains high levels of contamination

Answer:

(c) 0.46 million

Step-by-step explanation:

As provided immediate cash outlay = $8 million.

This will represent cash outflow at period 0, as it is made immediately, no time period has lapsed.

Cash inflows as provided and the respective present value factor are:

Year         Cash Inflow       Factor              Discounted Value

1                 $3 million         0.9091                $2,727,300

2                $4 million         0.8264               $3,305,600

3                 $2 million         0.7513                $1,502,600

Total present value of cash inflow = $7,535,500

Therefore, net present value = $7,535,500 - $8,000,000 = - $464,500

That is - 0.46 million

Correct option is

(c) 0.46 million

Answer:

Mid point will be 0.9887

Step-by-step explanation:

We have given a =0.9876 and b = 0.9887

And N = 2

We have to find midpoint

We know that formula for finding mid point that is

Midpoint [tex]=\frac{a+b}{2}[/tex]

So mid point will be

Midpoint [tex]=\frac{0.9876+0.9887}{2}=0.98815[/tex]

So the mid point between a = 0.9876 and b=0.9887 for N =2 will be 0.9887

Answer:

a) [tex]V = 3887.72 ft^{3}[/tex]

b)[tex]V = 104.97 m^{3}[/tex]

c)[tex]V = 104,968,468.538 cm^{3}[/tex]

Step-by-step explanation:

A tank has the format of a cylinder.

The volume of the cylinder is given by:

[tex]V = \pi r^{2}h[/tex]

In which r is the radius and h is the heigth.

The problem states that the diameter is measured to be 15.00 ft. The radius is half the diameter. So, for this tank

[tex]r = \frac{15}{2} = 7.50[/tex] ft

The height of the tank is 22 ft, so [tex]h = 22[/tex].

a) Volume of the tank in [tex]ft^{3}[/tex]:

[tex]V = \pi r^{2}h[/tex]

[tex]V = pi*(7.5)^2*22[/tex]

[tex]V = 3887.72 ft^{3}[/tex]

b) Volume of the tank in [tex]m^{3}[/tex]:

We must convert both the radius and the height to m.

Each feet has 0.30 m, so:

Radius:

1 feet - 0.30m

7.5 feet - r m

[tex]r = 7.5*0.30[/tex]

[tex]r = 2.25m[/tex]

Height

1 feet - 0.30m

22f - h m

[tex]h = 22*0.30[/tex]

[tex]r = 6.60m[/tex]

The volume is:

[tex]V = \pi r^{2}h[/tex]

[tex]V = pi*(2.25)^2*6.60[/tex]

[tex]V = 104.97 m^{3}[/tex]

c) Volume of the tank in [tex]cm^{3}[/tex]:

Each m has 100 cm.

So [tex]r = 2.25m = 225cm[/tex]

[tex]h = 6.60m = 660cm[/tex]

The volume is:

[tex]V = \pi r^{2}h[/tex]

[tex]V = pi*(225)^2*660[/tex]

[tex]V = 104,968,468.538 cm^{3}[/tex]

Answer:

the primes are all the even numbers in the equation

Step-by-step explanation:

Answer:

nth term of this sequence is [tex](197+(n+6)\times 3^{27})[/tex]

and 100th term is [tex](197+106\times 3^{27})[/tex].

Step-by-step explanation:

The given sequence is [tex](197+7\times 3^{27}),(197+8\times 3^{27}),(197+9\times 3^{27})[/tex]

Now we will find the difference between each successive term.

Second term - First term = [tex](197+8\times 3^{27})-(197+7\times 3^{27})[/tex]

                                         = [tex](8\times 3^{27}-7\times 3^{27})[/tex]

                                         = [tex]3^{27}(8-7)[/tex]

                                         = [tex]3^{27}[/tex]

Similarly third term - second term = [tex]3^{27}[/tex]

So there is a common difference of [tex]3^{27}[/tex].

It is an arithmetic sequence for which the explicit formula will be

[tex]T_{n}[/tex]=a + (n - 1)d

where [tex]T_{n}[/tex] = nth term of the arithmetic sequence

where a = first term of the arithmetic sequence

n = number of term

d = common difference in each successive term

Now we plug in the values to get the 100th term of the sequence.

[tex]T_{n}=(197+7\times 3^{27})+(n-1)\times 3^{27}[/tex]

               = [tex](197+(n+6)\times 3^{27})[/tex]

[tex]T_{100}=(197+7\times 3^{27})+(100-1)\times 3^{27}[/tex]

                   = [tex]197+7\times 3^{27}+99\times 3^{27}[/tex]

                   = [tex]197+106\times 3^{27}[/tex]

Therefore, nth term of this sequence is [tex](197+(n+6)\times 3^{27})[/tex]

and 100th term is [tex](197+106\times 3^{27})[/tex].