Solve the system of linear equations using the Gauss-Jordan elimination method.

(x,y,z)=__________________

2x + 2y − 3z = 16
2x − 3y + 2z = −4
4x − y + 3z =
−4

Answer:

  $2500

Step-by-step explanation:

The accrued interest is given by the formula ...

  i = Prt

i = interest; P = principal amount of the loan; r = annual interest rate; t = number of years.

__

Fill in the values and do the arithmetic.

  i = $10000×0.05×5 = $2500

The accrued interest on a $10,000 loan at a 5% annual interest rate for a term of 5 years is $2,500.

The accrued interest on a loan is calculated using the formula: Interest = Principal × rate × time. For a loan amount (Principal) of $10,000 at an annual interest rate (Rate) of 5% over a period (Time) of 5 years, the simple interest calculation would be:

Interest = Principal × rate × time
Interest = $10,000 × 0.05 × 5
Interest = $500 × 5
Interest = $2,500

Therefore, the accrued interest over the term of 5 years on a $10,000 loan at a 5% annual interest rate would be $2,500.

Answer:

a) 18 cm

b) 18

c) 3

The Earl's height is unusual because the  z score does not lies in the given range of usual i.e -2 and 2

Step-by-step explanation:

Given:

Mean height, μ = 174 cm

Standard deviation = 6 cm

height of Earl, x = 192 cm

a) The positive difference between Earl height and the mean = x - μ

= 192 - 174 = 18 cm

b) standard deviations is 18

c) Now,

the z score is calculated as:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

or

[tex]z=\frac{192-174}{6}[/tex]

or

z = 3

The Earl's height is unusual because the  z score does not lies in the given range of usual i.e -2 and 2

Final answer:

Earl's height is 18 cm above the mean, which is 3 standard deviations from the mean, resulting in a z-score of 3. Considering the usual z-score range of -2 to 2, Earl's height is hence considered unusual.

Explanation:

The problem at hand involves understanding the concepts of statistics, particularly regarding mean, standard deviation, and z-scores in the context of normal distribution.

a. Positive Difference Between Earl's Height and the Mean

The positive difference between Earl's height and the mean is simply calculated by subtracting the mean from Earl's height. If Earl's height is 192 cm and the mean height is 174 cm, the difference is:

192 cm - 174 cm = 18 cm

b. Standard Deviations from the Mean

The number of standard deviations from the mean is found by dividing the difference by the standard deviation. Since the standard deviation is 6 cm,
18 cm / 6 cm = 3 standard deviations

c. Z-score Conversion

The z-score is calculated using the formula:

z = (Earl's height - mean) / standard deviation

z = (192 cm - 174 cm) / 6 cm = 3

d. Usuality of Earl's Height

Since "usual" heights translate to z-scores between -2 and 2, a z-score of 3 indicates that Earl's height is unusual.

Answer:

E and O are equivalents.

Step-by-step explanation:

We will define the one-to-one correspondence this way

E           O

0           1

2           3

4           5

6           7

8           9

.................

2n        2n - 1

This way, every 2n even number will have its correspondence with a 2n-1 odd number.

[tex]\displaystyle\lim_{x\to-2^+}\frac{x-1}{x^2(x+2)}[/tex]

The limit is infinite because the denominator approaches 0 while the numerator does not, since [tex]x+2=0[/tex] when [tex]x=-2[/tex]. Which infinity it approaches (positive or negative) depends on the sign of the other terms for values of [tex]x[/tex] near -2.

Since [tex]x\to-2[/tex] from the right, we're considering values of [tex]x>-2[/tex]. For example, if [tex]x=-1.9[/tex], then [tex]\dfrac{x-1}{x^2}=\dfrac{-2.9}{1.9^2}<0[/tex]; if [tex]x=-1.99[/tex], then [tex]\dfrac{x-1}{x^2}=\dfrac{-2.99}{1.99^2}<0[/tex], and so on. We can keep picking values of [tex]x[/tex] that get closer and closer to -2, and we would see that [tex]\dfrac{x-1}{x^2}[/tex] contributes a negative sign every time. So the limit must be [tex]\boxed{-\infty}[/tex].

[tex]\displaystyle\lim_{x\to0}\frac{x-1}{x^2(x+2)}[/tex]

By similar reasoning above, we see that [tex]\dfrac{x-1}{x+2}[/tex] contributes a negative sign regardless of which side we approach 0 from. [tex]x-1[/tex] is always negative and [tex]x+2[/tex] is always positive, so the net effect is a negative sign and the limit from either side is [tex]\boxed{-\infty}[/tex].

[tex]\displaystyle\lim_{x\to2\pi}x\csc x=\lim_{x\to2\pi}\frac x{\sin x}[/tex]

Direct substitution gives 0 in the denominator. For [tex]x>2\pi[/tex] we have [tex]\sin x>0[/tex], and for [tex]x<2\pi[/tex] we have [tex]\sin x<0[/tex]. Meanwhile, the numerator stays positive, which means the limit is positive or negative infinity depending on the direction in which [tex]x[/tex] approaches [tex]2\pi[/tex], so this limit does not exist.

Answer:

The following system is not linear.

The following system is time-invariant

Step-by-step explanation:

To determine whether a system is linear, the following condition must be satisfied:

[tex]f(a) + f(b) = f(a+b)[/tex]

For [tex]y(t) = cos(3t)[/tex], we have

[tex]y(a) = cos(3at)[/tex]

[tex]y(b) = cos(3bt)[/tex]

[tex]y(a+b) = cos(3(a+b)t) = cos(3at + 3bt)[/tex]

In trigonometry, we have that:

[tex]cos(a+b) = cos(a)cos(b) - sin(a)sin(b)[/tex]

So

[tex]cos(3at + 3bt) = cos(3at)cos(3bt) - sin(3at)sin(3bt)[/tex]

[tex]y(a) + y(b) = cos(3at) + cos(3bt)[/tex]

[tex]y(a+b) = cos(3(a+b)t) = cos(3at + 3bt) = cos(3at)cos(3bt) - sin(3at)sin(3bt)[/tex]

Since [tex]y(a) + y(b) \neq y(a+b)[/tex], the system [tex]y(t) = cos(3t)[/tex] is not linear.

If the signal is not multiplied by time, it is time-invariant. So [tex]y(t) = cos(3t)[/tex]. Now, for example, if we had [tex]y(t) = t*cos(3t)[/tex] it would not be time invariant.

Answer: [tex]\dfrac{1}{6}[/tex]

Step-by-step explanation:

We know that the total number of outcomes for fair dice {1,2,3,4,5,6} = 6

Given : Favorable outcome = 5

i.e. Number of favorable outcomes =1

We know that the formula to find the probability for each event is given by :-

[tex]\dfrac{\text{Number of favorable outcomes}}{\text{Total outcomes}}[/tex]

Then, the probability that the six  faces cube will land with the number 5 face up will be :_

[tex]\dfrac{1}{6}[/tex]

Hence, the required probability = [tex]\dfrac{1}{6}[/tex]

Answer:

The percentage of her total daily allowance of fat is 68.57%

Step-by-step explanation:

The complete question is:

Isla wants to watch her intake of fat per day. She estimates that the dinner will have 48 grams of fat. What percentage of her total Daily allowance of fat Will the dinner be? Daily allowance is 70g of fat

Solution:

Estimated fat in dinner =  48 grams

Daily allowance of fat = 70 grams

We have to find the percentage of her total Daily allowance of fat = ?

%fat = Estimated fat in dinner × 100 / Daily allowance of fat

%fat = 48×100/70

%fat = 68.57 %

Thus the percentage of her total daily allowance of fat is 68.57%

Without knowing Isla's total daily fat allowance, we can't work out the percentage that her dinner represents. However, if we knew this, we would divide the grams of fat in her dinner by her daily allowance and then multiply by 100 to find the percentage.

To answer this question, we would need to have information on what Isla's total daily allowance of fat is. Without this information, it's impossible to calculate the percentage. If we knew her daily allowance, we would divide the amount of fat in her dinner (48 grams) by her total daily fat allowance and multiply by 100 to get the percentage. For example, if her daily allowance was 100 grams, her dinner would comprise 48% of her daily fat intake.

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Answer:

Infinite circles but with values (h,k,r)=(h,3h-3,√(10h²-30h+25))

Step-by-step explanation:

You are correct, if we only have two points, then we will have infinite possibilities, but all those circles will have to be attached to those two points. We have to find a condition to have infinite circles which pass through those points.

Let h be the x coordinate of the center of the circle

Let k be the y coordinate of the center of the circle

Let r be the radius of the circle. Having the circle equation:

[tex](x-h)^{2}+(y-k)^{2}=r^{2}[/tex]

And now expand every binomial and substitute the values of the points:

[tex](x-h)^{2}+(y-k)^{2}=r^{2}\\\\(0-h)^{2}+(2-k)^{2}=r^{2}\\\\(3-h)^{2}+(1-k)^{2}=r^{2}[/tex]

[tex]x^{2}-2xh+h^{2}+y^{2}-2yk+k^{2}=r^{2}\\\\h^{2}+4-4k+k^{2}=r^{2} (I)\\\\9-6h+h^{2}+1-2k+k^{2}=r^{2}[/tex]

Now substract the first and secod equation, and then the first and the third one:

[tex]x^{2}-2xh+y^{2}-4-2yk+4k=0\\\\x^{2}-9-2xh+6h+y^{2}-1-2yk+2k=0[/tex]

Now we substract the second equation from the first one:

[tex]-4+4k+9-6h+1-2k=0[/tex]

Reduce terms and we will have a linear equation, clear k:

[tex]2k-6h+6=0\\\\k=3h-3[/tex]

The last equation let us know the value of k if we asign any value to h.

Now we substitute h and k=3h-3 in I:

[tex]h^{2}+4-4(3h-3)+(3h-3)^{2}=r^{2}\\\\10h^{2}-30h+25=r^{2}\\\\r=\sqrt{10h^{2}-30h+25}[/tex]

The last equation let us know the value of r if we asign any value to h.

These equations (in fact functions), will let us know every set for h, k and r to form a circle (which will pass through the mentioned points)

[tex](h,k,r)=(h,3h-3,\sqrt{10h^{2}-30h+25})[/tex]

Where the indepent variable is h for any real, k is any real and the minimum r is √2.5:

[tex]h:R\\k:R\\r:(\sqrt{2.5},infinite)[/tex]

Step-by-step explanation:

To prove the identity we just manually compute the left hand side of it, simplify it and check that we do get the right hand side of it:

[tex](a+b)^2=(a+b)\cdot(a+b)[/tex] (that's the definition of squaring a number)

[tex]=a\cdot a + a\cdot b + b \cdot a +b \cdot b[/tex] (we distribute the product)

[tex]=a^2+ab+ba+b^2[/tex] (we just use square notation instead for the first and last term)

[tex]=a^2+ab+ab+b^2[/tex] (since product is commutative, so that ab=ba)

[tex]=a^2+2ab+b^2[/tex] (we just grouped the two terms ab into a single term)

Answer:

[tex]t = 9.05[/tex] weeks

Step-by-step explanation:

The mass of this particular substance can be modeled by the following exponential function:

[tex]m(t) = m(0)*e^{rt}[/tex]

In which [tex]m(t)[/tex] is the mass in function of time, [tex]m(0)[/tex] is the initial mass and r, in decimal, is the growth rate of the mass.

The problem states that:

The mass of a particular substance is known to grow exponentially at a rate of 17% per week. Its initial mass was 12 grams and, after t weeks, it weighed 56 grams. So:

[tex]r = 17% = 0.17[/tex]

[tex]m(0) = 12[/tex]

[tex]m(t) = 56[/tex]

We have to solve this equation for t. So:

[tex]m(t) = m(0)*e^{rt}[/tex]

[tex]56 = 12*e^{0.17t}[/tex]

[tex]e^{0.17t} = \frac{56}{12}[/tex]

[tex]e^{0.17t} = 4.67[/tex]

To solve for [tex]t[/tex], we put ln in both sides

[tex]ln e^{0.17t} = ln 4.67[/tex]

[tex]0.17t = 1.54[/tex]

[tex]t = \frac{1.54}{0.17}[/tex]

[tex]t = 9.05[/tex] weeks

Answer:

  76%

Step-by-step explanation:

Let f represent the score on the final. Then Inara wants the final grade to be ...

  0.25·74 +0.20·99 +0.55·x ≥ 80

  38.3 +0.55x ≥ 80

  0.55x ≥ 41.7

  x ≥ 41.7/0.55 ≈ 75.82

Inara needs a final exam grade of 76% or better to earn at least 80% in the course.

Answer:

The average sales per week for the first 16 weeks is $477 million dollars per week.

Step-by-step explanation:

Here, you have to find the average value of a continuous function over an interval.

Suppose you have a function [tex]f(x)[/tex] over an interval from a to b. The average of the function in this interval is given by:

[tex]\frac{1}{b-a}\int\limits^b_a {f(x)} \, dx[/tex]

Solution:

In this problem, the function is given by:

[tex]S(t) = 59t + 5[/tex]

The problem asks the average value for the first 16 weeks. It means that our interval goes from 0 to 16. So [tex]a = 0, b = 16[/tex].

The average value is given by the following integral:

[tex]A = \frac{1}{16}\int\limits^{16}_{0} {(59t + 5)} \, dt[/tex]

[tex]A = \frac{59t^{2}}{32} + \frac{5t}{16}, 0 \leq t \leq 16[/tex]

[tex]A =\frac{59*(16)^{2}}{32} + \frac{5*16}{16}[/tex]

[tex]A = $477[/tex]

The average sales per week for the first 16 weeks is $477 million dollars per week.

The line ups for the basketball team is 108.

The arrangement of the different things or numbers in a number of ways is called the combination.

It is given that a basketball team has 2 point guards, 3 shooting guards, 3 small forwards, 3 power forwards, and 2 centres.

The Line up will be:-

Combination = 2x3x3x3x2

Combination = 108.

Hence, the lineup will have 108 arrangements.

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Final answer:

The subject of the question is Mathematics, combinatorics specifically, and we're calculating the number of basketball line-ups possible. By multiplying the choices for each position (2 point guards, 3 shooting guards, 3 small forwards, 3 power forwards, 2 centers), we find that there are 108 possible line-ups.

Explanation:

The subject of the question is Mathematics, specifically combinatorics, which is concerned with counting the different ways in which objects can be arranged or combined. In this particular problem, we are looking to find the number of possible basketball line-ups given a certain number of players available for each position.

To solve this, we need to calculate the product of the different choices for each position: 2 choices for point guard, 3 choices for shooting guard, 3 choices for small forward, 3 choices for power forward, and 2 choices for center. Since we are choosing one player for each position to form a single line-up, we multiply the number of choices for each position together.

The calculation is as follows:

For Small Forward: 3 possible choices

For Center: 2 possible choices

Thus, the total number of possible line-ups is:

2 (Point Guards) * 3 (Shooting Guards) * 3 (Small Forwards) * 3 (Power Forwards) * 2 (Centers) = 108 possible line-ups.

Answer:

10.27

Step-by-step explanation:

Find the number in the hundredth place  7  and look one place to the right for the rounding digit 1 . Round up if this number is greater than or equal to  5  and round down if it is less than  5 . And the answer is 10.27 which is rounded to the nearest hundredth.

Here, we are required to round the number 10.2719 to the nearest hundredth.

While considering place values of numbers,

Therefore, according to the question, the digit with occupies the hundredth position is 7.

As such, to round off the number 10.2719, the digit which is after the digit 7 is considered.

If the digit is less than 5, it is rounded to 0, and if greater or equal to 5, it is rounded to 1 and ultimately added to the preceding number.

In this case, the number is 1 and since 1 is less than 5, it is rounded to 0 and added to 7.

Ultimately, the number 10.2719 rounded to the nearest hundredth is;

10.27

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Answer:

Option d - 204 m

Step-by-step explanation:

Given : The atmospheric pressures at the top and the bottom of a building are read by a barometer to be 96.0 and 98.0 kPa. If the density of air is 1.0 kg/m³.

To find : The height of the building ?

Solution :

We have given atmospheric pressures,

[tex]P_{\text{top}}=96\ kPa[/tex]

[tex]P_{\text{bottom}}=98\ kPa[/tex]

The density of air is 1.0 kg/m³ i.e. [tex]\rho_a=1\ kg/m^3[/tex]

Atmospheric pressure reduces with altitude,

The height of the building is given by formula,

[tex]H=\frac{\triangle P}{\rho_a\times g}[/tex]

[tex]H=\frac{P_{\text{bottom}}-P_{\text{top}}}{\rho_a\times g}[/tex]

[tex]H=\frac{(98-96)\times 10^3}{1\times 9.8}[/tex]

[tex]H=\frac{2000}{9.8}[/tex]

[tex]H=204\ m[/tex]

Therefore, Option d is correct.

The height of the building is 204 meter.

Using the barometric pressure formula and the given atmospheric pressures at the top and bottom of the building, the height is calculated to be approximately 204 meters, which matches option (d).

To calculate the height of the building using the difference in atmospheric pressure at the top and bottom, we can use the barometric pressure formula P = h ρ g, where P is the pressure, h is the height, ρ is the density of the fluid (or air in this case), and g is the acceleration due to gravity (approximately 9.8 m/s²).

The difference in pressure between the two points can be used to solve for h.

Given:

Difference in atmospheric pressure ΔP = 98.0 kPa - 96.0 kPa = 2.0 kPa

Density of air ρ = 1.0 kg/m3

Acceleration due to gravity g = 9.8 m/s²

We rearrange the formula to solve for h: h = ΔP / ( ρg)

h = (2.0 kPa) / (1.0 kg/m³ · 9.8 m/s²)

h = (2000 Pa) / (9.8 N/kg)

h = 204.08 m

The height of the building is therefore approximately 204 meters, making option (d) the correct answer.

Answer:

lim f(x)=(11x-40)/(x-4) // for all x's, except x=4

lim f(x) ≈ ∞    // if x=4

Step-by-step explanation:

lim f(x) - 4/(x - 4) = 11

lim f(x) = 11 + 4/(x - 4)= (11x-40)/(x-4)

at x=4, there is a indetermination point, the function tends to infinity at that point

So:

lim f(x)=(11x-40)/(x-4) // for all x's, except x=4

lim f(x) ≈ ∞    // if x=4

Answer:

Maximum profits are earned when x = 64 that is when 64 units are sold.

Maximum Profit = P(64) = 2,08,490.666667$

Step-by-step explanation:

We are given the following information:[tex]P(x) = 6400x - 18x^2 - \frac{x^3}{3} - 40000[/tex], where P(x) is the profit function.

We will use double derivative test to find maximum profit.

Differentiating P(x) with respect to x and equating to zero, we get,

[tex]\displaystyle\frac{d(P(x))}{dx} = 6400 - 36x - x^2[/tex]

Equating it to zero we get,

[tex]x^2 + 36x - 6400 = 0[/tex]

We use the quadratic formula to find the values of x:

[tex]x = \displaystyle\frac{-b \pm \sqrt{b^2 - 4ac} }{2a}[/tex], where a, b and c are coefficients of [tex]x^2, x^1 , x^0[/tex] respectively.

Putting these value we get x = -100, 64

Now, again differentiating

[tex]\displaystyle\frac{d^2(P(x))}{dx^2} = -36 - 2x[/tex]

At x = 64,  [tex]\displaystyle\frac{d^2(P(x))}{dx^2} < 0[/tex]

Hence, maxima occurs at x = 64.

Therefore, maximum profits are earned when x = 64 that is when 64 units are sold.

Maximum Profit = P(64) = 2,08,490.666667$

To solve the problem, we write the equation g + 300 - 2(700) = 1500, where g represents the initial weight of the mound of gravel. Simplifying the equation, we find that g is equal to 2600 pounds.

To write an equation that describes the situation, let g be the initial weight of the mound of gravel in pounds. Throughout the day, 300 pounds of gravel are added to the mound. Two orders of 700 pounds each are sold and the corresponding amounts are removed from the mound. At the end of the day, the mound has 1,500 pounds of gravel.

The equation representing this situation can be written as:

g + 300 - 2(700) = 1500

To solve for g, we simplify the equation:

g + 300 - 1400 = 1500

g - 1100 = 1500

g = 1500 + 1100

g = 2600 pounds

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g = 2600 pounds.

Let g represent the initial amount of gravel in pounds at the start of the day.

During the day, 300 pounds of gravel are added, so we add 300 to g.

Two orders of 700 pounds of gravel are sold.
Thus, we subtract a total of 2  times 700 (1400) pounds from the amount after the addition.

The mound of gravel at the end of the day is reported to be 1,500 pounds.

We can write the equation as,
g + 300 - 1400 = 1500
g - 1100 = 1500
g = 1500 + 1100

g = 2600

Therefore, the initial amount of gravel at the start of the day was 2600 pounds.

The answers are:

We know that if we have a set of N elements, the number of different groups of K elements that we can make, such that:

N ≥ K

Is given by:

[tex]C(N, K) = \frac{N!}{(N - K)!*K!}[/tex]

a) If we have 20 semiconductors and we make groups of 3, then we can have:

[tex]C(20, 3) = \frac{20!}{(20 - 3)!*3!} = \frac{20*19*18}{3*2} = 1,140[/tex]

b) The probability that a randomly picked chip is nonconforming is given by the quotient between the number of nonconforming chips and the total number of chips.

So, if the first selected chip is the nonconforming one, the probability of selecting it is:

P = 10/20 = 1/2

The next two ones work properly, the probability is computed in the same way, but notice that now there are 10 proper chips and 19 chips in total.

Q = 10/19

And for the last one we have:

K = 9/18

The joint probability is:

P*Q*K = (1/2)*(9/19)*(8/18) = 0.132

But this is only for the case where the first one is the nonconforming, then we must take in account the possible permutations (there are 3 of these) then the probability is:

p = 3*0.132 = 0.395

c) The probability of getting at least one nonconforming chip is equal to the difference between 1 and the probability of not getting a nonconforming chip.

That probability is computed in the same way as above)

P = (10/20)*(9/19)*(8/18) = 0.105

Then we have:

1 - P = 1 - 0.105 = 0.895

So the probability of getting at least one nonconforming chip is 0.895.

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The total number of different samples is 1140, the number of samples with exactly one nonconforming chip is 450, and the number of samples with at least one nonconforming chip is 1020. The problem involves calculating combinations related to semiconductor chip samples.

In this problem, we are dealing with combinations and probabilities. Let's denote the total number of chips as 20, the number of nonconforming chips as 10, and the sample size as 3.

a. How many different samples are possible?

We use the combination formula to find the total number of different samples of 3 chips that can be drawn from 20 chips.

b. How many samples of 3 contain exactly one nonconforming chip?

To get samples with exactly one nonconforming chip, we need to choose 1 nonconforming chip from 10, and 2 conforming chips from the remaining 10.

c. How many samples of 3 contain at least one nonconforming chip?

First, calculate the total number of samples that contain no nonconforming chips.

Using the complement rule:

Number of samples with at least one nonconforming chip: Total samples - Samples with no nonconforming chips

Given : Significance level : [tex]\alpha: 1-0.99=0.01[/tex]

Critical value : [tex]z_{\alpha/2}=2.576[/tex]

Margin of error : [tex]E=5[/tex]

Standard deviation : [tex]\sigma=6[/tex]

The formula to find the sample size :-

[tex]n=(\dfrac{z_{\alpha/2}\times\sigma}{E})^2[/tex]

Then, the sample size will be :-

[tex]n=(\dfrac{(2.576)\times6}{5})^2\\\\=(3.0912)^2=9.55551744\approx10[/tex]

The minimum final size sample required is 10.

If only 30 percent of households have a cat, then the proportion of households have a cat = 0.3

The formula to find the sample size :-

[tex]n=p(1-p)(\dfrac{z_{\alpha/2}}{E})^2[/tex]

Then, the sample size will be :-

[tex]n=0.3(1-0.3)(\dfrac{(2.576)}{5})^2\\\\=(0.21)(0.5152)^2=0.0557405184\approx1[/tex]

Hence, the initial number of households that need to be contacted =1

Final answer:

The final sample size required is 94 cat owners and the initial number of households that need to be contacted is approximately 314, accounting for the 30% ownership rate of cats among households.

Explanation:

To determine the final sample size required for a survey that aims to establish the average number of cans of cat food purchased by cat owners monthly, we use the formula for sample size in estimating a mean:

[tex]n = \left(\frac{Z \cdot \sigma}{E}\right)^2[/tex]

Where n is the sample size, Z is the Z-score associated with the confidence level, σ is the standard deviation, and E is the margin of error.

For a 99% confidence level, the Z-score is approximately 2.576 (from the Z-table). Given that the standard deviation (σ) is 6 units and the margin of error (E) is 5 units, the calculation is as follows:

[tex]n = (2.576 * 6 / 5)^2 \approx 9.68^2 \approx 93.7[/tex]

So, we would need at least 94 cat owners (rounding up since we cannot have a fraction of a person).

To calculate the initial number of households to be contacted, considering that only 30% have cats, you would divide the needed sample size by the proportion of households with cats:

Initial Households = Final Sample Size / Proportion of Cat Owners ≈ 94 / 0.30 ≈ 313.3

Therefore, approximately 314 households should be initially contacted to ensure that the final sample of cat owners is achieved.

Answer:

Answered

Step-by-step explanation:

It is a combbinatorics problem. let's think as we need to do 8 partitions  

of these 13 to separate the postcards of different types. So The number of

partitions of n=13 into r=8 terms counting 0's as terms as  C(n+r-1,r-1)​.

(a)

Here n=13 and r=8, put it in the above formula so we get C(13+8-1,8-1)= C(20,7)= 77520 selections.

b).

Here, either (i) we can choose none of type I or (ii) we choose one of type I

Case(i): r=7, n=12 (Here we have only 7 types to choose from​)

Case(ii): r=7, n=11 (Here we have only 11 cards to choose and only 7 types to choose them from)​

Case (i) + Case(ii) = ,C(12+7-1,7-1) + C(11+7-1,7-1) = C(18,6) + (17,6) = 18564+12376 = 30940 selections.

Answer:

[tex]l_n=3^n-1[/tex]

Step-by-step explanation:

We will prove by mathematical induction that, for every natural n,  

[tex]l_n=3^n-1[/tex]

We will prove our base case (when n=1) to be true:

Base case:

As stated in the qustion, [tex]l_1=2=3^1-1[/tex]

Inductive hypothesis:  

Given a natural n,  

[tex]l_n=3^n-1[/tex]

Now, we will assume the inductive hypothesis and then use this assumption, involving n, to prove the statement for n + 1.

Inductive step:

Let´s analyze the problem with n+1 stones. In order to move the n+1 stones from A to C we have to:

Then,

[tex]l_{n+1}=3l_n+2[/tex].

Therefore, using the inductive hypothesis,

[tex]l_{n+1}=3l_n+2=3(3^n-1)+2=3^{n+1}-3+2=3^{n+1}-1[/tex]

With this we have proved our statement to be true for n+1.    

In conlusion, for every natural n,

[tex]l_n=3^n-1[/tex]

Answer:

a, b and c= (-∞,∞) d = (-∞,3) U (-3,3) U (3,∞)

Step-by-step explanation:

Hi there!

1) Firstly, let's recap the sum of functions rule:

f(x) + g(x) = (f+g)(x)

Applying it to those functions, we have:

f(x)=5-x +g(x)=[tex]x^{2} -9[/tex] = [(5-x) +([tex]x^{2}[/tex]-9)](x)

(f+g)(x)=[tex]5-x+x^{2} -9[/tex]

2) To State the Domain is to state the set which is valid the quantities of x, of a function. In this case,

a) (f+g)(x)= 5-x+[tex]x^{2}[/tex]+9

Simplifiying

(f+g)(x) =[tex]x^{2} -x+4=0[/tex]

Since there are no restrictions neihter discontinuities, this function has a Domain which can expressed this way:

X may assume infinite quantities, negatives or positives one in the Real set.

(-∞< x <∞+) or simply (-∞,∞)

Or simply put, x ∈ R. Remember, ∞ is not a number, it's a notation meaning infinite values. That's why it's not a closed interval.

Check the graph below.

b) (f-g)(x) =(5-x) -(x²-9)

(f-g)(x)= 5-x-x²+9

Domain of (f-g)(x) =x²-x+14

Similarly to a) this function (f-g)(x) has not discontinuity, nor restrictions on its Domains.

Since there are no restrictions either discontinuities, this function has a Domain which can be expressed this way:

X may assume infinite quantities, negatives or positives one in the Real set.

(-∞< x <∞ +) then finally, the answer: (-∞,∞)

c) (f*g)(x)=(5-x)(x²-9)

(f*g)(x)=5x²-45-x³+9x

Again, this function has no discontinuities, nor restrictions in its Domain as you can check it on its graph.

Then, the Domain of (f*g)(x)=(5-x)(x²-9) is also (-∞,∞)

d) (f/g)(x) =(5-x)/(x²-9)

Highlighting the denominator, we can calculate the Domain.

We can see a restriction here. There is no denominator zero, defined for the Set of R.

Then, let's calculate

[tex]x^{2} -9>0\\ \sqrt{x^2} >\sqrt{9}\\x>3[/tex] and x < -3

In the Numerator, no restrictions.

So the Domain will be the union between the Numerator's Domain and the Denominator's Domain with Restrictions.

Check the graph below.

Finally

D = (-∞,3) U (-3,3) U (3,∞)

Final answer:

The sum, difference, product, and quotient of two functions f(x) = 5 - x and g(x) = x^2 - 9 can be found by adding, subtracting, multiplying, and dividing their corresponding terms. The domain of each resulting function can be determined by considering the domain of the original functions and any restrictions imposed by the operations involved.

Explanation:

(a) f + g

To find the sum of two functions, we simply add the corresponding terms. So, f + g = (5 - x) + (x^2 - 9).

Simplifying, we get f + g = x^2 - x - 4.

The domain of the function f + g is the intersection of the domains of f and g. Since both f and g are defined for all real numbers, the domain of f + g is also all real numbers, which can be represented as (-∞, ∞) in interval notation.

(b) f - g

To find the difference of two functions, we subtract the corresponding terms. So, f - g = (5 - x) - (x^2 - 9).

Simplifying, we get f - g = -x^2 + x + 14.

Similar to part (a), the domain of the function f - g is all real numbers, which can be represented as (-∞, ∞) in interval notation.

(c) fg

To find the product of two functions, we multiply the corresponding terms. So, fg = (5 - x)(x^2 - 9).

Simplifying, we get fg = -x^3 + 9x - 5x^2 + 45.

Similar to parts (a) and (b), the domain of the function fg is all real numbers, which can be represented as (-∞, ∞) in interval notation.

(d) f/g

To find the quotient of two functions, we divide the corresponding terms. So, f/g = (5 - x)/(x^2 - 9).

However, we need to consider the values of x that make the denominator zero, since division by zero is undefined. In this case, the denominator x^2 - 9 is equal to zero when x = ±3.

Therefore, the domain of the function f/g is all real numbers except x = ±3. In interval notation, this can be represented as (-∞, -3) ∪ (-3, 3) ∪ (3, ∞).

Answer:

Extensive property

Step-by-step explanation:

The intensive properties does not depend on the amount of mass of the system or the size of the system, for example the density [tex]\rho[/tex] is a quantity that is already defined for the system, because the density of water is equal for a drop or for a pool of water.

In the case of extensive properties the value of them is proportional to the size or mass of the system. For example mass is an extensive property because depend on the amount of substance. Other example could be the enegy.

In the case of weigth is a quantity that depends on mass value, then weigth is an extensive property.

Answer: The slope ($32) is giving the rate of each widgets.

Step-by-step explanation:

The equation of a line in intercept form is given by :-

[tex]y=mx+c,[/tex] where the coefficient of x is the slope of line.

Given   :  The total cost of making and selling x widgets is [tex]C = 32x + 560[/tex] is in intercept form, where the coefficient of x = 32.

Since the slope gives the rate of change of dependent variable w.r.t. independent variable.

Therefore, the meaning of the slope in this case is the the rate of change of cost per unit of widgets x.

i.e Slope gives the cost of each widgets =$32.

Answer:

17.21 × 10^7

Step-by-step explanation:

When writting in scientific notation, you only place the decimal as 17.21. Then, you count how many zeros are after of before the numbers. If it's before 17.21, then it would be 17.21^-7, if it is after 17.21, then it would be 17.21^7.

Answer:

[tex]1.721 \cdot 10^{10}[/tex]

Step-by-step explanation:

17,210,000,000 written in scientific notation

To write the number in scientific notation we move the decimal point after the first number.

To move the decimal point after 1, move 10 places to the left

So the number becomes

1.7210000000

Ignore all the zeros. we moved 10 places to the left so multiply with 10^10

[tex]1.721 \cdot 10^{10}[/tex]

Answer:

n = 135 tablets

Step-by-step explanation:

See in the picture.

Answer:

No. See explanation below.

Step-by-step explanation:

A random sample is a type of sample in which every item has the same probability of being selected. Equally, after we take one item, the remaining items keep having the same probability of being selected.

In this problem, the vitamin pills are firstly obtained from the pills produced in an hour at the Health Supply Company, therefore, not all pills had the same probability of being selected in the first place.

Answer:

Horizontal distance = 1218.41 ft

Step-by-step explanation:

Given data:

Slope distance = 1223.88 ft

Zenith angle is  = 95°25'14"

converting zenith angle into degree

Zenith angle is [tex]= 95 +\frac{25}{60} + \frac{14}{3600} =[/tex] 94.421°

Horizontal distance[tex]= S\times sin(Z)[/tex]

putting all value to get horizontal distance value

Horizontal distance [tex]= 1223.88\times sin(95.421)[/tex]

Horizontal distance = 1218.41 ft

Answer:

The number of bicycles is 19.

and the number of tricycles is 4.

Step-by-step explanation:

As we know that the number of seats in bicycles as well as in tricycles is 1.

The number of wheels in bicycles is 2

and the number of wheels in tricycles is 3.

Let the number of bicycles be x.

and the number of tricycles be y.

Thus using the given information we can make equation as,

x + y = 23

and, 2x + 3y = 50

Solving these two equations:

We get, x = 19 and y = 4

Thus the number of bicycles is 19.

and the number of tricycles is 4.