An urn contains 2 red marbles and 3 blue marbles. 1. One person takes two marbles at random from the urn and does not replace them. a) State the general ways in which the person could get a red marble and a blue marble b) State the number of ways this can occur. c) What is the probability the person gets a red and a blue marble? P(R & B) =

Answer:

The number of solutions of a system is given by the number of different variables in the system, this number has to be the same as the number of independent equations. The coefficients and the augmented matrix of the system show these values in a matrix form. A system has a unique solution when the rank of both matrixes and the number o variables in the system are the same.

Step-by-step explanation:

For example, the following system has 2 different variables, x and y.

[tex]x+y=1\\x+2y=5[/tex]

In order to find a unique solution to the system, the number of independent equations and variables in the system must be the same In the previous example, you have 2 independent equations and 2 variables, then the solution of the system is unique.  

The rank of a matrix is the dimension of the vector generated by the columns, in other words, the rank is the number of independent columns of the matrix.  

According to Rouché-Capelli Theorem, a system of equations is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. The inconsistency of the system is because you can't find a combination of the variables that will solve the system.

If the intravenous fluid is adjusted to deliver 15 mg of medication per hour, the dosage rate would be 0.5 mg of medication delivered per half minute.

First, let's convert the 1 hour dosage rate of 15 mg to minutes. Since there are 60 minutes in an hour, divide 15 mg by 60 to find the dosage rate per minute.

15 mg / 60 min = 0.25 mg/min

Next, convert the dosage rate from minutes to half minutes. Since there are 2 half minutes in a minute, multiply the dosage rate per minute by 2 to find the dosage rate per half minute.

0.25 mg/min * 2 = 0.5 mg/half min

Therefore, if the intravenous fluid is adjusted to deliver 15 mg of medication per hour, the dosage rate would be 0.5 mg of medication delivered per half minute.

Know more about unit conversion,

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If 15 mg of medication is delivered per hour, the rate per half-minute would be 0.125 mg. This is calculated by converting 15 mg/hour to 0.25 mg/minute, and then halving to find the dose per half-minute.

The student's question pertains to the calculation of dosage rates in medical treatment, specifically how much medication is delivered per half-minute given a rate of 15 mg of medication per hour. To figure this out, it's a simple conversion from hours to minutes and then to half-minutes.

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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: 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:

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]

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.

Answer:

You should make 200 quarts of Creamy Vanilla and 50 quarts of Continental Mocha to earn the largest profit.

Step-by-step explanation:

You are going to earn the largest profit when you manage to use all the eggs and cups of cream that you have in stock.

This problem can be solved by a first order equation

I am going to call x the number of quarts of Creamy Vanilla and y the number of quarts of Continental Mocha.

The problem states that each quart of Creamy Vanilla uses 2 eggs and each quart of Continental Mocha uses 1 egg. There are 450 eggs in stock, so:

2x + y = 450.

The problem also states that each quart of Creamy Vanilla uses 3 cups of cream and that each quart of Continental Mocha uses 3 cups of cream. There are 750 cups of cream in stock, so:

3x + 3y = 750

Now we have to solve the following system of equations

1) 2x + y = 450

2) 3x + 3y = 750

The first thing i am going to do is simplify the equation 2) by 3

2) 3(x+y)/3 = 750/3

2) x+y = 250

So now, we have the following system

1) 2x + y = 450

2) x + y = 250

I am going to write y as a function of x in 2), and replace in 1)

y = 250 - x

Replacing in 1)

2x + 250 - x = 450

2x - x = 450 - 250

x = 200

You should make 200 quarts of Creamy Vanilla

From 2), we have

y = 250 - x = 250 - 200 = 50

You should make 50 quarts of Continental Mocha

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]

The distribution function for the length of the time that the first to arrive has to wait for the other is given by a triangular shape. This comes from the continuous uniform distribution nature of their arrival time. More probable waiting times are represented by the peak of the triangle.

This problem relates to the mathematical concept of a continuous uniform distribution. In the meeting scenario of Mr. Warren Buffet and Mr. Zhao Danyang, each person arrives between 12 pm and 1 pm at random with a uniform probability. This means there is equal likelihood for each time interval within the hour for them to arrive.

We are interested in the length of time the first person to arrive has to wait for the second. Let's denote the time of arrival of Mr. Buffet by X and that of Mr. Zhao by Y, both from 0 (12:00 pm) to 1 (1:00 pm). The time the first to arrive has to wait is |X - Y|. The distribution function of this waiting time (W = |X - Y|) is a triangle with its peak at W = 0 (no waiting time). The two ends of the base of the triangle fall at -1 and 1. This triangular shape comes from the fact that shorter waiting times (near the peak) are more probable because there is more overlap between X and Y.

Therefore, the cumulative distribution function for the waiting time, W, is given by:

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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$

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.

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.

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:

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:

10 growth per 1000.

Step-by-step explanation:

Given,

Rate of birth = 30 births per 1000

Rate of death = 20 deaths per 1000

As the growth in population is the difference in the number of the child take birth and the person die.

As we are calculating the rate of birth and rate of growth in per thousands of members, so the growth rate will be also in per thousands.

As we can see on every one thousand people,

total birth = 30

total death = 20

so, total growth = total birth - total growth

                           = 30 - 20

                           = 10

As at every 1000 persons, there are 10 persons survive, so the rate of growth will be 10 growth per 1000.

Answer:

To multiply two numbers in scientific notation, multiply their coefficients and add their exponents. To divide two numbers in scientific notation, divide their coefficients and subtract their exponents. In either case, the answer must be converted to scientific notation.

Step-by-step explanation:

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:

[tex]x=-\frac{7}{26}, y=\frac{37}{26}, z=-\frac{21}{13}[/tex]

Step-by-step explanation:

The matrix representation of the system of linear equations is:

[tex]\left(\begin{array}{ccccc}15&13&4&\vdots&8\\11&13&9&\vdots&1\\3&5&7&\vdots&-5\\8&8&2&\vdots&6\\7&5&2&\vdots&2\end{array}\right)[/tex]

First apply the following row operations:

[tex]\begin{array}{c}R_{2}\to R_{2}+(-\frac{11}{15})R_{1}\\R_{3}\to R_{3}+(-\frac{1}{5})R_{1}\\R_{4}\to R_{4}+(-\frac{8}{15})R_{1}\\R_{5}\to R_{5}+(-\frac{7}{15})R_{1}\end{array}[/tex]

The resulting matrix is:

[tex]\left(\begin{array}{ccccc}15&13&4&\vdots&8\\0&\frac{52}{15}&\frac{91}{15}&\vdots&-\frac{73}{15}\\0&\frac{12}{5}&\frac{31}{5}&\vdots&-\frac{33}{5}\\0&\frac{16}{15}&-\frac{2}{15}&\vdots&\frac{26}{15}\\0&-\frac{16}{15}&\frac{2}{15}&\vdots&-\frac{26}{15}\end{array}\right)[/tex]

Then apply  the row operations:

[tex]\begin{array}{c}R_{3}\to R_{3}+(-\frac{12}{5}\cdot \frac{15}{52})R_{2}\\R_{4}\to R_{4}+(-\frac{-16}{15}\cdot \frac{15}{52})R_{2}\\R_{5}\to R_{5}+(\frac{16}{15}\cdot \frac{15}{52})R_{2}\end{array}[/tex]

The resulting matrix is:

[tex]\left(\begin{array}{ccccc}15&13&4&\vdots&8\\0&\frac{52}{15}&\frac{91}{15}&\vdots&-\frac{73}{15}\\0&0&2&\vdots&-\frac{42}{13}\\0&0&-2&\vdots&\frac{42}{13}\\0&0&2&\vdots&-\frac{42}{13}\end{array}\right)[/tex]

Now apply the row operations:

[tex]\begin{array}{c} R_{4}\to R_{4}+R_{3}\\R_{5}\to R_{5}+(-1)R_{3}\end{array}[/tex]

The resulting matrix is:

[tex]\left(\begin{array}{ccccc}15&13&4&\vdots&8\\0&\frac{52}{15}&\frac{91}{15}&\vdots&-\frac{73}{15}\\0&0&2&\vdots&-\frac{42}{13}\\0&0&0&\vdots&0\\0&0&0&\vdots&0\end{array} \right)[/tex]

The equivalent linear system associated to this matrix is

[tex]\begin{cases}15x+13y+4z=8\\\frac{52}{15}y+\frac{91}{15}z=-\frac{73}{15}\\2z=-\frac{42}{13}\end{cases}[/tex]

To Solve this last system is very simple by substitution. The solutions are:

[tex]x=-\frac{7}{26}, y=\frac{37}{26}, z=-\frac{21}{13}[/tex]

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.

Answer:

At x = 2 and 10.

Step-by-step explanation:

Given : The cost of producing x hundred items is given by the equation [tex]C(x) = x^2-3x + 7[/tex]

The revenue generated from sales of x hundred units is given by the equation [tex]R(x) = -x^2 + 21x-33[/tex]

To Find :What values of x will the company break even?

Solution:

Cost function : [tex]C(x) = x^2-3x + 7[/tex]

Revenue function : [tex]R(x) = -x^2 + 21x-33[/tex]

Now to find the company break even :

[tex]-x^2 + 21x-33= x^2-3x + 7[/tex]

[tex]24x= 2x^2+40[/tex]

[tex]12x= x^2+20[/tex]

[tex]x^2-12x+20=0[/tex]

[tex]x^2-10x-2x+20=0[/tex]

[tex]x(x-10)-2(x-10)=0[/tex]

[tex](x-2)(x-10)=0[/tex]

So, x = 2,10

Hence the company break even at x = 2 and 10.

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

[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:

[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:

$37685.56

Step-by-step explanation:

Given,

Total amount we want to accumulate,A = $500,000

Total time, we have,t = 30 years

Interest rate,r = 9%

We are asked to calculate how much money we should deposit to get the required amount after a certain time period.

So, according to compound interest formula,

[tex]A\ =\ P(1+r)^t[/tex]

Where, P = amount of money we need to deposit

[tex]=>\ 500,000\ =\ P(1+0.09)^{30}[/tex]

[tex]=>\ 500,000\ =\ P(1.09)^{30}[/tex]

[tex]=>\ 500,000\ =\ P\times 13.267[/tex]

[tex]=>\ \dfrac{500,000}{13.267}\ =\ P[/tex]

[tex]=>\ P\ =\ 37685.568[/tex]

So, we need to deposit total amount of $37,685.56.

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:

c. a strawberry or a red fruit

Step-by-step explanation:

Since, Banana and apple, banana and kiwi are different fruits. Also, it is given that kiwi is brown, thus kiwi or a red fruit are different sets. So they are mutually exclusive.

Since strawberry is red fruit. Thus a strawberry or a red fruit is not mutually exclusive.

Further, If two sets A and B is Mutually Exclusive, then any units of Set A is not a member of Set B and vice-versa.

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:

139,600.96

Step-by-step explanation:

We use the payment of a loan formula:

[tex] \displaystyle PMT = \frac{P \left(\displaystyle \frac{r}{n}\right)}{\left[ 1 - \left( 1 + \displaystyle \frac{r}{n}\right)^{-nt} \right]} [/tex]

P is the principal: $200,000. t is the number of years: 30, n is 12 since it is compounded monthly. And r is 0.039 which is 3.9% in decimal form (3.9/100)

So the formula becomes:  

[tex] \displaystyle PMT = \frac{200000 \left(\displaystyle \frac{0.039}{12}\right)}{\left[ 1 - \left( 1 + \displaystyle \frac{0.039}{12}\right)^{-12(30)} \right]} [/tex]

And using our calculator we get: PMT = $943.336

Then the total amount of money paid in the mortgage is:  

PMT*n*t = $943.336(12)(30) = $339,600.96

Therefore, the interest paid is:  

$339,600.96 - $200,000 = $139,600.96

You have to enter it without $ and rounded to the nearest cent so: 139,600.96

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:

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