Freedonia has 49 senators. Each senator is either honest or corrupt. Suppose you knowthat at least two of the Freedonian senators are honest and that, out of any threeFreedonian senators, at least one is corrupt. Based on these facts, can you determinehow many Freedonian senators are honest and how many are corrupt? if so, what is theanswer?

Answer:

108 one-bedroom units

72 two-bedroom units

36 three-bedroom units

Step-by-step explanation:

Let x, y, z the number of one-bedroom, two-bedroom and three-bedroom units respectively. Then  

1) x+y+z = 216

2)     y+z = x

3)         x = 3z

Multiplying equation 1) by -1 and adding it to 2), we get

-x = x-216 so, x = 216/2 = 108

x = 108

Replacing this value in 3) we get

z = 108/3 = 36

z = 36

Replacing now in 2)

y+36 = 108, y = 108-36 and

y = 72

In the planned apartment complex, there will be 0 one-bedroom units, 216 two-bedroom townhouses, and 0 three-bedroom townhouses.

Let x be the number of one-bedroom units. Since the number of two- and three-bedroom townhouses equals the number of one-bedroom units, let y be the number of two-bedroom townhouses and z be the number of three-bedroom townhouses. We know that x + y + z = 216. Additionally, x = 3z because the number of one-bedroom units will be 3 times the number of three-bedroom townhouses. Substituting x = 3z into the first equation gives 3z + y + z = 216. Simplifying this equation, we get 4z + y = 216.

Now, we can solve this system of equations to find the values of x, y, and z. Subtracting y from both sides of the equation 4z + y = 216 gives 4z = 216 - y. Let's call this equation (1). Substituting x = 3z and y = 216 - 4z into the equation x + y + z = 216 gives 3z + (216 - 4z) + z = 216. Simplifying this equation, we get 4z + 216 = 216. Subtracting 216 from both sides of the equation gives 4z = 0. Let's call this equation (2).

Since equation (1) and equation (2) both have 4z on the left side, we can equate the right sides of the equations. This gives 216 - y = 0. Solving for y, we find y = 216. Plugging this value of y into equation (1), we get 4z = 216 - 216, which simplifies to 4z = 0. Solving for z, we find z = 0. Finally, plugging the value of z into the equation x = 3z, we get x = 3(0), which simplifies to x = 0.

Therefore, there are 0 one-bedroom units, 216 two-bedroom townhouses, and 0 three-bedroom townhouses in the complex.

Answer: $957646.07

Step-by-step explanation:

The formula we use to find the accumulated amount of the annuity is given by :-

[tex]FV=m(\frac{(1+\frac{r}{n})^{nt})-1}{\frac{r}{n}})[/tex]

, where m is the annuity payment deposit, r is annual interest rate , t is time in years and n is number of periods.

Given : m= $2000 ; n= 12   [∵12 in a  year] ;   t= 20 years ;   r= 0.063

Now substitute all these value in the formula , we get

[tex]FV=(2000)(\frac{(1+\frac{0.063}{12})^{12\times20})-1}{\frac{0.063}{12}})[/tex]

i.e. [tex]FV=(2000)(\frac{(1+0.00525)^{240})-1}{0.00525})[/tex]

i.e. [tex]FV=(2000)(\frac{(3.51382093497)-1}{0.00525})[/tex]

i.e. [tex]FV=(2000)(\frac{2.51382093497}{0.00525})[/tex]

i.e. [tex]FV=(2000)(478.823035232)[/tex]

i.e. [tex]FV=957646.070464\approx957646.07\ \ \ \text{ [Rounded to the nearest cent]}[/tex]

Hence, the accumulated amount of the annuity= $957646.07

The future value or accumulated amount of an ordinary annuity is calculated using the formula where P is the periodic payment, r is the interest rate per period, n is the number of compounding periods per year, and t is the time in years. Given P = $2000, r = 6.3%, n = 12 and t =  20 years, substituting these values into the formula gives the accumulated amount

To find the future value or accumulated amount of an ordinary annuity, we use the formula: FV = P * (((1 + r)^nt - 1) / r), where P is the periodic payment, r is the interest rate per period, n is the number of compounding periods per year, and t is the time in years.

In the given problem, P = $2000, r = 6.3% or 0.063 (in decimal), n = 12 (since the payments are monthly), and t =  20 years.

Substituting these into the formula, FV = $2000 * (((1 + 0.063 /12)^(12*20) - 1) / (0.063/12)).

Calculating the equation, we'll get the accumulated amount to the nearest cent.

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Answer:

There are 12 ways to make change for a $50 bill using $5, $10 and $20 bills

Step-by-step explanation:

Let's write down every possibility starting by using the largest quantity of $20 bills and we'll go from there, everytime that we get a $10 bill we will split it in the next option into 2 $5 bills.

(20)(20)(10)

(20)(20)(5)(5)

(20) (10)(10)(10)

(20)(10)(10)(5)(5)

(20)(10)(5)(5)(5)(5)

(20)(5)(5)(5)(5)(5)(5)

Now we start with the largest quantity of $10 bills (5) and go from there, splitting them into two 5 dollar bills in the next option.

(10)(10)(10)(10)(10)

(10)(10)(10)(10)(5)(5)

(10)(10)(10)(5)(5)(5)(5)

(10)(10)(5)(5)(5)(5)(5)(5)

(10)(5)(5)(5)(5)(5)(5)(5)(5)

(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)

Answer:

12 ways

Step-by-step explanation:

20 x 20 x 10 x 20 x 20 x 5 x 5

Answer:

Timy earned $116.25 this week.

Step-by-step explanation:

Timy earns $7.75 from his part-time job at Walmart.

Let us suppose this is his hourly rate.

He worked 5 hours on Monday, 3 hours on Wednesday and 7 hours on Friday.

So, total hours he worked = [tex]5+3+7=15[/tex] hours

Now, his earnings will be = [tex]15\times7.75=116.25[/tex] dollars

Therefore, Timy earned $116.25 this week.

Answer:

$116.25

Step-by-step explanation:

Timy earns $7.75 from his part-time job at Walmart.

He worked on Monday = 5 hours

          On Wednesday  = 3 hours

            On Friday         = 7 hours

Total earning of this week = (7.75 × 5) + (7.75 × 3) + (7.75 × 7)

                                            = 38.75 + 23.25 + 54.25

                                            = $116.25

Timy earned $116.25 this week.

Answer:

The length of the diagonal is 22.6 ft.

Step-by-step explanation:

To find the length of the diagonal of a square, multiply the length of one side by the square root of 2:

If the length of one side is x, [tex]length = x\sqrt{2}[/tex] as you can see in the image attached.

This fact is a consequence of applying the Pythagoras' Theorem to find the length of the diagonal if we know the side length of the square.

[tex]length^{2}  = x^{2}+x^{2}  \\ length=\sqrt{x^{2}+x^{2}} \\ length=\sqrt{2x^{2} } \\ length=x\sqrt{2}[/tex]

We know that the length of one side is 16 ft so [tex]length = 16\sqrt{2}=22.627[/tex] and round to the nearest tenth is 22.6 ft

Answer:

There are 48 children.

Step-by-step explanation:

Given :There are 3 times as many boys as girls.

            There are 24 more boys than girls,

To Find : how many children are there?

Solution:

Let the number of girls be x

Now we are given that there are 3 times as many boys as girls.

So, no. of boys = 3x

Now we are given that there are 24 more boys than girls.

So, [tex]3x-x=24[/tex]

[tex]2x=24[/tex]

[tex]x=12[/tex]

So, no. of girls = 12

No. of boys = 3x = 3(12) = 36

Now the total no. of children = 12+36 = 48

Hence there are 48 children.

Answer:

Suppose the bowl is situated such that the rim of the bowl touches the x axis, and the semicircular cross section of the bowl lies below the x-axis (in (iii) and (iv) quadrant ). Then the equation of the cross section of the bowl would be [tex]x^2+y^2=144[/tex], where y≤ 0,

⇒ [tex]y=-\sqrt{144-x^2}[/tex]

Here, h represents the depth of water,

Thus, by using shell method,

The volume of the disk would be,

[tex]V(h) = \pi \int_{-12}^{-12+h} x^2 dx[/tex]

[tex]= \pi \int_{-12}^{-12+h} (144-y^2) dy[/tex]

[tex]= \pi |144y-\frac{y^3}{3}|_{-12}^{-12+h}[/tex]

[tex]=\pi [ (144(-12+h)-\frac{(-12+h)^3}{3}-144(-12)+\frac{(-12)^3}{3}}][/tex]

[tex]=\pi [ -1728 + 144h - \frac{1}{3}(-1728+h^3+432h-36h^2)+1728-\frac{1728}{3}][/tex]

[tex]=\pi [ 144h - \frac{1}{3}(h^3+432h-36h^2}{3}][/tex]

[tex]=\pi [ 144h - \frac{h^3}{3} - 144h + 12h^2][/tex]

[tex]=\pi ( 12h^2 - \frac{h^3}{3})[/tex]

Special cases :

If h = 0,

[tex]V(0) = 0[/tex]

If h = 12,

[tex]V(12) = \pi ( 1728 - 576) = 1152\pi [/tex]

There is a 99% likelihood that they will sell between 127.12 and 152.88 units.

There is a 95% likelihood that they will sell between 130.2 and 149.8 units.

There is a 68% likelihood that they will sell between 135 and 145 units.

Use the concept of the confidence interval of statistics defined as:

In statistics, a confidence interval describes the likelihood that a population parameter would fall between a set of values for a given percentage of the time. Confidence ranges that include 95% or 99% of anticipated observations are frequently used by analysts.

Given that,

The marketing team at Beth's Butter Works prefers the traditional plastic tub packaging.

They wanted a more refined estimate of potential sales.

They launched a third test at a regional level across 100 stores.

The average daily sales of these 100 stores during the test period was 140 units.

The standard deviation of daily sales across the 100 stores was 50 units.

To calculate the confidence intervals:

Consider the sample mean, sample standard deviation, and the desired level of confidence.

In this case,

Use the average daily sales of 140 units and the standard deviation of 50 units.

Now,

For a 99% confidence interval:

Use a z-score of 2.576 (corresponding to a 99% confidence level).

The formula for the confidence interval is:

Confidence Interval[tex]=\text{ Sample Mean} \pm (\text{Z-Score} \times (\text{Sample Standard Deviation} /\sqrt{\text{Sample Size}}))[/tex]

For a 99% confidence interval, the values are:

Lower bound [tex]= 140 - (2.576 \times (50 / \sqrt{100}))[/tex]

Lower bound = 127.12

Upper bound [tex]= 140 + (2.576 \times (50 / \sqrt{100}))[/tex]

Upper bound = 152.88

For a 95% confidence interval:

Use a z-score of 1.96 (corresponding to a 95% confidence level).

The values are:

Lower bound = [tex]140 - (1.96 \times (50 / \sqrt{100}))[/tex]

Lower bound = 130.2

Upper bound = [tex]140 + (1.96 \times (50 / \sqrt{100}))[/tex]

Upper bound = 149.8

For a 68% confidence interval:

Use a z-score of 1 (corresponding to a 68% confidence level).

The values would be:

Lower bound [tex]= 140 - (1 \times(50 / \sqrt{100}))[/tex]

Lower bound = 135

Upper bound [tex]= 140 + (1 \times(50 / \sqrt{100}))[/tex]

Upper bound = 145

Hence,

99% confidence interval:

There is a 99% likelihood that they will sell between 127.12 and 152.88 units.

95% confidence interval:

There is a 95% likelihood that they will sell between 130.2 and 149.8 units.

68% confidence interval:

There is a 68% likelihood that they will sell between 135 and 145 units.

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Final answer:

To calculate the confidence intervals for the average number of units sold by Beth's Butter Works, the formula for confidence intervals is used with a mean of 140 units and a standard deviation of 50 across 100 stores. The 99%, 95%, and 68% confidence levels correspond to confidence intervals of 127.12 to 152.88 units, 130.2 to 149.8 units, and 135 to 145 units, respectively.

Explanation:

To calculate the confidence intervals for the average number of units sold, we will use the formula for a confidence interval: mean ± (z * (standard deviation / √(sample size))). The mean daily sales are 140 units and the standard deviation is 50. Since the sample size is 100 stores, the standard error (standard deviation / √(sample size)) would be 50 / √(100) = 50 / 10 = 5. The z-scores for the different confidence levels are approximately 2.576 for 99%, 1.96 for 95%, and 1 for 68% (as this lies closest to one standard deviation from the mean).

For a 99% confidence interval, the calculation is:

140 ± (2.576 * 5) = 140 ± 12.88

The 99% confidence interval is therefore between 127.12 and 152.88 units.

For a 95% confidence interval, the calculation is:

140 ± (1.96 * 5) = 140 ± 9.8

The 95% confidence interval is therefore between 130.2 and 149.8 units.

To explain what a 95% confidence interval means for this study, it implies that, if we were to take many samples and build confidence intervals in the same way, 95% of them would contain the true average units sold across all possible stores.

For a 68% confidence interval, the calculation is:

140 ± (1 * 5) = 140 ± 5

The 68% confidence interval is therefore between 135 and 145 units.

Answer:

The next three terms are 1, 8 and 3.

Step-by-step explanation:

Consider the provided sequence,

2, 9, 4, 11, 6....

We need to find the next three terms.

It is given that we need to use a traditional clock face to determine the next three terms in the following sequence.

In the above sequence we are asked to add 7 hours to each time on the traditional clock face.

2 + 7 = 9

9 + 7 = 16 In traditional clock 16 is 4 O'clock

4 + 7 = 11

11 + 7 = 18 In traditional clock 18 is 6 O'clock

6 + 7 = 13 In traditional clock 13 is 1 O'clock

1 + 7 = 8

8 + 7 = 15 In traditional clock 15 is 3 O'clock

Hence, the next three terms are 1, 8 and 3.

Answer: Our required probability is 0.194.

Step-by-step explanation:

Since we have given that

Amount win for showing 6 = $6

Amount win for showing 5 = $3

Amount win for showing 4 = $1

Amount win for showing 1, 2, 3 = $0

So,we need to find the probability that he will win nothing on the two plays of the game.

so, the outcomes would be

(1,1), (1,2), (1,3), (2,1), (3,1),(2,2), (3,3)

So, Number of outcomes = 7

total number of outcomes = 36

So, Probability of wining nothing = [tex]\dfrac{7}{36}=0.194[/tex]

Hence, our required probability is 0.194.

Answer:

No, Jay is not correct.

Step-by-step explanation:

Quotient of powers property:

For any non-zero number a and any integer x and y:

[tex]\frac{a^x}{a^y}=a^{x-y}[/tex]

According to by the quotient of powers property

[tex]\frac{0^5}{0^2}=0^{5-2}\Rightarrow 0^3=0[/tex]

We need to check whether Jay is correct or not.

No, Jay is not correct because quotient of powers property is used for non-zero numbers.

[tex]\frac{0^m}{0^n}=\frac{0}{0}=unde fined[/tex]

Therefore, Jay is not correct.

Distance for which Aeroplane can be in contact with Airport B is = 396.34 km

In the question,

We have an Airport at point A and another at point B.

Now,

Airplane flying at the angle of 72° with vertical catches signals from point D.

Distance travelled by Airplane, AD = 495 km

Now, Let us say,

AB = x

So,

In triangle ABD, Using Cosine Rule, we get,

[tex]cos(90-72) =cos18= \frac{AB^{2}+AD^{2}-BD^{2}}{2.AD.AB}[/tex]

So,

On putting the values, we get,

[tex]cos18 = \frac{x^{2}+495^{2}-300^{2}}{2(495)(x)}\\0.951(990x)=x^{2}+245025-90000\\x^{2}-941.54x+155025=0\\[/tex]

Therefore, x is given by,

x = 212.696, 728.844

So,

The value of x can not be 212.696 as the length of LB (radius) itself is 300 km.

So,

x = 728.844 km

So,

AL = AB - BL

AL = x - 300

AL = 728.844 - 300

AL = 428.844 km

Now, in the circle from a property of secants we can say that,

AL x AM = AD x AC

So,

428.844 x (728.844 + 300) = 495 x AC

441213.576 = 495 x AC

AC = 891.34 km

So,

The value of CD is given by,

CD = AC - AD

CD = 891.34 - 495

CD = 396.34 km

Therefore, the distance for which the Aeroplane can still be in the contact with Airport B is 396.34 km.

Answer:

The amount of drug required = 44.44 mL

Diluent needed = 355.56 mL

Step-by-step explanation:

Data provided in the question:

Total volume of solution = 400 mL

Concentration of drug = 1 : 8

Now,

The ratio is interpreted as 1 part of drug and 8 part of diluent

Thus,

The amount of drug required = [tex]\frac{1}{1+8}\times\textup{Total volume of solution}[/tex]

or

The amount of drug required = [tex]\frac{1}{1+8}\times\textup{400 mL}[/tex]

or

The amount of drug required = 44.44 mL

and,

Diluent needed = [tex]\frac{8}{1+8}\times\textup{400 mL}[/tex]

or

Diluent needed = 355.56 mL

Final answer:

To make a 400mL solution with a 1:8 drug concentration, you need 44.4mL of the drug and 355.6mL of sterile water.

Explanation:

To prepare 400mL of a solution with a 1:8 concentration of a drug, using sterile water as the diluent, we should first calculate the amount of drug needed. A 1:8 concentration ratio means that for every 1 part drug, there are 8 parts diluent. Therefore, the total number of parts is 1 (drug) + 8 (diluent) = 9 parts.

To find the amount of drug needed:

To find the amount of diluent needed:

To summarize, you need 44.4mL of the drug and 355.6mL of sterile water to make a 400mL solution with a 1:8 drug concentration.

Answer:

See proof below.

Step-by-step explanation:

The Pigeonhole principle states that if we place n+1 objects in n places, then one of those n places must have more than one object. In theory, this may seem a very obvious principle but some of the problems which involve this principle can be more difficult than what you'd think of.

In this case we have to prove that given ANY set of integers, there are two of them having the property that either their sum or their difference is evenly divisible by 103.

This would translate to: if we have n and m integers in this set, we'd have one pair for which 103|(n+m) or 103|(n-m). This last condition gives us the clue of using modulos for this problem.

First, we're going to choose 52 pigeonholes (since we have 53 integers). Now, we're going to label the integers with numbers from 0 to 102 depending on their congruence modulo 103.

Once we've done this, we're going to place the integers in the pigeonhole according to their congruence, the pigeonholes will be numbered (0,103), (1,102), (2,101), (3,100)... (50,53), (51,52). (I.e: If the integer is congruent to 6 modulo 103, it will be placed in the (6,97) pigeonhole).

This way any two integers that are placed in one of these pigeonholes will be divisible by 103 (either their sum or their difference).

Note that we have 52 pigeonholes and 53 integers, therefore, one of the pigeonholes will have more than one number (two at least) and that's how we are sure it will satisfy the relation that their sum or their difference is evenly divisible by 103.

Answer:

d) $37800

Step-by-step explanation:

Cost of making x items = [tex]C(x)=25x + 300[/tex]

Cost of making [tex]1500[/tex] items = [tex]C(1500)=25(1500) + 300\\C(1500)= 37500 + 300\\C(1500)= 37800[/tex]

Cost of making [tex]1500[/tex] items = $37800

d) $37800 is the correct answer

Constraints are simply the subjects of an objective function.

The point of intersection is:  [tex]\mathbf{(x_1,y_1) = (9.54,-4.19)}[/tex]

The constraints are given as:

[tex]\mathbf{9x_1 + 7x_2 \ge 57}[/tex]

[tex]\mathbf{4x_1 + 6x_2 \ge 13}[/tex]

Express [tex]\mathbf{4x_1 + 6x_2 \ge 13}[/tex] as an equation

[tex]\mathbf{4x_1 + 6x_2= 13}[/tex]

Subtract 6x2 from both sides

[tex]\mathbf{4x_1 = 13 - 6x_2}[/tex]

Divide through by 4

[tex]\mathbf{x_1 = \frac{1}{4}(13 - 6x_2)}[/tex]

Substitute [tex]\mathbf{x_1 = \frac{1}{4}(13 - 6x_2)} \\[/tex] in [tex]\mathbf{9x_1 + 7x_2 \ge 57}[/tex]

[tex]\mathbf{9 \times \frac{1}{4}(13 - 6x_2) + 7x_2 \ge 57}[/tex]

Open brackets

[tex]\mathbf{29.25 - 13.5x_2 + 7x_2 \ge 57}[/tex]

[tex]\mathbf{29.25-6.5x_2 \ge 57}[/tex]

Collect like terms

[tex]\mathbf{-6.5x_2 \ge 57 - 29.25}[/tex]

[tex]\mathbf{-6.5x_2 \ge 27.25}[/tex]

Divide both sides by -6.5

[tex]\mathbf{x_2 \ge -4.19}[/tex]

Substitute -4.19 for x2 in [tex]\mathbf{4x_1 + 6x_2 \ge 13}[/tex]

[tex]\mathbf{4x_1 + 6 \times -4.19 \ge 13}[/tex]

[tex]\mathbf{4x_1 - 25.14 \ge 13}[/tex]

Add 25.14 to both sides

[tex]\mathbf{4x_1 \ge 38.14}[/tex]

Divide both sides by 4

[tex]\mathbf{x_1 \ge 9.54}[/tex]

Hence, the values are:

[tex]\mathbf{(x_1,y_1) = (9.54,-4.19)}[/tex]

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The solution of the problem involves finding the values of x1 and x2 which satisfy both inequalities when plotted on a graph. This can be done by simplifying  the equations and comparing them.

To solve this problem, we need to find where the two inequalities intersect. This means that we need to find the values of x1 and x2 which satisfy both inequalities.

Let's start with the first inequality '9x1 + 7x2 ≥ 57'. This means that the sum of 9 times x1 and 7 times x2 should be greater than or equal to 57. You can simplify this inequality by dividing the entire expression by the smallest coefficient which is 9, getting 'x1 + (7/9)x2 ≥ 57/9'.

Similarly, simplifying the second inequality '4x1 + 6x2 ≥ 13' by dividing by the smallest coefficient which is 4, we get 'x1 + (3/2)x2 ≥ 13/4'.

By comparing these two simplified inequalities, you should be able to identify the values of x1 and x2 where both inequalities are satisfied.

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Answer: The value of x is 3.

Step-by-step explanation:

Since we have given that

10,11, 12 and x

Average of above numbers = 9

As we know that

Average is given by

[tex]\dfrac{\text{Sum of observation}}{\text{Number of observation}}\\\\\\\dfrac{10+11+12+x}{4}=9\\\\10+11+12+x=9\times 4\\\\33+x=36\\\\x=36-33\\\\x=3[/tex]

Hence, the value of x is 3.

Answer:

$54,000

Amount of discount = $88.02

The new price = $978 - $88.02 = $889.98

Length of  shorter piece is 1 ft and longer piece is 5 ft

Step-by-step explanation:

Given:

Raise received = 3%

New salary = $55,620

Now,

New salary = old salary + 3% of old salary

or

$55,620 = old salary  + (0.03 × old salary)

or

$55,620 = Old salary × (1.03)

or

Old salary = $54,000

Given:

Price of the new fridge = $978

Discount offered = 9%

Thus,

Amount of discount = 9% of $978

or

Amount of discount = 0.09 × $978

or

Amount of discount = $88.02

And, the new price = Price of the fridge - Amount of discount

or

The new price = $978 - $88.02 = $889.98

Given:

Length of the of the board before cutting = 6 ft

Now,

According to the question

let the length of the shorter piece be 'x'

thus,

6 = x + (3x + 2)

or

6 = 4x + 2

or

4 = 4x

or

x = 1 ft

hence,

shorter piece is 1 ft long and longer piece is  (3x +2 = 5ft)

Answer:

0.692.

Step-by-step explanation:

This is a Binomial Probability of Distribution  with P(success) = 0.67.  Prob  success  >= 20) , 31 trials.

From  Binomial Tables we see that the required probability  = 0.692.

Answer:

For linear equations we use:

p=mx+b   ------  (1)

Now we have the following coordinates:

(x1,p1)= (1000,280) and (x2, p2)=(5000,400)

First we need slope (m)

m= [tex](400-280)/(5000-1000)[/tex]

= [tex]120/4000=0.03[/tex]

Now we will plug the value of m in the first equation

[tex]280=0.03(1000)+b[/tex]

=> [tex]280=30+b[/tex]

=> b = 250

Now plug into p=mx+b using only m=0.03 and b=250

[tex]p=0.03x+250[/tex]

When the unit price is $440, we can plug in 440 in for p;

[tex]440=0.03x+250[/tex]

=> [tex]0.03x=440-250[/tex]

=> [tex]0.03x=190[/tex]

=> x = 6333 refrigerators

The lowest price at which  a refrigerator will be marketed, we can find this by plugging x = 0 in p=mx+b.

[tex]p=0.03(0)+250[/tex]

=> p = $250

The linear equation relating the unit price p to the quantity supplied x is p = 0.03x + 250. When the unit price is $440, approximately 6333 refrigerators will be marketed. The lowest price at which a refrigerator will be marketed is $250.

To find the equation relating the unit price p of a refrigerator to the quantity supplied x when the relationship is known to be linear, we can use the two given points: (1000, 280) and (5000, 400).

First, we determine the slope (m) of the line:

m = (400 - 280) / (5000 - 1000) = 120 / 4000 = 0.03

Next, we use the point-slope form of the equation y - y₁ = m(x - x₁) where (x₁, y₁) is one of our points. We can use (1000, 280):

p - 280 = 0.03(x - 1000)

p = 0.03x + 250

Now, let's determine how many refrigerators will be marketed when the unit price is $440:

440 = 0.03x + 250

190 = 0.03x

x = 6333.33

So, approximately 6333 refrigerators will be marketed when the unit price is $440.

Lastly, we find the lowest price at which a refrigerator will be marketed by setting x to 0:

p = 0.03(0) + 250 = 250

The lowest price at which a refrigerator will be marketed is $250.

Answer:

1. The college (Arts and Science, Business, etc.) you are enrolled in

Nominal

2. The number of students in a statistics course  Ratio

3. The age of each of your classmates  Ratio

4. Your hometown  Nominal

Step-by-step explanation:

Nominal, ordinal, interval, or ratio data are the four fundamental levels of measurement scales that are used to capture data.

Nominal, are used for labeling variables, without any quantitative value.

Ordinal, the order of the values is what is significant, but the differences between each one is not really known.

Interval, we know both, the order and the exact differences between the values

Ratio, they have the order, the exact value between units, and have an absolute zero

Answer:

(E) None of these above are true.

Step-by-step explanation:

Married = 74% or 0.74

College graduates = 42% or 0.42

pr(married | college graduates) = 0.56

(A) These events are pairwise disjoint. This is false. Pairwise disjoint are also known as mutually exclusive events. Here we can see that both events are occurring at same time.

(B) These events are independent events. This is also false.

(C) These events are both independent and pairwise disjoint. False

(D) A worker is either married or a college graduate always. False

Here Probability(A or B) shall be 1

= Pr(A) + Pr(B) - Pr( A and B) = 0.74 + 0.42 - 0.56 * 0.42 = 0.9248

This is not equal to 1.

(E) None of these above are true. This is true.

Answer:

x-intercept: (2,0)

y-intercept: (0,3)

Step-by-step explanation:

We are asked to graph our given equation [tex]\frac{1}{2}x+\frac{1}{3}y=1[/tex].

To find x-intercept, we will substitute [tex]y=0[/tex] in our given equation.

[tex]\frac{1}{2}x+\frac{1}{3}(0)=1[/tex]

[tex]\frac{1}{2}x+0=1[/tex]

[tex]2*\frac{1}{2}x=2*1[/tex]

[tex]x=2[/tex]

Therefore, the x-intercept is [tex](2,0)[/tex].

To find y-intercept, we will substitute [tex]x=0[/tex] in our given equation.

[tex]\frac{1}{2}(0)+\frac{1}{3}y=1[/tex]

[tex]0+\frac{1}{3}y=1[/tex]

[tex]3*\frac{1}{3}y=3*1[/tex]

[tex]y=3[/tex]

Therefore, the y-intercept is [tex](0,3)[/tex].

Upon connecting these two points, we will get our required graph as shown below.

Answer:

The cost to rent a trailer for 2.8 hours is $21.4.

The cost to rent a trailer for 3 hours is $23.

The cost to rent a trailer for 8.5 hours is $67.

Step-by-step explanation:

Let x be the number of hours.

It is given that the charge to rent a trailer is $15 for up to 2 hours plus $8 per additional hour or portion of an hour.

The cost to rent a trailer for x hours is defined as

[tex]C(x)=\begin{cases}15 & \text{ if } x\leq 2 \\ 15+8(x-2) & \text{ if } x>2 \end{cases}[/tex]

For x>2, the cost function is

[tex]C(x)=15+8(x-2)[/tex]

We need to find the cost to rent a trailer for 2.8 hours, 3 hours, and 8.5 hours.

Substitute x=2.8 in the above function.

[tex]C(2.8)=15+8(2.8-2)=15+8(0.8)=21.4[/tex]

The cost to rent a trailer for 2.8 hours is $21.4.

Substitute x=3 in the above function.

[tex]C(3)=15+8(3-2)=15+8(1)=23[/tex]

The cost to rent a trailer for 3 hours is $23.

Substitute x=8.5 in the above function.

[tex]C(8.5)=15+8(8.5-2)=15+8(6.5)=67[/tex]

The cost to rent a trailer for 8.5 hours is $67.

Written all the ordered pairs in the form of (hours, cost).

(2.8,21.4), (3,23) and  (8.5,67)

Plot these points on coordinate plane.

Final answer:

To find the cost to rent a trailer for 2.8 hours, we consider the flat fee of $15 for the first 2 hours and add the additional cost of $8 for the partial hour beyond 2 hours, resulting in a total cost of $23.

Explanation:

The cost to rent a trailer for a given number of hours is determined by a flat fee of $15 for the first 2 hours and an additional cost of $8 for each extra hour or partial hour. For 2.8 hours, since this exceeds the initial 2-hour period, we calculate the cost as follows:

Therefore, the cost to rent a trailer for 2.8 hours is $23.

Answer:

1.69224581396 Kg/L

Step-by-step explanation:

We are given the measure of the block as 4.45 cm × 3.35 cm × 6.15 cm.

Volume of block = 4.45 cm × 3.35 cm × 6.15 cm = 91.681125 cm cube = 91.681125 × 0.001 L = 0.091681125 L

We did the above step to convert the volume of block into Liter.

Mass of block is given as 155.147 gram = 155.147 × 0.001 kg = 0.155147 kg

We converted the mass of block into kilograms because we need density in Kg/L.

Density is defined as mass per unit volume

Density = [tex]\frac{Mass}{Volume}[/tex]

             = [tex]\frac{0.155147 }{0.091681125}[/tex]]

             = 1.69224581396 Kg/L

The density is found to be approximately 1.688 kg/L.

To find the density of the block, we need to use the density formula:

Density = Mass / Volume

The given dimensions of the block are:

First, calculate the volume:

Volume = Length × Width × Height

Volume = 4.45 cm × 3.35 cm × 6.15 cm

Volume ≈ 91.88925 cubic centimeters (cm)

Next, convert mass to kilograms and volume to liters:

Finally, calculate the density in kg/L:

Density = Mass / Volume

Density ≈ 0.155147 kg / 0.09188925 L

Density ≈ 1.688 kg/L

Thus, the density of the block is approximately 1.688 kg/L.

Answer:

[tex]P(x) = x - x^2 + x^3 - x^4+x^5[/tex]

Step-by-step explanation:

Let us first remember how a Taylor polynomial looks like:

Given a differentiable function [tex]f[/tex] then we can find its Taylor series to the [tex]nth[/tex] degree as follows:

[tex]P(x) = f(x_{0}) + f'(x_{0}).(x-x_{0}) + \frac{f''(x_{0})}{2!}.(x-x_{0})^2+.....+\frac{f^n(x_{0})}{n!}.(x-x_{0})^n + R_{n}(x).(x-x_{0})^n[/tex]

Where [tex]R_{n}(x)[/tex] represents the Remainder and [tex]f^n(x)[/tex] is the [tex]nth[/tex] derivative of [tex]f[/tex].

So let us find those derivatives.

[tex]f(x) = \frac{x}{1+x}\\f'(x) = \frac{1}{(1+x)^2}\\f''(x) = \frac{-2}{(1+x)^3}\\f'''(x) = \frac{6}{(1+x)^4}\\f''''(x) = \frac{-24}{(1+x)^5}\\f'''''(x) = \frac{120}{(1+x)^6}[/tex]

The only trick for this derivatives is for the very first one:

[tex]f'(x) = \frac{1}{1+x} - \frac{x}{(1+x)^2}\\f'(x) = \frac{(1+x) - x}{(1+x)^2} = \frac{1}{(1+x)^2}\\[/tex]

Then it's only matter of replacing on the Taylor Series and replacing [tex]x_{0}=0[/tex]

Answer:    

[tex]x-1[/tex] is a factor of [tex]x^n - 1[/tex]

Step-by-step explanation:

[tex]x-1[/tex] is a factor of [tex]x^n - 1[/tex]

We will prove this with the help of principal of mathematical induction.

For n = 1, [tex]x-1[/tex] is a factor [tex]x-1[/tex], which is true.

Let the given statement be true for n = k that is [tex]x-1[/tex] is a factor of [tex]x^k - 1[/tex].

Thus, [tex]x^k - 1[/tex] can be written equal to  [tex]y(x-1)[/tex], where y is an integer.

Now, we will prove that the given statement is true for n = k+1

[tex]x^{k+1} - 1\\=(x-1)x^k + x^k - 1\\=(x-1)x^k + y(x-1)\\(x-1)(x^k + y)[/tex]

Thus, [tex]x^k - 1[/tex] is divisible by [tex]x-1[/tex].

Hence, by principle of mathematical induction, the given statement is true for all natural numbers,n.

Answer:

The probability is 9.80%.

Step-by-step explanation:

The u.s. senate consists of 100 members, 2 from each state.

A committee of five senators is formed.

P(at least one from Your state) = 1- [tex]\frac{98c5}{100c5}[/tex]

= 1- [tex]\frac{67910864}{75287520}[/tex]

= [tex]1-0.9020[/tex]

= 0.098

That is, 9.80%.

Answer:

D. A, B, and C

Step-by-step explanation:

Option (A) is true because when two straight lines intersect to each other we get two pair of vertically opposite angles and the angles opposite to each other is always equal.

Option (B) is also correct as If the sum of two angles is equal to 180°, then they are supplementary to each other.

Option (C) is also correct as If the sum of the two angles is equal to 90°, then they are Complementary to each other.

Hence, Option (D) is correct.

Answer:

The chances it will rain two days in a row is 12%

Step-by-step explanation:

The forecast calls for a 30% chance of snow today

So, chance of snowfall today = 30% = 0.3

A 40% chance of snow tomorrow.

So, chance of snowfall tomorrow= 40% = 0.4

The chances it will rain two days in a row = [tex]0.4  \times 0.3[/tex]

                                                                    = [tex]0.12[/tex]  

So, percent  it will rain two days in a row = [tex]0.12 \times 100 = 12\%[/tex]

Hence the chances it will rain two days in a row is 12%