What is the principal square root of -4

Answer:

Distance of the point from its image = 8.56 units

Step-by-step explanation:

Given,

Co-ordinates of point is (-2, 3,-4)

Let's say

[tex]x_1\ =\ -2[/tex]

[tex]y_1\ =\ 3[/tex]

[tex]z_1\ =\ -4[/tex]

Distance is measure across the line

[tex]\dfrac{x+2}{3}\ =\ \dfrac{2y+3}{4}\ =\ \dfrac{3z+4}{5}[/tex]

So, we can write

[tex]\dfrac{x-x_1+2}{3}\ =\ \dfrac{2(y-y_1)+3}{4}\ =\ \dfrac{3(z-z_1)+4}{5}\ =\ k[/tex]

[tex]=>\ \dfrac{x-(-2)+2}{3}\ =\ \dfrac{2(y-3)+3}{4}\ =\ \dfrac{3(z-(-4))+4}{5}\ =\ k[/tex]

[tex]=>\ \dfrac{x+4}{3}\ =\ \dfrac{2y-3}{4}\ =\ \dfrac{3z+16}{5}\ =\ k[/tex]

[tex]=>\ x\ =\ 3k-4,\ y\ =\ \dfrac{4k+3}{2},\ z\ =\ \dfrac{5k-16}{3}[/tex]

Since, the equation of plane is given by

x+y+z=3

The point which intersect the point will satisfy the equation of plane.

So, we can write

[tex]3k-4+\dfrac{4k+3}{2}+\dfrac{5k-16}{3}\ =\ 3[/tex]

[tex]=>6(3k-4)+3(4k+3)+2(5k-16)\ =\ 18[/tex]

[tex]=>18k-24+12k+9+10k-32\ =\ 18[/tex]

[tex]=>\ k\ =\dfrac{13}{8}[/tex]

So,

[tex]x\ =\ 3k-4[/tex]

   [tex]=\ 3\times \dfrac{13}{8}-4[/tex]

   [tex]=\ \dfrac{7}{4}[/tex]

[tex]y\ =\ \dfrac{4k+3}{2}[/tex]

   [tex]=\ \dfrac{4\times \dfrac{13}{8}+3}{2}[/tex]

   [tex]=\ \dfrac{19}{4}[/tex]

[tex]z\ =\ \dfrac{5k-16}{3}[/tex]

  [tex]=\ \dfrac{5\times \dfrac{13}{8}-16}{3}[/tex]

   [tex]=\ \dfrac{-21}{8}[/tex]

Now, the distance of point from the plane is given by,

[tex]d\ =\ \sqrt{(x-x_1)^2+(y-y_1)^2+(z-z_1)^2}[/tex]

   [tex]=\ \sqrt{(-2-\dfrac{7}{4})^2+(3-\dfrac{19}{4})^2+(-4+\dfrac{21}{8})^2}[/tex]

   [tex]=\ \sqrt{(\dfrac{-15}{4})^2+(\dfrac{-7}{4})^2+(\dfrac{9}{8})^2}[/tex]

   [tex]=\ \sqrt{\dfrac{225}{16}+\dfrac{49}{16}+\dfrac{81}{64}}[/tex]

   [tex]=\ \sqrt{\dfrac{1177}{64}}[/tex]

   [tex]=\ 4.28[/tex]

So, the distance of the point from its image can be given by,

D = 2d = 2 x 4.28

            = 8.56 unit

So, the distance of a point from it's image is 8.56 units.

Answer:

[tex]angle = 0.24 rad = 14 °[/tex]

Step-by-step explanation:

The angle of depression is the angle formed by the horizontal at the camera position and the line formed by the camera and the objective, therefore it can be calculated from the information provided as shown in the attached file:

First, you draw f' and f" as parallel lines to the floor at the height of the objective and the height of the camera respectively. Then draw the line o between camera and objective.

the blue angle created by o and f" is the depression angle, which is the same as the angle created by o and f' because angles between parallel lines.

You need to calculate b, as:

b = h - a = 9 - 6 = 3

Then, for the trigonometric function tangent as we have a rectangle triangle:

[tex] Tan(angle)=\frac{b}{f'}[/tex]

therefore:

[tex]angle = Tan^{-1}(\frac{b}{f'} ) = Tan^{-1}(\frac{3}{12} )[/tex]

[tex]angle = 0.24 rad = 14 °[/tex]

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.

Final answer:

Jordan received 27 text messages this week, which is 3 times more than the 9 messages she got last week.

Explanation:

Jordan received 9 text messages last week and this week she received 3 times more. To find out how many text messages Jordan received this week, multiply the number of messages from last week by 3.

9 text messages (last week) × 3 = 27 text messages (this week)

Therefore, Jordan received 27 text messages this week.

Jordan received 27 text messages this week. The average number of texts received per hour by a user is approximately 1.7292, based on the daily average. Precise probabilities for receiving exact or more than two messages per hour cannot be calculated without additional data.

Jordan received 9 text messages last week and three times more this week. To calculate how many text messages Jordan received this week, we multiply 9 by 3, which is 9 × 3 = 27 text messages this week.

To calculate the average texts received per hour, we divide the daily average by the number of hours in a day:

41.5 texts / 24 hours ≈ 1.7292 texts per hour.

a. The probability that a text message user receives or sends exactly two messages per hour is not provided in the given information and would typically require more data to calculate, like the distribution type. However, we know the average is 1.7292, so two messages is a little above average.

b. The probability of receiving more than two messages per hour involves determining the proportion of time users receive more than two messages, based on the average rate. Again, more information is needed to provide a precise probability.

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:

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:

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:

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:

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%.

Final answer:

There are 21 different sets of five marbles that can be made from one transparent, three yellow, and three orange marbles, with no red or green marbles included.

Explanation:

The student has asked how many possible sets of five marbles there are, with the restriction that none of the marbles in a set can be red or green. Considering the available marbles, the student can only choose from one transparent, three yellow, and three orange marbles.

Since there is no replacement and the colors do not matter beyond not being red or green, the student is creating combinations of seven unique marbles taken five at a time. This can be calculated using the combination formula C(n, k) = n! / (k!(n - k)!), where n is the total number of items to choose from, k is the number needed for the set, and ! denotes factorial.

For this problem, n is 7 (1 transparent + 3 yellow + 3 orange) and k is 5. So, we calculate C(7, 5) = 7! / (5!(7 - 5)!) = 7! / (5!2!) = (7 × 6) / (2 × 1) = 21. There are 21 different sets of five marbles where none are red or green.

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:

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:

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:

See explanation below.

Step-by-step explanation:

To prove that the greatest common divisor of two numbers is 1, we use the Euclidean algorithm.

1. In this case, and applying the algorithm we would have:

(14a + 3, 21a + 4) = (14a + 3, 7a + 1)  = (1, 7a + 1) = 1

2. Other way of proving this statement would be that we will need to find two integers x and y such that 1 = (14a + 3) x + (21a + 4) y

Let's make x = 3  and y = -2

Then we would have:

[tex](14a+3)(3) + (21a+4)(-2)\\=42a+9-42a-8\\=1[/tex]

Therefore, (14a + 3, 21a + 4) = 1

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:

8

Step-by-step explanation:

Maximum amount of Acetaminophen that can be taken = 4 g per day

Weight of acetaminophen tablet = 500 mg

let the number of tablets that can be taken be 'x'

therefore,

x × 500 mg ≤ 4 g

also, 1 g = 1000 mg

thus,

x × 500 ≤ 4000

or

x ≤  8

hence,

the maximum numbers of tablets that can be taken per day is 8

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.

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:

325/790 rounded to the nearest hundredths place is 0.41

Step-by-step explanation:

The given fraction is :

[tex]\frac{325}{790}[/tex]

Dividing by 5:

[tex]\frac{65}{158}[/tex]

= 0.41139

We can see that we have a 1 after the hundredth place, so we will not round the 41 as 42.

Now, rounding this to the nearest hundredths place, we get 0.41.

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:

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:

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:  374416

Step-by-step explanation:

Given : A test requires that you answer first Part A and then either Part B or Part C.

Part A consists of 4 true false questions, Part B consists of 6 multiple-choice questions with one correct answer out of five, and Part C consists of 5 multiple-choice questions with one correct answer out of six.

i.e. 2 ways to answer each question in Part A.

For 4 questions, Number of ways to answer Part A = [tex]2^4[/tex]

5 ways to answer each question in Part B.

For 6 questions, Number of ways to answer Part B = [tex]5^6[/tex]

6 ways to answer each question in Part C.

For 5 questions, Number of ways to answer Part C = [tex]6^5[/tex]

Now, the number of ways to  completed answer sheets are possible :_

[tex]2^4\times5^6+2^4\times6^5\\\\=2^4(5^6+6^5)\\\\=16(15625+7776)\\\\=16(23401)=374416[/tex]

Hence, the number of ways to  completed answer sheets are possible = 374416

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:

$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:

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:

[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:

7438.

Step-by-step explanation:

Let x be the attendance before the drop.

We have been attendance dropped 16% this year, to 6248. We are asked to find the attendance before the drop.

The attendance after drop would be 84% (100%-16%) of x.

[tex]\frac{84}{100}\cdot x=6248[/tex]

[tex]0.84x=6248[/tex]

[tex]\frac{0.84x}{0.84}=\frac{6248}{0.84}[/tex]

[tex]x=7438.095[/tex]

[tex]x\approx 7438[/tex]

Therefore, the attendance before the drop is 7438.