Please help me with geometry. Theres only 3 questions i need help with

Answer: [tex]\dfrac{3}{4}[/tex]

Step-by-step explanation:

Given : Line k lies in the xy-plane.

The x-intercept of line K is -4. i.e. line k is intersecting the x-axis at (-4,0).

We know that the mid point of a line passing through any two points (a,b) and (c,d) is [tex](\dfrac{a+c}{2},\dfrac{b+d}{2})[/tex].

Then, the  midpoint of the line segment whose endpoints are (2, 9) and (2, 0) will be :-

[tex](\dfrac{2+2}{2},\dfrac{9+0}{2})=(2,4.5)[/tex]

The slope of line passing through (p,q) and (r,s) will be :-

[tex]m=\dfrac{s-q}{r-p}[/tex]

Then, the slope of line passing through (-4,0) and (2,4.5) will be :-

[tex]m=\dfrac{4.5-0}{2-(-4)}=\dfrac{4.5}{2+4}=\dfrac{4.5}{6}\\\\=\dfrac{45}{60}=\dfrac{3}{4}[/tex]

Hence, the slope of line k [tex]=\dfrac{3}{4}[/tex]

Answer: In fraction : [tex]\dfrac{373}{500}[/tex]

In Decimal :  0.746

Step-by-step explanation:

Given : A ream of paper contains 500 sheets of paper.

Norm has 373 sheets of paper left from a ream.

Then, the fraction of ream Norm has will be :-

[tex]\dfrac{\text{Number of sheets left from ream }}{\text{Total sheets in a ream }}\\\\=\dfrac{373}{500}[/tex]

To convert in decimal we divide 373 by 500, we get 0.746.

Hence, The portion of a ream Norm has as = [tex]\dfrac{373}{500}[/tex] or 0.746

Step-by-step explanation:

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

[tex](x-y)(x^{n}+x^{n-1}y+...+xy^{n-1}+y^{n})=x^{n+1}-y^{n+1}[/tex]

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

Base case:

[tex](x-y)(x^{n}+x^{n-1}y+...+xy^{n-1}+y^{n})=(x-y)(x^{1}+y^{1})=x^2-y^2=x^{1+1}-y^{1+1}[/tex]

Inductive hypothesis:  

Given a natural n,  

[tex]x^{n+1}-y^{n+1}=(x-y)(x^{n}+x^{n-1}y+...+xy^{n-1}+y^{n})[/tex]

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

Inductive step:

Observe that, for y=0 the conclusion is clear. Then we will assume that [tex]y\neq 0.[/tex]

[tex](x-y)(x^{n+1}+x^{n}y+...+xy^{n}+y^{n+1})=(x-y)y(\frac{x^{n+1}}{y}+x^{n}+...+xy^{n-1}+y^{n})=(x-y)y(\frac{x^{n+1}}{y})+(x-y)y(x^{n}+...+xy^{n-1}+y^{n})=(x-y)y(\frac{x^{n+1}}{y})+y(x^{n+1}-y^{n+1})=(x-y)x^{n+1}+y(x^{n+1}-y^{n+1})=x^{n+2}-yx^{n+1}+yx^{n+1}-y^{n+2}=x^{n+2}-y^{n+2}\\[/tex]

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

In conlusion, for every natural n,

[tex](x-y)(x^{n}+x^{n-1}y+...+xy^{n-1}+y^{n})=x^{n+1}-y^{n+1}[/tex]

The confidence interval for population mean is given by :-

[tex]\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]

, where [tex]\hat{p}[/tex] is the sample proportion, n is the sample size , [tex]z_{\alpha/2}[/tex] is the critical z-value.

The  values needed to calculate a confidence interval at the 99% confidence level are :

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

Sample size : n=450

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

Sample proportion: [tex]\hat{p}=\dfrac{280}{450}\approx0.62[/tex]

Now, the  99% confidence level will be :

[tex]\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\\\\=0.62\pm(2.576)\sqrt{\dfrac{0.62(1-0.62)}{450}}\\\\\approx0.62\pm0.023\\\\=(0.62-0.023,\ 0.62+0.023)=(0.597,\ 0.643)[/tex]

Hence, the  99% confidence interval is [tex](0.597,\ 0.643)[/tex]

To calculate a 99% confidence interval for the true proportion, find the values needed and apply the formula. In this case, the confidence interval is between 0.5601 and 0.6811.

1. Sample proportion [tex](\( \hat{p} \))[/tex]:

[tex]\[ \hat{p} = \frac{x}{n} = \frac{280}{450} \approx 0.6222 \][/tex]

2. Margin of error  E :

For a 99% confidence level, z = 2.576.

[tex]\[ E = z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]\[ E = 2.576 \times \sqrt{\frac{0.6222 \times (1-0.6222)}{450}} \]\[ E = 2.576 \times \sqrt{\frac{0.6222 \times 0.3778}{450}} \]\[ E = 2.576 \times \sqrt{\frac{0.235052}{450}} \]\[ E \approx 2.576 \times \sqrt{0.00052233} \]\[ E \approx 2.576 \times 0.022854 \]\[ E \approx 0.058893 \][/tex]

3. Confidence interval:

[tex]\[ \text{Confidence interval} = \hat{p} \pm E \]\[ \text{Confidence interval} = 0.6222 \pm 0.0589 \][/tex]

Therefore, the confidence interval for the percentage of elementary school children who have a social media account at the 99% confidence level is approximately (0.5633, 0.6811) .

Answer:

Step-by-step explanation:

We can solve this using a Venn Diagram,

First we write down the 91 tourists that didn't visit any of the theme parks.

Then we focus on the 38 tourists that visited all three theme parks and write that down in the intersection of the 3 circles.

80 tourists visited both LEGOLAND and Universal Studios but we already know that 38 visited the three of them, so 80-38= 42 tourists visited only those two parks.

97 tourists visited both Magic Kingdom and Universal Studios, but again, we know that 38 visited the three of them, so 97-38 = 59 tourists visited only those two parks.

94 tourists visited both Magic Kingdom and LEGOLAND, but since 38 visited the three of them, we subtract 94-38 = 56 tourists visited those two parks.

292 tourists visited LEGOLAND so we'll focus on that circle 292 - (56+38+42) = 156 tourists visited only LEGOLAND

280 tourists visited Magic Kingdom so we focus now on the Magic Kingdom circle and we have 290 - (56+38+59) = 127 only visited Magic Kingdom.

Now we will sum up all the quantities we have and subtract them from the total amount of tourists (1116) to find out how many people visit Universal Studios

1116 - (91+156+42+38+56+59+127) = 547

Answer:

The equation to find the number of gallons purchased is:

[tex]C(n) = 3.05n[/tex]

You purchased 12 gallons of gasoline

Step-by-step explanation:

This problem can be modeled by the following first order function

[tex]C(n) = P_{G}n[/tex]

Where C(n) is the cost in function to the number of gallons, P is the price of the gallon and n is the number of gallons

The problem states that the price of gasoline is $3.05 per gallon, so P = 3.05

The equation to find the number of gallons purchased is:

[tex]C(n) = 3.05n[/tex]

If you pay $36.60, you have C = 36.60, and want to find n, so:

[tex]36.60 = 3.05n[/tex]

[tex]n = \frac{36.60}{3.05}[/tex]

[tex]n = 12[/tex]

You purchased 12 gallons of gasoline

Answer:

y = [tex]C_{1}e^{2x} + C_{2}e^{-2x}[/tex]

Step-by-step explanation:

We are given the differential equation: y'' - 4y = 0

We have to find the general solution.

The auxiliary equation for the above differential equation can be written as:

m² - 4 = 0

We solve for m.

⇒m² = 4

⇒m = ±2

⇒[tex]m_{1}[/tex] = +2 and [tex]m_{2}[/tex] = -2

Thus, we have two distinct roots or we have two distinct values of m.

Thus, the general solution will be of the form:

y = [tex]C_{1}e^{m_{1}x} + C_{2}e^{m_{2}x}[/tex]

y = [tex]C_{1}e^{2x} + C_{2}e^{-2x}[/tex]

Answer:

y+0.5x-2=0

Step-by-step explanation:

Given,

X-intercept of line = 4

So, the line intersect the x axis at (4,0)

Y-intercept of the line = 2

So, the line intersect the y axis at (0,2)

So, the slope of the line can be given by

[tex]m\ =\ \dfrac{y_2-y_1}{x_2-x_1}[/tex]

    [tex]=\ \dfrac{2-0}{0-4}[/tex]

      = -0.5

Hence, the equation of line can be given by

y=mx+c

where, m=slope of the line

             c= y-intercept of line

So, after putting the value of m and c in above equation, the equation of line will be

y = -0.5x + 2

=> y+0.5x-2=0

So, the equation of line will be y+0.5x-2=0.

Answer: 0.9575465

Step-by-step explanation:

Let the random variable X is normally distributed with mean [tex]\mu=50[/tex] and standard deviation[tex]\sigma=7[/tex] .

Using the formula , [tex]z=\dfrac{x-\mu}{\sigma}[/tex] , we have the z-value for x= 34

[tex]z=\dfrac{34-50}{7}\approx-2.29[/tex]

For x= 63

[tex]z=\dfrac{63-50}{7}\approx1.86[/tex]

P-value : P(34<x<63)=P(-2.29<z<1.86)

[tex]=P(z<1.86)-P(z<-2.29)\\\\=0.9685572-(1-P(z<2.29))\\\\1=0.9685572-(1-0.9889893)\\\\=0.9575465[/tex]

Hence, the required probability = 0.9575465

Answer: 200 mm

Step-by-step explanation:

The perimeter of rectangle is given by :-

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

Given : Two sides of a rectangle are 4 cm in length. The other two sides are 6 cm in length.

The perimeter of the rectangle will be :_

[tex]P=2(4+6)=2(10)=20\ cm[/tex]

We know that 1 cm = 10 mm

Therefore,  perimeter of the rectangle = [tex]20\times10=200\ mm[/tex]

Answer:

The output for NAND gate is false if all the inputs are true. it is true if either A or B is False.

Step-by-step explanation:

Step1

NAND operator:

NAND is a logic or Boolean expression and is a combination of NOT and AND Gate.

NAND gate is built using various junction diodes and transistors.

Step2

The output of NAND can be 0 or 1. Here, LOW (0) and HIGH (1). 0 or 1 is commonly named as FALSE or TRUE respectively.

It is basically the complement of AND gate. If all its inputs are True or 1, then the output is False or 0 else if either A or B is 0 or 1 then it will give the output as TRUE.

Step3

Here, A and B are 2 inputs used to make the truth table for NAND.

In 1st row, when A and B are both 0, the output for NAND is 1.

Likewise, in 2st row and 3rd row, when either A or B is 0 or 1, the output for NAND is 1 that is TRUE.

But in last row, when both inputs A and B are true, then the output for NAND gate is FALSE or 0.

The diagram and truth table for NAND is shown below.

Final answer:

A truth table for the NAND logical operator lists all possible binary input combinations of variables A and B, applies the AND operation, and then negates the output to present the NAND result. The resulting truth table will show that a NAND gate outputs true for all cases except when both inputs are true.

Explanation:

Constructing a truth table for the logical operator NAND is a way to visualize how this operator works with different input values.

The NAND operator is essentially the negation of the AND operator, meaning that it outputs the opposite of what an AND gate would. Here is how you construct a truth table for NAND:

The resulting truth table for the NAND operation is as follows:

A B A AND B A NAND B

false false false true

false true false true

true false false true

true true true false

Answer:

Considering the equation f(x) = 64-7x+ 4x^2 this are the answers

a) Equation 1:

[tex]y=8x-7[/tex]

b) Equation 2:

[tex]y=\frac{64}{x}-7+4x[/tex]

c) Intersection of Equation 1 and Equation 2:

The lines intersects in the points:

P(x,y)=(4,25) and P(x,y)=(-4,-39)

Step-by-step explanation:

a) Drivate f(x) to find Equation 1:

y=f'(x)

y=0-7+8x

y=8x-7

b) Equation 2 is f(x)/x

y=f(x)/x

[tex]y=\frac{64-7x+4x^2}{x}[/tex]

[tex]y=\frac{64}{x}-7+4x[/tex]

c) The intersection between the two equations is the only point that they have in common, this means that the points (x,y) satisfies both equations

To find it lets write the equations side by side

[tex]\left \{ {{y=8x-7} \atop {y=\frac{64}{x}-7+4x}} \right.[/tex]

Given the fact that the y points are the same for both equations, you can replace the equation 1 into equation 2, this means, instead of write y, write equation 2:

[tex]8x-7=\frac{64}{x}-7+4x[/tex]

now you can solve this for x

[tex]8x-4x-7+7=\frac{64}{x}\\4x=\frac{64}{x}\\4x^2=64[/tex]

[tex]x^2=\frac{64}{4}\\x=\sqrt{16}\\ x=±4[/tex]

With the values of x, you can find the values of y by putting it into equation 1:  

y=8*(+4)-7 and y=8*(-4)-7

y=25 and y=-39

Finally, the points where these two equations intersect are P(x,y)=(4,25) and P(x,y)=(-4,-39).

Answer:

$ 30

Step-by-step explanation:

Let x be the cost of tie ( in dollars ),

∵ The  shirt costs $21 more than the tie,

So, the cost of shirt = x + 21

Thus, the cost of a tie and a shirt = x + x + 21 = 2x + 21,

According to the question,

2x + 21 = 57

2x = 57 - 21

2x = 36

x = 18

Hence, the cost of the shirt = 18 + 12 = $ 30

The problem is solved using Polya's four-step problem-solving process. By setting up two equations based on the problem, we determine that the tie costs $18 and the shirt costs $39 which is confirmed by substitution into the original problem statement.

According to Polya's four-step problem solving strategy, we first need to understand the problem. We know that a shirt and tie cost $57 together, and the shirt costs $21 more than the tie.

Step two of Polya's strategy involves devising a plan. In this case, we can write two expressions to represent the cost of the two items: S (the cost of the shirt) and T (the cost of the tie). The total cost (S+T) equals $57. We also know that S (the cost of the shirt) is T (the cost of the tie) plus $21.

For step three, we carry out the plan. We can substitute S from the second expression into the first expression: (T + $21) + T = $57. Solving for T gives us T = $18. Hence, the tie costs $18.

Finally, for step four, we review our answer. We can plug T back into the second expression to get S = T + $21 = $18 + $21 = $39. So, the shirt costs $39, which sounds reasonable given that the shirt is supposed to be more expensive than the tie.

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Answer: There are 4.8 grams in 100 mL of Drug A

Step-by-step explanation:

In order to determinate how many grams are in 100 mL of Drug A you can multiply and divide the expression of the concentration by 10 (To obtain 100 mL in the denominator)

Notice that the value remains unaltered.

(475 mg/10 mL )(10/10) = 4750 mg/ 100 mL

But the question is how many grams are in 100 mL, so you have to convert the value from mg to g.

The prefix m (mili) is equivalent to 0,001 so you can use 0.001 instead of the prefix

4750(0.001) g/ 100 mL

4.75 g/ 100 mL

The rounded result is 4.8 g/ 100 mL

Answer:

A simple random sample.

Step-by-step explanation:

A simple random sample is an statistical sample in which each member of a group has the same probability of being chosen. Since the researcher doesn't really have specific characteristics added to the sample other than being from public or private universities, this would be a simple random sample.

Answer:

1mL should be administered to provide a 25-mg dose of aminophylline.

Step-by-step explanation:

The problem states that one 20-mL ampul contains 0.5 g of aminophylline, and asks how many milliliters should be administered to provide a 25-mg dose of aminophylline.

The first step is converting 0.5g to mg.

Each g has 1000mg, so:

1g - 1000mg

0.5g - xmg

x = 1000*0.5

x = 500mg.

Now we have that one 20-mL ampul contains 500mg of aminophylline. How many milliliters should be administered to provide a 25-mg dose of aminophylline?

As the dose increases, so does the quantity of aminophylline. It means that we have a direct rule of a three, there is a cross multiplication. So:

20mL - 500mg

x mL - 25mg

500x = 500

[tex]x = \frac{500}{500}[/tex]

x = 1 mL

1mL should be administered to provide a 25-mg dose of aminophylline.

Final answer:

1 mL of the aminophylline solution should be administered to provide a 25-mg dose. This calculation was made by first converting the total amount of aminophylline to milligrams and then establishing the amount per milliliter to solve for the necessary volume.

Explanation:

To determine how many milliliters of aminophylline should be administered to provide a 25-mg dose, we need to use the following information:

First, we need to convert 0.5 g of aminophylline to milligrams:

0.5 g × 1000 mg/g = 500 mg

Now that we have the total amount of aminophylline in milligrams, we can determine the amount per milliliter:

500 mg / 20 mL = 25 mg/mL

To find the volume that contains 25 mg, we set up a proportion:

(25 mg of aminophylline) / (X mL) = (25 mg/mL)

By solving for X, we find:

X = 1 mL

So, 1 mL of the aminophylline solution should be administered to provide a 25-mg dose.

Step-by-step explanation:

Consider the provided information.

We have given that H and k are the subgroups of orders 5 and 8, respectively.

We need to prove that H∩K = {e}.

As we know "Order of element divides order of group"

Here, the order of each element of H must divide 5 and every group has 1 identity element of order 1.

1 and 5 are the possible order of 5 order subgroup.

For subgroup order 8: The possible orders are 1, 2, 4 and 8.

Now we want to find the intersection of these two subgroups.

Clearly both subgroup H and k has only identity element in common.

Thus, H∩K = {e}.

Answer: 0.1038

Step-by-step explanation:

We assume that oil in each container is filled will normal distribution.

Given : Population mean : [tex]\mu=12[/tex]

Standard deviation: [tex]\sigma=0.25[/tex]

Sample size : [tex]n=40[/tex]

Let x be the random variable that denotes the amount of oil filled in container.

z-score : [tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

For x= 12.05

[tex]z=\dfrac{12.05-12}{\dfrac{0.25}{\sqrt{40}}}=1.26491106407\approx1.26[/tex]

Now by using the standard normal table for z, we have the probability that the sample mean,X is greater  then 12.05:-

[tex]P(z>1.26)=1-0.8961653=0.1038347\approx0.1038[/tex]

Hence, the probability that the sample mean,X is greater  then 12.05 = 0.1038

Answer:  21

Step-by-step explanation:

Given : A bag containing five red marbles, two green ones, one transparent one, four yellow ones, and two orange ones .

Total marbles other than red or green = 1+4+2=7

Now, the number of combinations to select five marbles from the set of 7 will be :-

[tex]7C_5=\dfrac{7!}{5!(7-5)!}=\dfrac{7\times6\times5!}{5!\times2!}=21[/tex]

Hence, the number of  possible sets of five marbles are there in which none of them are red or green =21

Answer:

No, whole numbers are numbers that are whole, which would mean that they are not a decimal or a fraction. This is because a fraction or a decimal is a portion of a number, which would mean that the decimal or fraction is not complete/ not whole!

Answer:

Step-by-step explanation:

Given that the cost function is linear.

Fixed charges = 4 dollars

Variable charges perhalf hour = 65 cents = 0.65 dollars

Suppose a man leaves for x hours say

then we have he would be charged 4 dollars besides 0.65 for 2x half hours.

Hence

[tex]C(x) = 5+4(2x)\\C(x) = 8x+5[/tex] where x is the number of hours.

The variable charge is 1.3 dollars per hour and the fixed cost is 4 then the linear cost equation is C(x) = 1.3x + 4.

It is a system of an equation in which the highest power of the variable is always 1. A one-dimension figure that has no width. It is a combination of infinite points side by side.

Given

A parking garage charges 4 dollars plus 65 cents per half-hour.

Let C(x) be the linear cost.

The fixed charge is 4 dollars

The variable charge per half-hour is 0.65 dollars.

Suppose the men leave for x half-hours for one hour the x will be 2x.

Then the equation will be

[tex]\rm C(x) = 0.65*2x + 4\\\\C(x) = 1.3x + 4[/tex]

Thus, the equation is [tex]\rm C(x) = 1.3x + 4[/tex].

More about the linear system link is given below.

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Answer:

a) You can buy 282.88 British pounds.

b) you should use 17.5 cups of flour.

Step-by-step explanation:

a) You arrive in London with $400. How many British pounds can you purchase with your US dollars if the conversion is $1.414 per British pound?

We can solve this problem by using a rule of three

1.414 dollars --- 1 British pound

400 dollars  ---  x British pounds

We solve for x,

x = 400/1.414 = 282.88

We can buy 282.88 British pounds.

b) Your goal is to prepare a recipe for a gathering of 21 people. If the original recipe serves 6 people and requires 5 cups of flour, how much flour should you use?

Again, we can use a rule of three.

6 people --- 5 cups of flour

21 people --- x cups of flour

We solve for x:

x= (21)(5)/6 = 105/6=17.5

We should use 17.5 cups of flour.

Answer:

[tex]\frac{25}{8}[/tex]

Step-by-step explanation:

A fraction is a part of a whole .

A proper fraction is a fraction whose numerator is less than denominator .

For example [tex]\frac{2}{3}\,,\,\frac{3}{4}[/tex]

An improper fraction is a fraction whose numerator is greater than denominator .

For example [tex]\frac{5}{4}\,,\,\frac{8}{7}[/tex]

A mixed fraction is made up of whole number and a proper fraction .

Given : [tex]4\frac{1}{6}\,,1\frac{1}{3}[/tex]

Solution :

[tex]4\frac{1}{6}=\frac{4\times 6+1}{6}=\frac{25}{6}\\1\frac{1}{3}=\frac{3\times 1+1}{3}=\frac{4}{3}[/tex]

We need to divide [tex]4\frac{1}{6}[/tex] by [tex]1\frac{1}{3}[/tex] .

[tex]4\frac{1}{6}\div 1\frac{1}{3}=\frac{25}{6}\div\frac{4}{3} \\\Rightarrow 4\frac{1}{6}\div 1\frac{1}{3}= \frac{25}{6}\times \frac{3}{4}=\frac{25}{8}[/tex]

Answer:

  The digit 6 in February is worth 1/100 of its value in January.

Step-by-step explanation:

The value of a given digit in a number can be found by setting the other digits to zero.

In February, the digit 6 has a value of 00600 = 600.

In January, the digit 6 has a value of 060000 = 60,000.

The ratio of the value in February to the value in January is ...

  600/60000 = 1/100

The digit 6 in February is one hundredth of the digit 6 in January.

Answer: I'm sure its 2/21

Step-by-step explanation: you just need to multiply cross sides then divide by any number that works on both of them.

I hope that I answer your question.

Given : An urn contains 2 red marbles and 3 blue marbles.

Total marbles = 2+3=5

a) The general ways in which the person could get a red marble and a blue marble are :

1)  He draws red marble first and then second marble as blue.

2)  He draws blue marble first marble and then second marble as red.

b)  The number of ways to get one red and one blue marble is given by :-

[tex]^2C_1\times^3C_1=2\times3=6[/tex]                          (i)

c) Number of ways to get 2 marbles from 5 is given by :-

[tex]^5C_2=\dfrac{5!}{2!(5-2)!}=\dfrac{5\times4\times3!}{3!\times2}=10[/tex]     (ii)

Now, The probability the person gets a red and a blue marble will be :-

[tex]P(R\ \&B)=\dfrac{6}{10}=0.6[/tex]       [Divide (i) by (ii)]

Hence, the  probability the person gets a red and a blue marble= 0.6

Answer:

  (x, y) = (4, -3)

Step-by-step explanation:

Relative to straight east with angles measured CCW, the vector is ...

  5∠-36.9° = 5(cos(-36.9°), sin(-36.9°)) = 5(0.8, -0.6) = (4, -3)

The components of vector A that has a magnitude of 5.00 and is at a 36.9 degrees angle south of east, are 3.96 towards east (x-direction), and -3.00 towards south (y-direction).

The components of a vector can be calculated by using trigonometry. Where the x-component or the horizontal component can be found using the cosine of the angle and the y-component or the vertical component can be found using the sine of the angle. However, there are different coordinate systems and conventions. In this case, positive x-direction is to the east and positive y-direction is to the north, and the angle is measured from the positive x-axis (east) but the vector points towards the negative y-axis (south). So you need to apply a negative to the sine of the angle.

For the vector A⃗:

Therefore, the components of vector A⃗ are 3.96 in x-direction (east) and -3.00 in y-direction (south).

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Answer:

Net Present Value = - $99,360

Step-by-step explanation:

As provided,

Cash outlay = $65,000 each year for 10 years

Since the first outlay will be immediately, the cumulative discounting factor for cash outlay will be @ 16% = 1 for year 0 + 4.606 for 9 years = 5.606

Therefore, cumulative present value of total cash outlay = $65,000 [tex]\times[/tex] 5.606 = $364,390

Cash inflows beginning in year 11 = $170,000 for another continuous 20 years.

these cash flow will occur in the beginning of year 11 and end of year 10

Discounting factor will be [tex]\frac{1}{(1+0.16)^1^0}[/tex] = 0.2267

For, consecutive 20 years = 1.559

Therefore, value of inflows = $170,000 [tex]\times[/tex] 1.559 = $265,030

Net Present Value = Present Value of Cash Inflows - Present Value of Cash Outflows = $265,030 - $364,390 = - $99,360

Express the matrix [tex]I[/tex] like [tex](\delta_{ij})_{n\times n}[/tex], where [tex]\delta_{ij}=1[/tex] if [tex]i=j[/tex] and [tex]\delta_{ij}=0[/tex] if [tex]i\neq j[/tex]. ([tex]\delta_{ij}[/tex] is known as kronecker's delta)

In the same form we express the matrix [tex]A=(a_{ij})_{n\times n}[/tex].

The firs index indicate the row and the second the column.

By the multiplication of matrices we have [tex]AI=(c_{ij})_{n\times n}[/tex], where

[tex]c_{ij}=\sum_{k=1}^n a_{ik}\delta _{kj} = a_{ij}[/tex]

because only [tex]\delta_{jj}[/tex] is non-zero in the last sum.

therfore we have [tex]AI=(c_{ij})_{n\times n}=(a_{ij})_{n\times n}=A[/tex].

In the same manner we have

[tex]IA=(d_{ij})_{n\times n}[/tex], where

[tex]d_{ij}=\sum_{k=1}^n \delta _{ik}a_{kj} = a_{ij}[/tex]

And so, [tex]IA=A[/tex]

Matrix multiplication with the identity matrix leaves the original matrix unchanged. This is analogous to multiplying a number by 1, which leaves the number unchanged. Thus, for any square matrix A, AI = IA = A.

The n×n identity matrix, denoted by I, is a special square matrix that has ones on its main diagonal and zeros elsewhere. It's called the 'identity' matrix because multiplication with it leaves a matrix unchanged, similar to how multiplying a number by 1 leaves it unchanged.

Let's prove that for any n×n matrix A, we have AI=IA=A. As A is an n×n matrix, suppose it has elements aij. Because of the definition of matrix multiplication, element at ith row and jth column of the product of two matrices (say A and I) is given by summation of the products of corresponding elements in ith row of first matrix and jth column of the second matrix.

In the case of AI, for any element in the result, we get it by summation of product of corresponding elements in ith row of A and ith column of I. Now, since except for the ith position, rest of the elements in ith column of I are zero, we only get aii in the summation. Hence, AI = A. Similar argument can be made for IA. Therefore, AI=IA=A.

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Answer:

a) There are 3,628,800 different strings.

b) There are 28,800 different strings if the letters ad digits must alternate.

c)There are 86,400 different string if all five letters must be adjacent in each string.

Step-by-step explanation:

There are 10 digits.

Our strings have the following format:

C1 - C2 - C3 - C4 - C5 - C6 - C7 - C8 - C9 - C10

repitition of letters is not allowed.

a) How many different strings are there?

C1 can be any of the 10, C2 can be 9, C3 can be 8, ...

So in total there are:

[tex]T = 10*9*8*7*6*5*4*3*2*1 = 3,628,800[/tex]

There are 3,628,800 different strings.

b) How many different strings are there if the letters, i.e. A, E, I, O, U and the digits, i.e. 1,3, 5, 7, 9 must alternate?

There are the following possiblities:

5(l) - 5(d) - 4(l) - (4d) - ...

Or

5(d) - 5(l) - 4(d) - 4(l) - ...

So:

[tex]T = 2*(5*5*4*4*3*3*2*2*1*1) = 28,800[/tex]

There are 28,800 different strings if the letters ad digits must alternate.

c) How many different strings are there if all five letters must be adjacent in each string?

L - L - L - L - L - D - D - D - D - D

D - L - L - L - L - L - D - D - D - D

D - D - L - L - L - L - L - D - D - D

D - D - D - L - L - L - L - L - D - D

D - D - D - D - L - L - L - L - L - D

D - D - D - D - D - L - L - L - L - L

There are [tex]T = 6*(5*5*4*4*3*3*2*2*1*1) = 86,400[/tex]

There are 86,400 different string if all five letters must be adjacent in each string.