Compound Interest Application

Compound interest is given by the formula A = P ( 1 + r ) t . Where A is the balance of the account after t years, and P is the starting principal invested at an annual percentage rate of r , expressed as a decimal.

Wyatt is investing money into a savings account that pays 2% interest compounded annually, and plans to leave it there for 15 years. Determine what Wyatt needs to deposit now in order to have a balance of $40,000 in his savings account after 15 years.

Wyatt will have to invest $___________ now in order to have a balance of $40,000 in his savings account after 15 years. Round your answer UP to the nearest dollar.

Answer: a) i-1; b) 4+i/13

Step-by-step explanation:

The complex number [tex]i[/tex] is defined as the number such that [tex]i^{2}=-1[/tex];

We use the propperty to notice that [tex]i^1=i \quad i^2 = -1 \quad i^3=-i \quad i^4=1 \quad i^5=1 \quad i^6=-1 \quad i^7=-i \quad i^8=1 \quad i^9=i \quad i^10= -1 etc...[/tex].

a) We notice that [tex]i^{21}=i \quad \text{ and \text} \quad i^{32}=1[/tex]. Hence, [tex]i^{21}-i^{32}=i-1[/tex].

b) We multiply the expression by [tex]1=\frac{3+2\cdot i}{3 + 2 \cdot i}[/tex]. Then we get that

[tex]\frac{2-i}{3-2 \cdot i}=\frac{2-i}{3-2\cdot i}\cdot\frac{3+2 \cdot i}{3 + 2\cdot i } = \frac{(2-i)\cdot(3+2 \cdot i)}{3^2+2^2}= \frac{6+4i-3i+2i^2}{13}=\frac{6+i-2}{13} = \frac{4+i}{13}[/tex]

Answer and Step-by-step explanation:

Since we have given that

Number of children in his state were eligible for Medicaid = 100,000

Number of children in his state were not eligible for Medicaid = 200,000

After a large expansion in Medicaid, we get that

Number of children in his state were eligible for Medicaid = 150,000

Number of children in his state were not eligible for Medicaid = 150,000

According to question, , “the number of children without access to health care fell by one-quarter".

So, we check whether it is correct or not.

Difference between previous and current data who were not eligible is given by

[tex]200000-150000\\\\=50000[/tex]

Percentage of decrement is given by

[tex]\dfrac{50000}{200000}=\dfrac{1}{4}[/tex]

Yes , it is fell by one quarter.

Answer:

d)9

Step-by-step explanation:

first , you have to take 1125and divide it by 5 , then the answer you should get is 225 ,second you need to take 225and divide it by the 25 children , then you get 9 , when you divide 225÷25

1125/25= 45

45/ 5= 9

Answer is 9 days - d)

Answer:

given y=2.69+0.0138x the slope is the coefficient of x slope=0.0138   this means that the dive duration is expected to increase by about 0.0138 per minute

Step-by-step explanation:

Answer:

Hi!

A) The intercept of the regression line is 2.56.

B)The slope of the regression line is 0.0135.

C)For every increase of 1-meter depth, the mean dive duration increases by exactly 0.0135 minutes.

Explanation:

The definition of the regression line of a function is the value of Y(x) when x=0.

The slope of a function is the value that configures the ratio of change vertically on the regression.

In this case is represented by 0.0135 multiplied by the x.

If we are talking about depth, and the values of x are all positives values.

Y(50)=2.56+0.0135(50)=3.235

So, when the penguins go deeper, they remain more time diving.

Answer:

TT means pi and e means Euler

Step-by-step explanation:

The probability of getting an order from Restaurant A is [tex]0.321[/tex]

The probability of getting an order accurate in all the Restaurants is [tex]0.873[/tex]

The Probability of getting an order accurate in the Restaurant A is [tex]0.909[/tex]

Selecting an order from Restaurant A and selecting an accurate order are disjoint​ events

Total number of orders [tex]=321+276+235+126+32+56+40+11=1097[/tex]

The probability of getting an order from Restaurant A (the accurate +the not accurate) is   [tex]\dfrac{321+32}{1097}=0.321[/tex]  

Total Order Accurate  [tex]=321+276+235+126=958[/tex]

Probability of getting an order accurate is  [tex]\dfrac{958}{1097}=0.873[/tex]

Probability of getting an order accurate in the Restaurant A is   [tex]\dfrac{321}{353}=0.909[/tex]

Learn more about probability.

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To find the probability of getting an order from Restaurant A or an accurate order, sum the values of Restaurant A and the accurate orders and divide it by the total number of orders. The events of selecting an order from Restaurant A and selecting an accurate order are not disjoint.

To find the probability of getting an order from Restaurant A or an accurate order, we need to sum the values of Restaurant A and the accurate orders and divide it by the total number of orders.

Probability of getting an order from Restaurant A = (321+32)/(321+276+235+126+32+56+40+11)

Probability of getting an accurate order = (321+276+235+126)/(321+276+235+126+32+56+40+11)

Since both events can occur simultaneously (an order can be from Restaurant A and accurate), they are not disjoint. The final probability is the sum minus the probability of their intersection.

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Answer:

The probability that the committee will consist of 1 from each type of monster is 0.1318 and Suppose that the Pokemon refuse to be on the same committee as a the Trolls.So,the probability that this type of committee is formed is 0.5357        

Step-by-step explanation:

No. of Trolls = 3

No. of  Nazguls =4

No. of Ents = 4

No. of Pokemon = 5

Total Monsters = 16

We are given that A committee of 4 is to be picked to represent the group at the Monster Bash.

(a) Find the probability that the committee will consist of 1 from each type of monster.

So, the probability that the committee will consist of 1 from each type of monster:

= [tex]\frac{^3C_1 \times ^4C_1 \times ^4C_1 \times ^5C_1}{^{16}C_4}[/tex]

= [tex]\frac{\frac{3!}{1!(3-1)!} \times \frac{4!}{1!(4-1)!} \times \frac{4!}{1!(4-1)!}\times \frac{5!}{1!(5-1)!}}{\frac{16!}{4!(16-4)!}}[/tex]

= [tex]\frac{\frac{3!}{1!(2)!} \times \frac{4!}{1!(3)!} \times \frac{4!}{1!(3)!}\times \frac{5!}{1!(4)!}}{\frac{16!}{4!(12)!}}[/tex]

= [tex]0.1318[/tex]

b)Suppose that the Pokemon refuse to be on the same committee as a the Trolls.

So,  the probability that this type of committee is formed

= [tex]\frac{(^3C_3 \times ^8C_1)+(^3C_2 \times ^8C_2)+(^3C_1 \times ^8C_3)+(^3C_0 \times ^8C_4)+(^5C_4 \times ^8C_0)+(^5C_3 \times ^8C_1)+(^5C_2 \times ^8C_2)+(^5C_1 \times ^8C_3)}{^{16}C_4}[/tex]

= [tex]\frac{(\frac{3!}{3!(3-3)!} \times \frac{8!}{1!(8-1)!})+(\frac{3!}{2!(3-2)!} \times\frac{8!}{2!(8-2)!})+(\frac{3!}{1!(3-1)!} \times\frac{8!}{3!(8-3)!})+(\frac{3!}{0!(3-0)!} \times \frac{8!}{4!(8-4)!})+(\frac{5!}{4!(5-4)!} \times \frac{8!}{0!(8-0)!})+(\frac{5!}{3!(5-3)!} \times \frac{8!}{1!(8-1)!})+(\frac{5!}{2!(5-2)!} \times\frac{8!}{2!(8-2)!})+(\frac{5!}{1!(5-1)!} \times \frac{8!}{3!(8-3)!})}{\frac{16!}{4!(16-4)!}}[/tex]

= [tex]0.5357[/tex]

Hence The probability that the committee will consist of 1 from each type of monster is 0.1318 and Suppose that the Pokemon refuse to be on the same committee as a the Trolls.So,the probability that this type of committee is formed is 0.5357          

The probability that the committee will consist of 1 member from each type of monster is 1. If the Pokemon refuse to be on the same committee as a Troll, the probability that this type of committee is formed is 4/5 or 0.8.

(a) To find the probability that the committee will consist of 1 member from each type of monster, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes. There are 3 choices for a Troll, 4 choices for a Nazgul, 4 choices for an Ent, and 5 choices for a Pokemon. So the total number of possible outcomes is 3 * 4 * 4 * 5 = 240. Now, we need to calculate the number of favorable outcomes. Since we want 1 member from each type of monster, we choose 1 Troll from 3, 1 Nazgul from 4, 1 Ent from 4, and 1 Pokemon from 5. So the number of favorable outcomes is 3 * 4 * 4 * 5 = 240. Therefore, the probability is 240/240, which simplifies to 1.

(b) If the Pokemon refuse to be on the same committee as a Troll, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes. To do this, we need to consider two cases: one where a Troll is chosen and one where a Troll is not chosen. Case 1: If a Troll is chosen, then there are 3 choices for a Troll, 4 choices for a Nazgul, 4 choices for an Ent, and 4 choices for a Pokemon (since one Pokemon is excluded). So the total number of outcomes in this case is 3 * 4 * 4 * 4 = 192. Case 2: If a Troll is not chosen, then there are 0 choices for a Troll, 4 choices for a Nazgul, 4 choices for an Ent, and 5 choices for a Pokemon. So the total number of outcomes in this case is 0 * 4 * 4 * 5 = 0. Therefore, the total number of favorable outcomes is 192 + 0 = 192. The total number of possible outcomes is still 240. So the probability is 192/240, which simplifies to 4/5 or 0.8.

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Answer:

[tex]f (-2) =-\frac{8}{3}[/tex]

[tex]f (4) =\frac{4}{3}[/tex]

[tex]f (1) = - 4[/tex]

Step-by-step explanation:

For this case it has a piecewise function composed of two functions.

To evaluate the piecewise function observe the condition.

[tex]f (x) = \frac{1}{3}x ^ 2 -4[/tex] when [tex]x \neq 1[/tex]

[tex]f (x) = -4[/tex] when [tex]x = 1[/tex]

We start by evaluating [tex]f(-2)[/tex], note that [tex]x = -2\neq 1[/tex]. Then we use the quadratic function:

[tex]f (-2) = \frac{1}{3}(-2) ^ 2 -4 = -\frac{8}{3}[/tex]

Now we evaluate [tex]f(4)[/tex] note that [tex]x = 4\neq 1[/tex]. Then we use the quadratic function:

[tex]f (4) = \frac{1}{3}(4) ^ 2 -4 = \frac{4}{3}[/tex]

Finally we evaluate [tex]f(1)[/tex] As [tex]x = 1[/tex]  then

[tex]f (1) = - 4[/tex]

Answer:

(-21)+(-20)+(-19)+...+50 is equal to 1044

Step-by-step explanation:

Let's divide the number series and find its solution.

The original number series is:

(-21)+(-20)+(-19)+...+50 which is the same as:

{(-1)*(21+20+19+18+....+0)} + (1+2+3+...+50) which is

-A+B where:

A=(21+20+19+18+....+0)=(0+1+2+3+...+21)

B=(1+2+3+...+50)

For this problem, we can use the Gauss method, which establishes that for a continuos series of numbers starting in 1, we can find the sum by:

S=n*(n+1)/2 where n is the last value of the series, so:

Using the method for A we have:

S=n*(n+1)/2

S(A)=(21)*(21+1)/2

S(A)=231

Using the method for B we have:

S=n*(n+1)/2

S(B)=(50)*(50+1)/2

S(B)=1275

So finally,

-A+B=-231+1275=1044

In conclusion, (-21)+(-20)+(-19)+...+50 is equal to 1044.

The sum of the sequence from -21 to 50 is calculated using the arithmetic series formula, resulting in a total sum of 1044.

The question asks for the sum of a sequence of integers starting from -21 and ending at 50. To find this sum, you can either add each number consecutively or use the formula for the sum of an arithmetic series.

In this case, the series is arithmetic because each term increases by 1 from the previous term. The formula for the sum of an arithmetic series is S = n/2 * (a_1 + a_n), where S is the sum of the series, n is the number of terms, a_1 is the first term, and a_n is the last term.

First, determine the number of terms in the series. Since our first term is -21 and our last term is 50, the series has 50 - (-21) + 1 = 72 terms in total. Now, using the formula, we can calculate the sum of the series:

S = 72/2 * (-21 + 50)

S = 36 * 29

S = 1044

Therefore, the sum of the integers from -21 to 50 is 1044.

Answer:The mean number of checks written per​ day = 0.3370

The standard deviation = 0.5805

The variance = 0.3370

Step-by-step explanation:

Let the random variable x represent the number of checks he wrote in one​ day.

Given : The number of checks written in last year = 123

Let the number of days in the year must be 365.

Now, the mean number of checks written per​ day will be  :-

[tex]\lambda=\dfrac{123}{365}=0.33698630137\approx0.3370[/tex]

We know that in Poisson distribution , the variance is equals to the mean value .

[tex]\text{Thus , Variance }=\sigma^2= 0.3370[/tex]

[tex]\Rightarrow\ \sigma=\sqrt{0.3370}=0.580517010948\approx0.5805[/tex]

Thus,  Standard deviation = 0.5805

Answer:

[tex]y=-\frac{158}{579}[/tex]

Step-by-step explanation:

To find the matrix A, took all the numeric coefficient of the variables, the first column is for x, the second column for y, the third column for z and the last column for w:

[tex]A=\left[\begin{array}{cccc}1&1&2&2\\-7&-3&5&-8\\4&1&1&1\\3&7&-1&1\end{array}\right][/tex]

And the vector B is formed with the solution of each equation of the system:[tex]b=\left[\begin{array}{c}3\\-3\\6\\1\end{array}\right][/tex]

To apply the Cramer's rule, take the matrix A and replace the column assigned to the variable that you need to solve with the vector b, in this case, that would be the second column. This new matrix is going to be called [tex]A_{2}[/tex].

[tex]A_{2}=\left[\begin{array}{cccc}1&3&2&2\\-7&-3&5&-8\\4&6&1&1\\3&1&-1&1\end{array}\right][/tex]

The value of y using Cramer's rule is:

[tex]y=\frac{det(A_{2}) }{det(A)}[/tex]

Find the value of the determinant of each matrix, and divide:

[tex]y==\frac{\left|\begin{array}{cccc}1&3&2&2\\-7&-3&5&-8\\4&6&1&1\\3&1&-1&1\end{array}\right|}{\left|\begin{array}{cccc}1&1&2&2\\-7&-3&5&-8\\4&1&1&1\\3&7&-1&1\end{array}\right|} =\frac{158}{-579}[/tex]

[tex]y=-\frac{158}{579}[/tex]

Answer:

( )

Step-by-step explanation:

( ) shorthand label indicates an embedded design in mixed method research. It indicates that one form of data collection is embedded within another.Mostly ( ) is used used when data collection is embedded into larger data. So we can say that ( ) shorthand label indicates an embedded design in mixed research method

Answer:  The correct option is (B) 1625.2 miles.

Step-by-step explanation:  Given that at a given speed a car can travel 95.6 miles on 4 gallons of gasoline.

We are to find the distance that the car can travel on 68 gallons at the same speed.

We will be using the UNITARY method to solve the given problem.

Distance traveled by the car on 4 gallons of gasoline = 95.6 miles.

So, the distance traveled by the car on 1 gallon of gasoline will be

[tex]\dfrac{95.6}{4}=23.9~\textup{miles}.[/tex]

Therefore, the distance traveled by the car in 68 gallons of gasoline is given by

[tex]23.9\times68=1625.2~\textup{miles}.[/tex]

Thus, the required distance traveled by the car on 68 gallons of gasoline is 1625.2 miles.

Option (B) is CORRECT.

Answer:

3

Step-by-step explanation:

given: contribution margin=$450,000 and net profit= $150,000

To find operating leverage

Now, operating leverage is a cost accounting formula that measures the degree to which a firm or project can increase operating income by increasing revenue. A business that generates sales with a high gross margin and low variables costs has high leverage.

[tex]operating leverage=\frac{contribution margin}{net profit}[/tex]

therefore, operating leverage=[tex]\frac{450000}{150000}[/tex] =3

operating leverage is 3

The probability of all three dice landing on 4 is 1/216, calculated by multiplying the individual probabilities of each die landing on 4.

The probability of all three dice landing on 4 can be calculated using the concept of independent events. Each die has a 1/6 probability of landing on 4, so the probability of all three landing on 4 is (1/6) x (1/6) x (1/6) = 1/216.

Answer:

the total number of fur seal pups in Rookery A (a) is  45524

Step-by-step explanation:

Given data

in early August (b) = 5498

late August, a sample (c) = 1300

previously tagged (d ) = 157

to find out

the total number of fur seal pups in Rookery A (a)

solution

we will apply here proportion method

that is

a:b :: c :d

a/b = c/d

put all value and find a

a = c/d × b

a = 1300/157 × 5498

a = 45524.84

the total number of fur seal pups in Rookery A (a) is  45524

Answer: The answer is 135 murders.

Step-by-step explanation: The report tells us that statistically 67.5% of murders are committed using a firearm. It follows therefore that in a sample of 200 randomly selected murders, one would expect that 67.5% of those would be by a firearm. [tex]\frac{67.5}{100}[/tex] * 200 = 135.

It would certainly be higher that the expected value based on previous data collected but it would not be unusual because one sample may have a higher than "normal" amount of murders by firearm. Statistics aren't going to be exact for every sample.

Using the binomial distribution, it is found that:

a) The expected value is of 135.

b) Unusual, as 153 is more than 2.5 standard deviations above the mean.

It is the probability of exactly x successes on n repeated trials, with p probability of a success on each trial.

The expected value of the binomial distribution is:

[tex]E(X) = np[/tex]

The standard deviation of the binomial distribution is:

[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]

In this problem, the parameters are p = 0.675 and n = 200.

Item a:

E(X) = np = 200 x 0.675 = 135.

135 would be expected to be committed with a​ firearm.

Item b:

The standard deviation is given by:

[tex]\sqrt{V(X)} = \sqrt{200(0.675)(0.325)} = 6.624[/tex]

Then:

[tex]E(X) + 2.5\sqrt{V(X)} = 135 + 2.5(6.624) = 151.56 < 153[/tex]

Since 153 is more than 2.5 standard deviations above the mean, it would be unusual observe 153 murders by firearm in a random sample of 200 ​murders.

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Answer:

The center is (3,1) and the radius is 3.

Step-by-step explanation:

The goal is to write in [tex](x-h)^2+(y-k)^2=r^2[/tex] because this tells us the center (h,k) and the radius r.

So we are going to need to complete the square 2 times here, once for x and the other time for y.

I'm going to use this formula to help me to complete the square:

[tex]x^2+bx+(\frac{b}{2})^2=(x+\frac{b}{2})^2[/tex].

So first step:

I'm going to group my x's together and my y's.

[tex]x^2-6x+\text{ ___ }+y^2-2y+\text{ ___ }+1=0[/tex]

Second step:

I'm going to go ahead and subtract that one on both sides. Those blanks are there because I'm going to fill them in with a number so that I can write the x part and y part as a square.  Remember whatever you add on one side you must add on the other.  So I'm going to put 2 more blanks to fill in on the opposite side of the equation.

[tex]x^2-6x+\text{ ___ }+y^2-2y+\text{ ___ }=-1+\text{ ___ }+\text{ ___}[/tex]

Third step:

Alright first blank I'm putting (-6/2)^2 due to my completing the square formula.  That means this value will also go on the other side in on of those blanks.

In the second blank I'm going to put (-2/2)^2 due to the completing the square formula.  This must also go on one of the blanks on the other side.

So we have:

[tex]x^2-6x+(\frac{-6}{2})^2+y^2-2y+(\frac{-2}{2})^2=-1+(\frac{-6}{2})^2+(\frac{-2}{2})^2[/tex]

Fourth step:

Don't make this more hurtful than it already is. Just use the formula drag down the things inside the square. Remember this:

[tex]x^2+bx+(\frac{b}{2})^2[/tex]

equals

[tex](x+\frac{b}{2})^2[/tex].

We are applying that left hand side there (that bottom thing I just wrote).

Let's try it:

[tex](x+\frac{-6}{2})^2+(y+\frac{-2}{2})^2=-1+(\frac{-6}{2})^2+(\frac{-2}{2})^2[/tex]

Fifth step:

The hard part is out of the way.

This is just a bunch of simplifying now:

[tex](x-3)^2+(y-1)^2=-1+9+1[/tex]

[tex](x-3)^2+(y-1)^2=9[/tex]

The center is (3,1) and the radius is 3.

Answer:

190° rotation = c

80° rotation = a

Step-by-step explanation:

b = 180° rotation

d = 360° rotation

Answer:

The correct option is c.

Step-by-step explanation:

If the direction of rotation is not mentioned, then it is considered as counterclockwise rotation.

It is given that the figure XYZ rotated 190° and a 80° rotation(composition), both about point y.

It means figure is rotated 80° counterclockwise about the point y after that the new figure is rotated 190° counterclockwise about the point y.

[tex]80^{\circ}+190^{\circ}=270^{\circ}[/tex]

It means the figure XYZ rotated 270° counterclockwise about the point y.

In figure (a), XYZ rotated 90° counterclockwise about the point y.

In figure (b), XYZ rotated 180° counterclockwise about the point y.

In figure (c), XYZ rotated 270° counterclockwise about the point y.

In figure (d), XYZ rotated 360° counterclockwise about the point y.

Therefore the correct option is c.

Answer: 0.001

Step-by-step explanation:

Given : Number of true–false questions  =3

Choices of answer for true–false questions =2

Then , probability that the answer is correct for a true–false question =[tex]\dfrac{1}{2}=0.5[/tex]

Also, Number of multiple choice questions  =3

Choices of answer for multiple choice questions = 5

Then , probability that the answer is correct for a multiple choice question =[tex]\dfrac{1}{5}=0.2[/tex]

Now, the probability that the student will correctly answer all six questions :-

[tex](0.2)^3\times(0.5)^3=0.001[/tex]

Hence, the probability that the student will correctly answer all six questions = 0.001

The probability that a student blindly guessing will correctly answer all six questions on a pop quiz consisting of three true/false questions and three multiple choice questions is 0.001, or 0.1%.

The probability of correctly guessing the answer to a true/false question is 0.5, because there are two options, true or false. Therefore, the probability of correctly guessing all three true/false answers is 0.5^3, which is 0.125.

For a multiple choice question with five answers, the probability of guessing correctly is 0.2 (or 1/5). Therefore, the probability of correctly answering all three multiple choice questions is 0.2^3, which is 0.008.

To find the total probability of correctly answering all six questions, we multiply these two probabilities together: 0.125 * 0.008 = 0.001. So, the probability that a student guessing blindly will correctly answer all six questions is 0.001, or 0.1% when expressed as a percentage.

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Answer with explanation:

The given differential equation is

x²y" -7 x y' +1 6 y=0---------(1)

  Let, y'=z

y"=z'

[tex]\frac{dy}{dx}=z\\\\y=zx[/tex]

Substitution the value of y, y' and y" in equation (1)

→x²z' -7 x z+16 zx=0

→x² z' + 9 zx=0

→x (x z'+9 z)=0

→x=0 ∧ x z'+9 z=0

[tex]x \frac{dz}{dx}+9 z=0\\\\\frac{dz}{z}=-9 \frac{dx}{x}\\\\ \text{Integrating both sides}\\\\ \log z=-9 \log x+\log K\\\\ \log z+\log x^9=\log K\\\\\log zx^9=\log K\\\\K=zx^9\\\\K=y'x^9\\\\K x^{-9}d x=dy\\\\\text{Integrating both sides}\\\\y=\frac{-K}{8x^8}+m[/tex]

is another independent solution.where m and K are constant of integration.

Answer:

[tex]y_2=x^4lnx[/tex]

Step-by-step explanation:

We are given that a differential equation

[tex]x^2y''-7xy'+16y=0[/tex]

And one solution is [tex]y_1=x^4[/tex]

We  have to find the other independent solution by using reduction order method

[tex]y''-\frac{7}{x}y'+\frac{16}{x^2}y=0[/tex]

Compare with the equation

[tex]y''+P(x)y'+Q(x)y=0[/tex]

Then we get P(x)=[tex]-\frac{7}{x}['/tex] Q(x)=[tex]\frac{16}{x^2}[/tex]

[tex]y_2=y_1\int\frac{e^{-\intP(x)dx}}{y^2_1}dx[/tex]

[tex]y_2=x^4\int\frac{e^{\frac{7}{x}}dx}}{x^8}dx[/tex]

[tex]y_2=x^4\int\frac{e^{7ln x}}{x^8}dx[/tex]

[tex]y_2=x^4\int\frac{x^7}{x^8}dx[/tex]

[tex]e^{xlny}=y^x[/tex]

[tex]y_2=x^4\int frac{1}{x}dx[/tex]

[tex]y_2=x^4lnx[/tex]

A secant is a line that intersects the curve of a circle at two points.

The line in the picture that passes through two points would be Line EF.

Answer:

EF is the secant line

Step-by-step explanation:

A tangent line of a circle is a line that touches the circle at any point on the circumference of the circle.

A secant line is a line that lies inside the circles . secant line crosses the circumference of the circle at two points. The secant line has not end points.

In the given diagram, secant line is EF because it has no end points and it intersects the circle at two points

EF is the secant line

I'm guessing you were originally told to find the inverse of

[tex]A=\begin{bmatrix}6&7\\8&9\end{bmatrix}[/tex]

and you've found the inverse to be

[tex]A^{-1}=\begin{bmatrix}-\frac92&\frac72\\4&-3\end{bmatrix}[/tex]

I'm also guessing that "product of elementary matrices" includes the decomposition of [tex]A^{-1}[/tex] into lower and upper triangular as well as diagonal matrices.

First thing I would do would be eliminate the fractions by multiplying the first row of [tex]A^{-1}[/tex] by 2. In matrix form, this is done by multiplying [tex]A^{-1}[/tex] by

[tex]\begin{bmatrix}2&0\\0&1\end{bmatrix}[/tex]

which you can interpret as "multiply the first row by 2 and leave the second row alone":

[tex]\begin{bmatrix}2&0\\0&1\end{bmatrix}\begin{bmatrix}-\frac92&\frac72\\4&-3\end{bmatrix}=\begin{bmatrix}-9&7\\4&-3\end{bmatrix}[/tex]

Next, we make the matrix on the right side upper-triangular by eliminating the entry in row 2, column 1. This is done via the product

[tex]\begin{bmatrix}1&0\\4&9\end{bmatrix}\begin{bmatrix}2&0\\0&1\end{bmatrix}\begin{bmatrix}-\frac92&\frac72\\4&-3\end{bmatrix}=\begin{bmatrix}-9&7\\0&1\end{bmatrix}[/tex]

which you can interpret as "leave the first row alone, and replace row 2 by 4(row 1) + 9(row 2)".

Lastly, multiply both sides by the inverses of all matrices as needed to isolate [tex]A^{-1}[/tex] on the left side. That is,

[tex]\left(\begin{bmatrix}1&0\\4&9\end{bmatrix}\begin{bmatrix}2&0\\0&1\end{bmatrix}\right)A^{-1}=\begin{bmatrix}-9&7\\0&1\end{bmatrix}[/tex]

[tex]\left(\begin{bmatrix}1&0\\4&9\end{bmatrix}\begin{bmatrix}2&0\\0&1\end{bmatrix}\right)^{-1}\left(\begin{bmatrix}1&0\\4&9\end{bmatrix}\begin{bmatrix}2&0\\0&1\end{bmatrix}\right)A^{-1}=\left(\begin{bmatrix}1&0\\4&9\end{bmatrix}\begin{bmatrix}2&0\\0&1\end{bmatrix}\right)^{-1}\begin{bmatrix}-9&7\\0&1\end{bmatrix}[/tex]

[tex]A^{-1}=\left(\begin{bmatrix}1&0\\4&9\end{bmatrix}\begin{bmatrix}2&0\\0&1\end{bmatrix}\right)^{-1}\begin{bmatrix}-9&7\\0&1\end{bmatrix}[/tex]

For two invertible matrices [tex]X[/tex] and [tex]Y[/tex], we have [tex](XY)^{-1}=Y^{-1}X^{-1}[/tex], so that

[tex]A^{-1}=\begin{bmatrix}2&0\\0&1\end{bmatrix}^{-1}\begin{bmatrix}1&0\\4&9\end{bmatrix}^{-1}\begin{bmatrix}-9&7\\0&1\end{bmatrix}[/tex]

Compute the remaining inverses:

[tex]\begin{bmatrix}2&0\\0&1\end{bmatrix}^{-1}=\begin{bmatrix}\frac12&0\\0&1\end{bmatrix}[/tex]

[tex]\begin{bmatrix}1&0\\4&9\end{bmatrix}^{-1}=\begin{bmatrix}1&0\\-\frac49&\frac19\end{bmatrix}[/tex]

So we have

[tex]\begin{bmatrix}-\frac92&\frac72\\4&-3\end{bmatrix}=\begin{bmatrix}\frac12&0\\0&1\end{bmatrix}\begin{bmatrix}1&0\\-\frac49&\frac19\end{bmatrix}\begin{bmatrix}-9&7\\0&1\end{bmatrix}[/tex]

Answer:

Epistemology can be defined as the branch of philosophy which focuses on the theory of knowledge. It is the study of the nature of knowledge, and  analyses its relation concepts like truth, justifications and belief. It deals with the rationality of belief(logic or rationale behind a belief).

Yes, it is important to critical thinking as critical thinking is based on beliefs and actions that have reasons, logic and rationale behind them. But these logic and reasons are philosophically problematic as how one evaluate 'reason, 'logic', etc. These are abstract and difficult but question like these are a central part to epistemology.

Answer:

The quadratic function is:

[tex]f (x) = x ^ 2 + 11x +30[/tex]

Step-by-step explanation:

The zeros of a function are all values of x for which the function is equal to zero.

If a function has two zeros then it is a quadratic function.

If the zeros are -6 and -5. Then the function will have the following form:

[tex]f (x) = (x + 6) (x + 5)[/tex]

We can expand the expression by applying the distributive property, and we obtain

[tex]f (x) = x ^ 2 + 5x + 6x +30[/tex]

[tex]f (x) = x ^ 2 + 11x +30[/tex]

Answer:

Step-by-step explanation:

Since we are talking about compounded annual interest, we can use the Exponential Growth Formula to calculate the answer for this question.

[tex]y = a* (1+r)^{t}[/tex]

Where:

First we need to calculate the total after 2 years with a 9% interest.

[tex]y = 1000* (1+0.09)^{2}[/tex]

[tex]y =  1000* (1.09)^{2}[/tex]

[tex]y = 1000* 1.1881[/tex]

[tex]y = 1188.1[/tex]

So after 2 years there will be £1,188.10 in the account. Now we can add £3000 to that and use the new value as the initial amount, and calculate the new total in 5 years.

[tex]y = (1188.1+3000)* (1+0.09)^{5}[/tex]

[tex]y = 4188.1* (1.09)^{5}[/tex]

[tex]y = 4188.1* 1.5386 [/tex]

[tex]y = 6443.91[/tex]

So now we can subtract the £4000 purchase from the amount currently in the account, and calculate one more year of interest with the new initial amount.

[tex]y = (6443.91-4000)* (1+0.09)^{1}[/tex]

[tex]y = (2443.91)* 1.09[/tex]

[tex]y = 2663.86[/tex]

So at the end you would have £2,662.86 in the account one year after the purchase.

To calculate the amount left in the account one year after the purchase, calculate the interest earned on the initial and subsequent deposits, add them to the account balance, and subtract the purchase amount.

To calculate the amount left in the account one year after the purchase, we need to consider the interest earned on the initial deposit and the subsequent deposit in two years. Let's break it down step by step:

Therefore, there is a deficit of P2010 in the account one year after the purchase.

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Answer:

[tex]y=A+Bx^{2}[/tex]

Step-by-step explanation:

The given Cauchy-Euler equation is: [tex]x^2y''-xy'=0[/tex]

Comparing to the general form: [tex]ax^2y''+bxy'+cy=0[/tex], we have a=1,b=-1 and c=0

The auxiliary solution is given by: [tex]am(m-1)+bm+c=0[/tex]

[tex]\implies m(m-1)-m=0[/tex]

[tex]\implies m(m-1-1)=0[/tex]

[tex]\implies m(m-2)=0[/tex]

[tex]\implies m=0\:\:or\:\:m=2[/tex]

The general solution to this is of the form [tex]y=Ax^{m_1}+Bx^{m_2}[/tex], where A and B are constants.

[tex]y=Ax^{0}+Bx^{2}[/tex]

Therefore the general solution is;

[tex]y=A+Bx^{2}[/tex]

Let [tex]y_1=A[/tex] and [tex]y_2=Bx^2[/tex]

Since we CANNOT express the two solutions as constant multiple of each other, we say the two solutions are linearly independent.

[tex]y_1\neCy_2[/tex], where C is a constant.

The solutions to the differential equation x^2y" - xy' = 0 are y = 1 and y = x, which are linearly independent. The general solution is y = C1 + C2x, where C1 and C2 are constants.

For the Cauchy-Euler equation x^2y" - xy' = 0, we find solutions by substituting y = x^r. Differentiating yields y' = rx^{r-1} and y" = r(r-1)x^{r-2}. Substituting these into the equation and simplifying gives us r(r-1)x^r - rx^r = 0, which simplifies to the characteristic equation r^2 - r = 0. Solving this equation, we get the roots r1 = 0 and r2 = 1. Therefore, the two solutions are y1 = x^0 = 1 and y2 = x^1 = x. To show that they are linearly independent, we evaluate the Wronskian determinant W(y1,y2) = |1  x| = x which is nonzero for x > 0. The general solution is a combination of the two: y = C1y1 + C2y2, where C1 and C2 are arbitrary constants.

Given:

area of square, A = 16 [tex]cm^{2}[/tex]

error in area, dA = 0.03 cm^{2}

Step-by-Step Explanation:

Let 'a' be the side of the square

area of square, A = [tex]a^{2}[/tex]                 (1)

A = 16 =  [tex]a^{2}[/tex]

Therefore, a = 4 cm

for max tolerable error in length 'da', differentiate eqn (1) w.r.t 'a':

dA = 2a da

[tex]0.03 = 2\times 4\times da[/tex]

da = [tex]\frac{0.03}{8}[/tex]

da = 0.0375 cm

The side length of the square is 4 cm, the maximum error that can be tolerated in the length of each side to ensure that the area is within the specified tolerance is 0.0375 cm (or 0.0375 mm).

Given specifications:

Desired Area (A) = 16 cm²

Tolerance (ΔA) = 0.03 cm²

The formula for the area of a square is:

A = side length (L) * side length

Calculate the derivative of the area formula with respect to the side length (L):

dA/dL = 2L

Now, we want to find the maximum error in the side length (ΔL) that can be tolerated while keeping the area within the specified tolerance:

ΔA = (dA/dL) * ΔL

Plug in the values we have:

0.03 cm² = (2L) * ΔL

Solve for ΔL:

ΔL = 0.03 cm² / (2L)

To ensure that the area is within the tolerance, the error in the side length should be no more than ΔL.

Now, let's calculate ΔL using the formula above and for a given side length (L):

ΔL = 0.03 cm² / (2L)

If we assume a side length of L = 4 cm (to achieve the desired area of 16 cm²), we can calculate ΔL:

ΔL = 0.03 cm² / (2 * 4 cm) = 0.03 cm² / 8 cm = 0.00375 cm = 0.0375 mm

So, if the side length of the square is 4 cm, the maximum error that can be tolerated in the length of each side to ensure that the area is within the specified tolerance is 0.0375 cm (or 0.0375 mm).

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Answer:

16.2

Step-by-step explanation:

Since, if the sum of two angles is 90° then they are called complementary angles.

Here, the given angle is 73.8°,

Let x be the complementary angle of 73.8°,

Thus by the above definition,

x + 73.8° = 90°,

By subtraction property of equality,

⇒ x = 16.2°,

Hence, the complementary angle of 73.8° is 16.2°.