For the equation x^2y" - xy' = 0, find two solutions, show that they are linearly independent and find the general solution. Equations of the form ax^2y" + bxy' + cy = 0 are called Euler's equations or Cauchy-Euler equations. They are solved by trying y = x^r and solving for r (assume that x greaterthanorequalto 0 for simplicity).

Answer:

see attachment

Step-by-step explanation:

The range of [tex]\(y = \frac{4}{5}\sin x\)[/tex] for [tex]\(\pi \leq x \leq \frac{3\pi}{2}\)[/tex] is [tex]\(-\frac{4}{5} \leq y \leq 0\)[/tex] ( Option C).

To find the range of [tex]\(y = \frac{4}{5}\sin x\)[/tex] for [tex]\(\pi \leq x \leq \frac{3\pi}{2}\)[/tex], we need to determine the minimum and maximum values of sin x in the given interval and then scale them using [tex]\(\frac{4}{5}\)[/tex].

In the interval [tex]\(\pi \leq x \leq \frac{3\pi}{2}\)[/tex], the sine function is negative since it corresponds to the third and fourth quadrants on the unit circle. The minimum value of sin x in this interval is -1, and the maximum value is 0.

Now, scale these values using [tex]\(\frac{4}{5}\)[/tex]:

[tex]\(-1 \times \frac{4}{5} = -\frac{4}{5}\) (minimum)\\\\\(0 \times \frac{4}{5} = 0\) (maximum)[/tex]

Therefore, the range of [tex]\(y = \frac{4}{5}\sin x\) for \(\pi \leq x \leq \frac{3\pi}{2}\)[/tex] is [tex]\(-\frac{4}{5} \leq y \leq 0\)[/tex]. The correct choice is:

-4/5 <= y <= 0

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Answer:  The proof is done below.

Step-by-step explanation:  Given that U and V are subspaces of a vector space W.

We are to prove that the intersection U ∩ V is also a subspace of W.

(a) Since U and V are subspaces of the vector space W, so we must have

0 ∈ U and 0 ∈ V.

Then, 0 ∈ U ∩ V.

That is, zero vector is in the intersection of U and V.

(b) Now, let x, y ∈ U ∩ V.

This implies that x ∈ U, x ∈ V, y ∈ U and y ∈ V.

Since U and V are subspaces of U and V, so we get

x + y ∈ U  and  x + y ∈ V.

This implies that x + y ∈ U ∩ V.

(c) Also, for a ∈ R (a real number), we have

ax ∈ U and ax ∈ V (since U and V are subspaces of W).

So, ax ∈ U∩ V.

Therefore, 0 ∈ U ∩ V and for x, y ∈ U ∩ V, a ∈ R, we have

x + y and ax ∈ U ∩ V.

Thus, U ∩ V is also a subspace of W.

Hence proved.

bearing in mind that perpendicular lines have negative reciprocal slopes, let's find the slope of 5x -  y = 9 then.

[tex]\bf 5x-y=9\implies -y=-5x+9\implies y=\stackrel{\stackrel{m}{\downarrow }}{5}x-9\leftarrow \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\bf \stackrel{\textit{perpendicular lines have \underline{negative reciprocal} slopes}} {\stackrel{slope}{5\implies \cfrac{5}{1}}\qquad \qquad \qquad \stackrel{reciprocal}{\cfrac{1}{5}}\qquad \stackrel{negative~reciprocal}{-\cfrac{1}{5}}}[/tex]

so then, we're really looking for the equation of a line whose slope is -1/5 and runs through (3,-10).

[tex]\bf (\stackrel{x_1}{3}~,~\stackrel{y_1}{-10})~\hspace{10em} slope = m\implies -\cfrac{1}{5} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-(-10)=-\cfrac{1}{5}(x-3)\implies y+10=-\cfrac{1}{5}x+\cfrac{3}{5} \\\\\\ y=-\cfrac{1}{5}x+\cfrac{3}{5}-10\implies y=-\cfrac{1}{5}x+\cfrac{53}{5}[/tex]

and it looks like the one in the picture below.

Answer:

200π cubic units.

Step-by-step explanation:

Use the general method of integrating the area of the surface  generated by an arbitrary cross section of the region  taken parallel to the axis of revolution.

Here the axis  x = 9 is parallel to the y-axis.

The height of  one cylindrical shell = 8 - x^3.

The radius = 9 - x.

                                               2

The volume generated =  2π∫   (8 - x^3) (9 - x) dx

                                               0

= 2π ∫ ( 72 - 8x - 9x^3 + x^4) dx

             2

=      2 π [    72x - 4x^2 - 9x^4/4 + x^5 / 4  ]

            0

= 2 π  ( 144 - 16  - 144/4 + 32/4)

= 2 π * 100

= 200π.

The correct option is to reject the null hypothesis. The mean life spans for whites and non-whites born in 1900 in the certain county are not the same.

To conduct the hypothesis test, we will use a two-sample z-test for the difference in two means. The null hypothesis (H0) states that there is no difference in the mean life spans between whites and non-whites, while the alternative hypothesis (Ha) states that there is a difference.

The formula for the z-test statistic is:

[tex]\[ z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \][/tex]

where:

- [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_2\)[/tex] are the sample means for whites and non-whites, respectively.

- [tex]\(\mu_1\)[/tex] and [tex]\(\mu_2\)[/tex] are the population means for whites and non-whites, respectively.

- [tex]\(\sigma_1\)[/tex] and [tex]\(\sigma_2\)[/tex] are the population standard deviations for whites and non-whites, respectively.

- [tex]\(n_1\)[/tex] and [tex]\(n_2\)[/tex] are the sample sizes for whites and non-whites, respectively.

Given:

- [tex]\(\bar{x}_1 = 45.3\) years, \(s_1 = 12.7\) years, \(n_1 = 124\)[/tex] (for whites)

- [tex]\(\bar{x}_2 = 34.1\)[/tex] years, [tex]\(s_2 = 15.6\) years, \(n_2 = 82\)[/tex] (for non-whites)

- [tex]\(\mu_1 = 47.6\)[/tex] years (for whites)

- [tex]\(\mu_2 = 33.0\)[/tex] years (for non-whites)

Since we do not have the population standard deviations, we will use the sample standard deviations as an estimate. This is appropriate given the sample sizes are large enough (generally [tex]\(n > 30\)[/tex] is considered sufficient).

First, we calculate the standard error (SE) of the difference in means:

[tex]\[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{12.7^2}{124} + \frac{15.6^2}{82}} \][/tex]

[tex]\[ SE = \sqrt{\frac{161.29}{124} + \frac{243.36}{82}} \][/tex]

[tex]\[ SE = \sqrt{1.299 + 2.968} \][/tex]

[tex]\[ SE = \sqrt{4.267} \][/tex]

[tex]\[ SE \approx 2.066 \][/tex]

Now, we calculate the z-statistic:

[tex]\[ z = \frac{(45.3 - 34.1) - (47.6 - 33.0)}{2.066} \][/tex]

[tex]\[ z = \frac{11.2 - 14.6}{2.066} \][/tex]

[tex]\[ z = \frac{-3.4}{2.066} \][/tex]

[tex]\[ z \approx -1.646 \][/tex]

Next, we find the p-value for this z-statistic. Since we are conducting a two-tailed test, we will look at the probability of a z-score being less than -1.646 or greater than 1.646. Using a standard normal distribution table or a calculator, we find that the p-value is approximately 0.100.

Finally, we compare the p-value to our significance level (commonly denoted as [tex]\(\alpha\))[/tex]. If the p-value is less than [tex]\(\alpha\)[/tex], we reject the null hypothesis. If we choose [tex]\(\alpha = 0.05\)[/tex], then since [tex]\(0.100 > 0.05\)[/tex], we fail to reject the null hypothesis.

However, if we choose a significance level of [tex]\(\alpha = 0.10\)[/tex], then since [tex]\(0.100 \leq 0.10\)[/tex], we would reject the null hypothesis, indicating that there is a statistically significant difference in the mean life spans between whites and non-whites in the county.

Given the p-value is on the boundary of common significance levels, the conclusion may vary depending on the chosen level of significance. However, the correct option based on the provided conversation is to reject the null hypothesis, suggesting that the mean life spans are not the same for whites and non-whites in the county.

The amount of salt in the tank changes with rate according to

[tex]Q'(t)=\left(1\dfrac{\rm lb}{\rm gal}\right)\left(4\dfrac{\rm gal}{\rm min}\right)-\left(\dfrac{Q(t)}{300+(4-1)t}\dfrac{\rm lb}{\rm gal}\right)\left(1\dfrac{\rm gal}{\rm min}\right)[/tex]

[tex]\implies Q'+\dfrac Q{300+3t}=4[/tex]

which is a linear ODE in [tex]Q(t)[/tex]. Multiplying both sides by [tex](300+3t)^{1/3}[/tex] gives

[tex](300+3t)^{1/3}Q'+(300+3t)^{-2/3}Q=4(300+3t)^{1/3}[/tex]

so that the left side condenses into the derivative of a product,

[tex]\big((300+3t)^{1/3}Q\big)'=4(300+3t)^{1/3}[/tex]

Integrate both sides and solve for [tex]Q(t)[/tex] to get

[tex](300+3t)^{1/3}Q=(300+3t)^{4/3}+C[/tex]

[tex]\implies Q(t)=300+3t+C(300+3t)^{-1/3}[/tex]

Given that [tex]Q(0)=100[/tex], we find

[tex]100=300+C\cdot300^{-1/3}\implies C=-200\cdot300^{1/3}[/tex]

and we get the particular solution

[tex]Q(t)=300+3t-200\cdot300^{1/3}(300+3t)^{-1/3}[/tex]

[tex]\boxed{Q(t)=300+3t-2\cdot100^{4/3}(100+t)^{-1/3}}[/tex]

Answer:

Horizontal axis

Step-by-step explanation:

By convention, the independent variables is arrayed along the Horizontal axis (abscissa) in a scattergram.

The stcattergram has two dimensions

If the first card drawn is a diamond, the total number of cards reduces to 51 for the second draw. Out of these, 25 are red (13 hearts + 12 diamonds). So, the probability of drawing a red card on the second draw, if the first drawn card was a diamond, is 25/51 approximated to 0.4902.

The subject of this question is probability in mathematics, specifically related to a scenario that involves sampling without replacement from a deck of playing cards. Firstly, let us familiarize ourselves with the composition of a standard deck of cards. It consists of 52 cards divided into four suits: clubs, diamonds, hearts, and spades. Clubs and spades are black cards, while diamonds and hearts are red cards. Each suit has 13 cards.

If the first card drawn is a diamond, the total count of cards in the deck reduces to 51 (because we are drawing without replacement), and, since a diamond card has been withdrawn, the count of remaining red cards is 25 (13 hearts and 12 remaining diamonds). Thus, to find the probability of drawing a red card on the second draw, we simply count the remaining red cards and divide by the remaining total cards, giving us a probability of 25/51 or approximately 0.4902 when rounded to four decimal places.

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The probability of choosing a red card for the second card drawn, if the first card was a diamond, is 0.4902.

So, the calculated probabilities are:

- The probability of drawing a diamond first: 0.25
- The probability of drawing a red card second, given that the first card drawn was a diamond: 0.4902

Therefore, the probability of choosing a red card for the second card drawn, if the first card was a diamond, is approximately 0.4902.

The monthly payment for a $17,500 loan at 6% APR over 25 years is about $113.36. The total amount paid over the duration of the loan is $34,008, of which about 51.46% goes towards the principal and 48.54% towards interest.

To answer this question, we need to use the formula for calculating the monthly payment for a loan, which is P[r(1 + r)^n]/[(1 + r)^n - 1], where P is the principal loan amount, r is the monthly interest rate (annual rate divided by 12), and n is the number of payments (years times 12).

a. Monthly payment:

First, we have to calculate r: 6% APR implies a yearly interest rate of 6%/12 = 0.005 per month. Plugging P = $17,500, r = 0.005, and n = 25*12 = 300 into the formula, we get the monthly payment of approximately $113.36.

b. Total amount paid:

The total amount paid over the duration of the loan is simply the monthly payment times the total number of payments, so $113.36*300 = $34,008.

c. Percentages of principal and interest:

The percentage paid towards the principal is the original loan amount divided by the total payment amount times 100, so $17,500/$34,008*100 = approximately 51.46%. Therefore, the percentage paid for interest is 100 - 51.46 = 48.54%.

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The monthly payment for the loan is approximately $114.08. Over 25 years, the total amount paid will be about $34,224. About 51.11% of this amount goes toward the principal and 48.89% goes toward interest.

To solve this problem, we need to calculate the monthly payment, total amount paid, and the percentages of principal and interest paid for a student loan of $17,500 at a fixed APR of 6% for 25 years.

a. Calculate the monthly payment

We use the formula for the monthly payment on an amortizing loan:

→ [tex]M = P [r(1 + r)^n] / [(1 + r)^{n - 1}][/tex]

where:

→ P = loan principal ($17,500)

→ r = monthly interest rate (annual rate / 12)

    = 6% / 12

    = 0.005

→ n = total number of payments (years * 12)

     = 25 * 12

     = 300

Substituting the values into the formula:

→ [tex]M = 17500 [0.005(1 + 0.005)^{300}] / [(1 + 0.005)^{300 – 1}][/tex]

→ M = 17500 [0.005(4.2918707)] / [4.2918707 – 1]

→ M = 17500 [0.02145935] / [3.2918707]

→ M = 375.53965 / 3.2918707

→ M ≈ $114.08

b. Determine the total amount paid over the term of the loan

→ Total amount paid = monthly payment * total number of payments

→ Total amount paid = 114.08 * 300

                                 ≈ $34,224

c. Of the total amount paid, what percentage is paid toward the principal and what percentage is paid for interest

→ Principal: $17,500

→ Interest: Total amount paid - Principal

→ Interest = 34,224 - 17,500

                ≈ $16,724

→ Percentage toward principal = (Principal / Total amount paid) * 100

                                                   ≈ (17500 / 34224) * 100

                                                   ≈ 51.11%

→ Percentage toward interest = (Interest / Total amount paid) * 100

                                                 ≈ (16724 / 34224) * 100

                                                 ≈ 48.89%

Answer:

He did better on biology test.

Step-by-step explanation:

For comparing the scores we need to find z-scores for both subjects,

We know that,

z-score or standard score for x is,

[tex]z=\frac{x-\mu}{\sigma}[/tex]

Where, [tex]\mu[/tex] is mean,

[tex]\sigma[/tex] is standard deviation,

In history test,

[tex]x=77[/tex]

[tex]\mu=70[/tex]

[tex]\sigma = 6[/tex]

Thus, the z-score would be,

[tex]z_1=\frac{77-70}{6}[/tex]

[tex]\approx 1.167[/tex]

In biology test,

[tex]x=79[/tex]

[tex]\mu = 70[/tex]

[tex]\sigma = 4[/tex],

Thus, the z-score would be,

[tex]z_2=\frac{79-70}{4}[/tex]

[tex]= 2.25[/tex]

∵ [tex]z_2>z_1[/tex]

Hence, he did better on biology test.

Answer:

$148.02

Step-by-step explanation:

In the question

Principal = $100, rate(R) = 4% compounded annually, time(T)= 10 years

we know that the formula for compound interest

A=[tex]P\times_(1+\frac{R}{100} )^{T}[/tex]   where A is amount

now putting values in the above formula we get

A=[tex]100\times_(1+\frac{4}{100} )^{10}[/tex]

therefore A= $148.024428

rounding off to the nearest penny we get amount as $148.02 and compound interest will be $48.02

Answer:

Step-by-step explanation:

Let x and y represent the numbers of churches and unions contacted in the month, respectively. Then Jayanta's limit on letter writing hours is ...

  2x +2y ≤ 20

and her limit on follow-up call hours is ...

  x + 3y ≤ 14

Graphing these inequalities (see below) results in a feasible region with vertices at (x, y) = (0, 4 2/3), (8, 2), and (10, 0). Of these, the mixture of groups producing the most money is ...

  8 churches and 2 unions.

The money she can raise from that mixture is ...

  8×$125 +2×$175 = $1350 in a month

Answer:

Sqdancefan's answer is correct.

Step-by-step explanation:

I  misread the question.

Answer:

a. 3.53 b. 3.61 c. 3.91 d. 2.04

Step-by-step explanation:

a. 3.534 take 4 out

b. 3.669 nine is higher then 5 so, take six makes out of one

c. 3.969 nine is higher then 5 so, take six makes out of one

d. 2.040 take zero out

above that is the answer.... hope i helped this....

Answer:

11 < n < 29

Step-by-step explanation:

The  smallest the third side can be is just bigger than the difference of the other two sides

20-9 < n

11<n

The largest the third side can be is just smaller than the sum of the other two sides

20+9 > n

29 >n

Putting this together

11<n<29

Answer:

200 Creamy Vanilla

300 Continental Mocha

Step-by-step explanation:

3 cups cream for each:

500 Quarts total

Creamy Vanilla requires 400 eggs

Continental Mocha requres 300 eggs for a total of 700 eggs

The answer is:

[tex]\[ \boxed{V = 200, M = 300} \][/tex]

To solve this problem, we need to set up a system of equations based on the given information and then solve for the number of quarts of each flavour of ice cream that should be made.

Let [tex]\( V \)[/tex] represent the number of quarts of Creamy Vanilla ice cream and [tex]\( M \)[/tex] represent the number of quarts of Continental Mocha ice cream.

From the information given, we can derive the following equations:

For the eggs:

Each quart of Creamy Vanilla requires 2 eggs, so [tex]\( 2V \)[/tex] eggs are used for Creamy Vanilla.

Each quart of Continental Mocha requires 1 egg, so [tex]\( M \)[/tex] eggs are used for Continental Mocha.

Since there are 700 eggs in total, we have the equation:

[tex]\[ 2V + M = 700 \][/tex]

For the cream:

Each quart of Creamy Vanilla requires 3 cups of cream, so [tex]\( 3V \)[/tex] cups of cream are used for Creamy Vanilla.

Each quart of Continental Mocha also requires 3 cups of cream, so [tex]\( 3M \)[/tex] cups of cream are used for Continental Mocha.

Since there are 1500 cups of cream in total, we have the equation:

[tex]\[ 3V + 3M = 1500 \][/tex]

Now we have a system of two equations with two variables:

[tex]\[ \begin{cases} 2V + M = 700 \\ 3V + 3M = 1500 \end{cases} \][/tex]

To solve this system, we can simplify the second equation by dividing every term by 3:

[tex]\[ V + M = 500 \][/tex]

Now we have a simpler system:

[tex]\[ \begin{cases} 2V + M = 700 \\ V + M = 500 \end{cases} \][/tex]

We can subtract the second equation from the first to eliminate [tex]\( M \)[/tex] and solve for [tex]\( V \)[/tex] :

[tex]\[ (2V + M) - (V + M) = 700 - 500 \][/tex]

[tex]\[ V = 200 \][/tex]

Now that we have the value of [tex]\( V \)[/tex], we can substitute it back into the simplified second equation to find [tex]\( M \)[/tex]:

[tex]\[ 200 + M = 500 \][/tex]

[tex]\[ M = 500 - 200 \][/tex]

[tex]\[ M = 300 \][/tex]

Therefore, the factory should make 200 quarts of Creamy Vanilla and 300 quarts of Continental Mocha to use up all the eggs and cream.

The final answer is:

[tex]\[ \boxed{V = 200, M = 300} \][/tex]

Answer:

See below.

Step-by-step explanation:

The set of  all positive integers N is countable so we need to show that there is a 1 to 1 correspondence between the elements in N  with the set of all odd positive integers. This is the case as shown below:

1 2 3 4 5 6 ...

|  |  |   |   |  | ....

1 3 5 7 9 11....

There are 24 hours to a day, so you can write this as

704 days, 11 hours = 704 + 11/24 days

Then dividing by 29 gives

704/29 + 11/696 days

We have

704 = 24*29 + 8

so that the time is equal to

24 + 8/29 + 11/696 days

24 + (192 + 11)/696 days

24 + 7/24 days

which in terms of days and hours is

24 days, 7 hours

Answer:

The probability is 0.0643

Step-by-step explanation:

* Lets revise some definition to solve the problem

- The standard deviation of the distribution of sample means is called σM

- σM = σ/√n , where σ is the standard deviation and n is the sample size

- z-score = (M - μ)/σM, where M is the mean of the sample , μ is the mean

 of the population  

* Lets solve the problem

- The mean of the washing machine is 9.3 years

∴ μ = 9.3

- The standard deviation is 1.1 years

∴ σ = 1.1

- There are 70 washing machines randomly selected

∴ n = 70

- The mean replacement time less than 9.1 years

∴ M = 9.1

- Lets calculate z-score

∵ σM = σ/√n

∴ σM = 1.1/√70 = 0.1315

∵ z-score = (M - μ)/σM

∴ z-score = (9.1 - 9.3)/0.1315 = - 1.5209

- Use the normal distribution table of z to find P(z < -1.5209)

∴ P(z < -1.5209) = 0.06426

∵ P(M < 9.1) = P(z < -1.5209)

∴ P(M < 9.1) = 0.0643

* The probability is 0.0643

the total annual costs associated with Sam's current ordering policy amount to $1,215.

The student has provided the necessary information to calculate the total annual costs associated with Sam's current ordering policy at his auto shop. Sam orders 650 oil filters at a time and the shop uses 150 filters per week. The key components involved in this calculation are the order cost, annual demand, holding cost, and the size of each order.

The total annual demand (D) is the weekly demand (d) multiplied by the number of weeks per year:

D = d × 52
D = 150 filters/week × 52 weeks/year
D = 7,800 filters/year

The order cost (S) is given as $20 per order and the annual holding cost per unit (H) is $3 per filter. The size of each order (Q) is 650 filters.

The total annual ordering cost (AOC) can be calculated as the annual demand divided by the order size, multiplied by the order cost:

AOC = (D/Q)  × S
AOC = (7,800/650) × $20
AOC = 12 × $20
AOC = $240

The total annual holding cost (AHC) can be calculated as the average inventory level (which is Q/2 for consistent orders) multiplied by the holding cost per unit:

AHC = (Q/2) × H
AHC = (650/2) × $3
AHC = 325 × $3
AHC = $975

The total annual costs (TAC) are the sum of the total annual ordering cost and the total annual holding cost:

TAC = AOC + AHC
TAC = $240 + $975
TAC = $1,215

Therefore, the total annual costs associated with Sam's current ordering policy amount to $1,215.

Answer with explanation:

We have to prove that, For any integer m, 2 m×(3 m + 2) is divisible by 4.

We will prove this result with the help of Mathematical Induction.

⇒For Positive Integers

For, m=1

L HS=2×1×(3×1+2)

      =2×(5)

       =10

It is not divisible by 4.

⇒For Negative Integers

For, m= -1

L HS=2×(-1)×[3×(-1)+2]

      =-2×(-3+2)

       = (-2)× (-1)

       =2

It is not divisible by 4.

False Statement.

Answer:

1) 0.3456

2) 0.6544.

Step-by-step explanation:

Let X represents the event of recognizing the brand,

Given,

The probability of recognizing the brand, p = 40​% = 0.40,

Thus, the probability of not recognizing the brand, q = 1 - 0.40 = 0.60,

Since, the binomial distribution formula,

[tex]P(x) = ^nC_r (p)^r(q)^{n-r}[/tex]

Where,

[tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]

1) Thus, the probability that exactly 2 of the 5 consumers recognize the brand name is,

[tex]P(X=2)=^5C_2 (0.40)^2 (0.60)^{5-2}[/tex]

[tex]=10 (0.40)^2 (0.60)^3[/tex]

[tex]=0.3456[/tex]

2) Also, the probability that the number who recognize the brand name is not 2 = 1 - P(X=2) = 1 - 0.3456 = 0.6544.

Answer:  The average of the first two test scores is 97.

Step-by-step explanation:  Given that Kristie has taken five tests in science class. The average of all five of Kristie's test scores is 94 and the average of her last three test scores is 92.

We are to find the average score of her first two tests.

Let a1, a2, a3, a4 and a5 be teh scores of Kristle in first , second, third, fourth and fifth tests respectively.

Then, according to the given information, we have

[tex]\dfrac{a1+a2+a3+a4+a5}{5}=94\\\\\Rightarrow a1+a2+a3+a4+a5=94\times5\\\\\Rightarrow a1+a2+a3+a4+a5=470~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]

and

[tex]\dfrac{a3+a4+a5}{3}=92\\\\\Rightarrow a3+a4+a5=92\times3\\\\\Rightarrow a3+a4+a5=276~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)[/tex]

Subtracting equation (ii) from equation (i), we get

[tex](a1+a2+a3+a4+a5)-(a3+a4+a5)=470-276\\\\\Rightarrow a1+a2=194\\\\\Rightarrow \dfrac{a1+a2}{2}=\dfrac{194}{2}\\\\\Rightarrow \dfrac{a1+a2}{2}=97.[/tex]

Thus, the average of the first two test scores is 97.

Answer:

Thus, the average of the first two test scores is 97.

Step-by-step explanation:

Using the disk method, the volume is

[tex]\displaystyle\pi\int_0^\infty e^{-2x}\,\mathrm dx=\boxed{\frac\pi2}[/tex]

Alternatively, using the shell method, the volume is

[tex]\displaystyle2\pi\int_0^1y(-\ln y)\,\mathrm dy=\frac\pi2[/tex]

Using the shell method, the volume is

[tex]\displaystyle2\pi\int_0^\infty xe^{-x}\,\mathrm dx=\boxed{2\pi}[/tex]

Alternatively, using the disk method, the volume is

[tex]\displaystyle\pi\int_0^1(-\ln x)^2\,\mathrm dx=2\pi[/tex]

The area of the region is 1 square unit. The volume of the solid generated by revolving the region about the x-axis can be found by integrating π(y^2) dx from x = 0 to x = ∞.

To find the area of the region, we need to find the intersection points between the two curves. In this case, the curves are y = e^(-x) and y = 0. Since y ≥ 0, the region will lie between the x-axis and the curve y = e^(-x). The intersection point is where y = 0, which occurs at x = 0. To find the area, we integrate y = e^(-x) from x = 0 to x = ∞:

A = ∫0∞ e^(-x) dx = [-e^(-x)]0∞ = -[e^0 - 0]
               = -[1 - 0] = 1

The area of the region is 1 square unit.

To find the volume of the solid generated by revolving the region about the x-axis, we use the disk method. The radius of each disk is given by y = e^(-x), and the height of each disk is given by dx. The volume can be found by integrating π(y^2) dx from x = 0 to x = ∞:

V = π∫0∞ (e^(-x))^2 dx = π∫0∞ e^(-2x) dx

Answer:

4) 50,000

5) 900,000

Step-by-step explanation:

In Front end rounding what we do is we focus on first two digit of the number if the second digit number is greater than or equal to 5 we add the first digit by 1.

4) 50,987

here we can clearly see that the first two digit is 50 and second digit is not equal to 5 then rounding will be equal to 50,000.

5) 851,004

here we can clearly see that the first two digit is 85 and second digit is equal to 5 then we will round it  8+1=9

hence rounded number will be 900,000

Final answer:

Front-end rounding of 50,987 is 50,000 and for 851,004 it's 900,000, based on the most significant digits, using zeros as placeholders after rounding.

Explanation:

Front-end rounding involves rounding numbers based on the most significant digits. Let's apply this method to the numbers provided.

When rounding, if the digit immediately after the place you are rounding to is 5 or greater, you round up. Otherwise, you round down. Placeholder zeros are used to maintain the value's place in the number system.

Answer: 0.4302

Step-by-step explanation:

Given : Mean : [tex]\mu=\text{38.7 miles }[/tex]

Standard deviation : [tex]\sigma=\text{6.7 miles }[/tex]

Sample size : [tex]n=60[/tex]

Also, these distances are normally distributed.

Then , the formula to calculate the z-score is given by :-

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

For x=32.5

[tex]\\\\ z=\dfrac{32.5-38.7}{6.7}=-0.925373134\approx-0.93[/tex]

For x=40.5

[tex]\\\\ z=\dfrac{40.5-38.7}{6.7}=0.268656\approx0.27[/tex]

The p-value = [tex]P(-0.93<z<0.27)[/tex]

[tex]=P(0.27)-P(-0.93)=0.6064198- 0.1761855=0.4302343\approx0.4302[/tex]

Hence, the required probability :-0.4302

Answer: The price per unit is $48, when 30 units are demanded.

Step-by-step explanation:

Since we have given that

At price of $12 per unit, the number of units demanded = 40 units

At price of $18 per unit, the number of units demanded = 25 units.

So, the coordinates would be

(40,12) and (25,18)

As we know that x- axis denoted the quantity demanded.

y-axis denoted the price per unit.

So, the slope would be

[tex]m=\dfrac{y_2-y_1}{x_2-x_1}\\\\m=\dfrac{18-12}{25-40}\\\\m=\dfrac{-6}{15}\\\\m=\dfrac{-2}{5}[/tex]

So, the equation would be

[tex]y-y_1=m(x-x_1)\\\\y-12=\dfrac{-2}{5}(x-40)\\\\5(y-12)=-2(x-40)\\\\5y-60=-2x+80\\\\5y+2x=80+60\\\\5y+2x=140[/tex]

So, if 30 units are demanded, the price per unit would be

[tex]5y=140+2x\\\\5y=140+2\times 30\\\\5y=140+60\\\\5y=240\\\\y=\dfrac{240}{5}\\\\y=\$48[/tex]

Hence, the price per unit is $48, when 30 units are demanded.

The linear demand equation is Qd = 100 - 5P. In this equation, if we want to find the price per unit when 30 units are demanded, substitute Qd with 30 to get P = 14.

To find the linear demand equation, we can use the data given: consumers will buy 40 units of a product when the price is $12 per unit and 25 units when the price is $18 per unit. The general formula for a linear demand equation is Qd = a - bP, where Qd is the quantity demanded, P is the price per unit, a is the intercept, and b is the slope of the demand curve.

Let's use the two points (12, 40) and (18, 25) to formulate two equations with a and b as unknowns. We find that a = 100 and b = 5, therefore the demand equation is Qd = 100 - 5P. Now if we want to find the price per unit when 30 units are demanded, we just need to substitute Qd with 30 in the demand equation, hence P = 14.

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Answer with explanation:

[tex]\rightarrow 4x^2y+8x y'+y=x\\\\\rightarrow 8xy'+y(1+4x^2)=x\\\\\rightarrow y'+y\times\frac{1+4x^2}{8x}=\frac{1}{8}[/tex]

--------------------------------------------------------Dividing both sides by 8 x

This Integration is of the form ⇒y'+p y=q,which is Linear differential equation.

Integrating Factor

 [tex]=e^{\int \frac{1+4x^2}{8x} dx}\\\\e^{\log x^{\frac{1}{8}+\frac{x^2}{2}}\\\\=x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}[/tex]

Multiplying both sides by Integrating Factor  

[tex]x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}\times [y'+y\times\frac{1+4x^2}{8x}]=\frac{1}{8}\times x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}\\\\ \text{Integrating both sides}\\\\y\times x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}=\frac{1}{8}\int {x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}} \, dx \\\\8y\times x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}=\int {x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}} \, dx\\\\8y\times x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}=-[x^{\frac{9}{8}}]\times\frac{ \Gamma(0.5625, -x^2)}{(-x^2)^{\frac{9}{16}}}\\\\8y\times x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}=(-1)^{\frac{-1}{8}}[ \Gamma(0.5625, -x^2)]+C-----(1)[/tex]

When , x=1, gives , y=9.

Evaluate the value of C and substitute in the equation 1.

Answer:

Step-by-step explanation:

Put hypotheses as:

[tex]H_0: mu =40 mg\\H_a: mu \neq 40 mg.[/tex]

(Two tailed test at 4%)

Since population std deviation is  given we can do Z test.

[tex]Sample size n =30\\Sample mean = 42.3 mg\\Mean diff = 2.3 mg\\Std error of sample = \frac{7.1}{\sqrt{30} } =1.296[/tex]

Test statistic t = mean diff/se = [tex]\frac{2.3}{1.296} =1.775[/tex]

p value=0.075

Since p >0.04 our alpha, we accept null hypothesis.

At alphaequals0.04​, we cannot reject the​ company's claim

Step-by-step explanation:

Put hypotheses as:

\begin{gathered}H_0: mu =40 mg\\H_a: mu \neq 40 mg.\end{gathered}

H

0

:mu=40mg

H

a

:mu



=40mg.

(Two tailed test at 4%)

Since population std deviation is given we can do Z test.

\begin{gathered}Sample size n =30\\Sample mean = 42.3 mg\\Mean diff = 2.3 mg\\Std error of sample = \frac{7.1}{\sqrt{30} } =1.296\end{gathered}

Samplesizen=30

Samplemean=42.3mg

Meandiff=2.3mg

Stderrorofsample=

30

7.1

=1.296

Test statistic t = mean diff/se = \frac{2.3}{1.296} =1.775

1.296

2.3

=1.775

p value=0.075

Since p >0.04 our alpha, we accept null hypothesis.

At alphaequals0.04, we cannot reject the company's claim

Answer:

[tex]\dfrac{21\pi}{10},\ -\dfrac{19\pi}{10}[/tex]

Step-by-step explanation:

Any of the angles (in radians) π/10 +2kπ (k any integer) will be co-terminal with π/10. The angles listed in the answer above have k=1, k=-1.

_____

Comment on the last answer choice

The answer choices π/10+360° and π/10-360° amount to the same thing as the answer shown above, but use mixed measures. 1 degree is π/180 radians, so 360° is 2π radians. Then π/10+360° is fully equivalent to 21π/10 radians.

Answer:

C. P69,256.82

Step-by-step explanation:

We know that,

The amount formula in compound interest is,

[tex]A=P(1+\frac{r_1}{n_1})^{n_1t_1} (1+\frac{r_2}{n_2})^{n_2t_2}.......[/tex]

Where, P is the principal amount,

[tex]r_1, r_2....[/tex] are the annual rate for the different periods,

[tex]t_1, t_2,.....[/tex] are the number of year for different periods,

[tex]n_1, n_2, n_3...[/tex] are the number of periods,

Given,

A = P 200,000,

[tex]r_1=10%=0.1[/tex], [tex]n_1=4[/tex], [tex]t_1=5[/tex],[tex]r_2=12%=0.12[/tex], [tex]n_2=1[/tex], [tex]t_2=5[/tex]

Thus, by the above formula the final amount would be,

[tex]200000=P(1+\frac{0.1}{4})^{4\times 5}(1+\frac{0.12}{1})^{1\times 5}[/tex]

[tex]200000=P(1+0.025)^{20}(1+0.12)^5[/tex]

[tex]200000=P(1.025)^{20}(1.12)^5[/tex]

[tex]\implies P=69,256.824\approx 69,256.82[/tex]

Option C is correct.

The correct option is C. P69,256.82. The father should invest approximately P69,256.82 today to ensure his son receives P200,000 in ten years.

To determine the amount the father needs to invest today to give his son P200,000 ten years from now, we will break the problem into two phases due to different interest rates and compounding periods.

Phase 1: First Five Years (10% compounded quarterly)

→ Future Value (FV) needed after 10 years: P200,000

→ Future Value (FV) after first five years at 12% annual interest for the next five years:

Let's use the formula for compound interest to calculate amount required after the first five years.

Here,

→ n is the number of times the interest is compounded per year

→ t is time in years.

→ [tex]FV = PV*(1 + r/n)^{(nt)[/tex]

After the first five years, the amount needs to grow at 12% compounded annually for 5 years to reach P200,000.

We can calculate the present value (PV) at the end of the first five years needed to achieve P200,000 after next 5 years.

→ P200,000 = [tex]PV * (1 + 0.12/1)^{(1*5)[/tex]

→ P200,000 = [tex]PV * (1.12)^5[/tex]

Calculating PV:

→ PV = P200,000 / (1.7623)

        ≈ P113,477.57

Phase 2: First Five Years Investment Calculation

Now, we need to find out the amount the father should invest today to reach P113,477.57 after five years with 10% interest compounded quarterly.

Here,

→ r is the quarterly rate,

→ nt is the total number of quarters.

We use the same compound interest formula:

→ [tex]FV = PV * (1 + r/n)^{(nt)[/tex]

→ P113,477.57 = [tex]PV * (1 + 0.10/4)^{(4*5)[/tex]

→ P113,477.57 = [tex]PV * (1.025)^{20[/tex]

→ [tex](1.025)^{20} \approx 1.6386[/tex]

Calculating PV:

→ PV = P113,477.57 / 1.6386

        ≈ P69,256.82

So, the father needs to invest approximately P69,256.82 today.