Find an equation for the line that passes through the points (-6, -1) and (4, 5)

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:

Step-by-step explanation:

The x-intercept exists when y = 0; the coordinate takes on the form (    , 0)

The y-intercept exists when x = 0; the coordinate takes on the form (0,   ).

We will fill in the blanks in a sec.

If the x-intercept exists when y = 0, then we fill in 0 for y and solve for x:

-3(0) - x = -9 so

-x = -9 and

x = 9

The coordinate is (9, 0) for the x-intercept.

If the y-intercept exists when x = 0, then we fill in 0 for x and solve for y:

-3y - 0 = -9 so

-3y = -9 and

y = 3

The coordinate is (0, 3) for the y-intercept.

Answer:

[tex]L\left(te^{2t }sin3t\right)=\frac{6s-12}{(s^2-4s+13)^2}[/tex].

Step-by-step explanation:

If F(s)= L{f(t)}

Then [tex]L\left\{(t^nf(t)\right\}=(-1)^n\frac{\mathrm{d^n}F(s)}{\mathrm{d^n}s}[/tex]

[tex]L\left\{te^{2t}sin3t\right\}[/tex]

f(t)=[tex]e^{2t}sin3t[/tex]

[tex]L\left\{e^{at}sinbt\right\}=\frac{b}{(s-a)^2+b^2}[/tex]

Therefore,[tex] L\left\{e^{2t}sin3t\right\}=\frac{3}{(s-2)^2+(3)^2}[/tex]

[tex]L\left\{e^{2t}sin3t\right\}=\frac{3}{s^2-4s+13}[/tex]

[tex]L\left\{te^{2t}sin3t\right}=-\frac{\mathrm{d}F(s)}{\mathrm{d}s}[/tex]

=-[tex]\frac{\mathrm{d}e^{2t}sin3t}{\mathrm{d}s}[/tex]

[tex]L\left\{te^{2t}sin3t\right\}[/tex]

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

[tex]L\left\{te^{2t}sin3t\right\}=\frac{6s-12}{(s^2-4s+13)^2}[/tex].

The Laplace transform of te2tsin(3t) is 3(-1)(1)/(s2+9)2.

According to Theorem 7.4.1, the Laplace transform of a function tnf(t) can be calculated as (−1)n * dn/dsn * F(s), where F(s) is the Laplace transform of f(t). In this case, we need to evaluate the Laplace transform of te2tsin(3t).

First, we need to find the Laplace transform of te2t. Applying the formula F(s) = ℒ{f(t)} = ℒ{te2t}:

F(s) = -d/ds[2/(s-2)2]

Next, we need to find the Laplace transform of sin(3t). Applying the formula F(s) = ℒ{f(t)} = ℒ{sin(3t)}:

F(s) = 3/(s2+9)

Finally, we can evaluate the Laplace transform of te2tsin(3t) as the product of the Laplace transforms obtained:

F(s) = (-1)*(d1/d1s)*(-d1/d1s)[3/(s2+9)]

F(s) = 3(-1)(1)/(s2+9)2

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Answer:

314.16 in²

Step-by-step explanation:

If the diameter is 20", then the radius is 10" (1/2 the diameter).

Area of a circle is [tex]\pi r[/tex]²

So, π 10² = 100π = 314.16 in²

Answer:

  16.053 inches

Step-by-step explanation:

Many such probability questions are easily answered by a suitable calculator or spreadsheet.

Answer:$500

Step-by-step explanation:

Answer:

Step-by-step explanation:

Infinitely many.

Infinitely many.

A non-homogeneous system of linear equations can have a unique solution, no solution, or infinitely many solutions and a homogeneous system always has at least the trivial solution, and may have infinitely many non-trivial solutions depending on the rank of A.

A non-homogeneous system of linear equations (where the vector b on the RHS is non-zero) can have three types of solutions:

In contrast, a homogeneous system of linear equations (where the vector b on the RHS is zero) always has at least the trivial solution, where all variables equal zero. Depending on the rank of matrix A:

These rules are concisely summarized by the Rouche-Capelli theorem.

Answer:

10% of the school are blonde haired boys.

Step-by-step explanation:

Let x be the total number of students,

Number of students who have blonde hair = 40% of the total students

= 40% of x

= 0.4x

Also, the ratio of three times as many girls as boys have blond hair,

So, the ratio of blond hair boys and blond hair girls = 1 : 3

Let the number of boys who have blonde hair = y

And, the number of girls who have blonde hair = 3y

So, y + 3y = 0.4x

4y = 0.4x ⇒ y = 0.1x

Thus, the number of boys who have blonde hair = 0.1x

Hence, the percentage of the school are blonde haired boys

= [tex]\frac{0.1x}{x}\times 100=10\%[/tex]

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:

650 milliliters.

Step-by-step explanation:

Amount in the Jug = 1.5 * 1000 = 1,500 mls.

She pours out 250 mls and 0.6 * 1000 = 600 mls

The amount left = 1500 - 250 - 600

= 650 milliliters.

The amount of chocolate milk left in the jug is; D: 650 milliliters

We are told;

Amount of chocolate milk in the jug = 1.5 litres

She fills one cup with 250 mL = 0.25 liters

She fills another cup with 0.6 liters

Thus;

Total amount used = 0.6 +  0.25 = 0.85 litres

Amount left = 1.5 - 0.85 = 0.65 Liters

Converting to mL gives; 650 milliliters

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Answer:

M = ¼ k π R⁴

zG = 8/15 R

Step-by-step explanation:

Note: I'm using lower case r as the radius of each plate and upper case R as the radius of the hemisphere.

The mass of each plate is density times volume:

dm = ρ dV

Each plate has a radius r and a thickness dz.  So the volume of each plate is:

dV = π r² dz

Substituting:

dm = ρ π r² dz

We're told that ρ = kz.  Substituting:

dm = kz π r² dz

Next, we need to write the radius r in terms of the height z.  To do that, we need to look at the cross section (see image below).

The height z and the radius r form a right triangle, where the hypotenuse is the radius of the hemisphere R.

Using Pythagorean theorem:

z² + r² = R²

r² = R² − z²

Substituting:

dm = kπ z (R² − z²) dz

We now have the mass of each plate as a function of its height.  To find the total mass, we integrate between z=0 and z=R.

M = ∫ dm

M = ∫₀ᴿ  kπ z (R² − z²) dz

M = kπ ∫₀ᴿ (R² z − z³) dz

M = kπ (½ R² z² − ¼ z⁴) |₀ᴿ

M = kπ (½ R⁴ − ¼ R⁴)

M = ¼ k π R⁴

Next, to find the center of gravity, we use the weighted average:

zG = (∫ z dm) / (∫ dm)

zG = (∫ z dm) / M

We already found M, we just have to evaluate the other integral:

∫ z dm

∫₀ᴿ kπ z² (R² − z²) dz

kπ ∫₀ᴿ (R² z² − z⁴) dz

kπ (⅓ R² z³ − ⅕ z⁵) |₀ᴿ

kπ (⅓ R⁵ − ⅕ R⁵)

²/₁₅ k π R⁵

Plugging in:

zG = (²/₁₅ k π R⁵) / (¼ k π R⁴)

zG = ⁸/₁₅ R

The distribution of X is a normal distribution with a mean of 66 inches and a standard deviation of 2.5 inches. To find the probability that a person's height is between 64 and 69 inches, we need to determine the z-scores for each height value and use a z-table or calculator to find the probabilities.

A) Give the distribution of X.
X ~ Normal(66, 2.5)
B) Find the probability that the person is between 64 and 69 inches.
P(64 ≤ X ≤ 69)

To find this probability, we need to calculate the z-score for each height value using the formula z = (X - μ) / σ. Then, we can use a z-table or a calculator to find the corresponding probabilities.

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The height X of Asian adult males follows a normal distribution with parameters X ~ N(66, 2.5). The probability that one randomly chosen male is between 64 and 69 inches tall can be found using Z-scores and the area under the normal curve.

The distribution of the height X for Asian adult males according to the given study is X ~ N(66, 2.5), where 66 is the mean height in inches and 2.5 is the standard deviation.

Probability that the height is between 64 and 69 inches

To find the probability that the height of a randomly chosen Asian adult male is between 64 and 69 inches, we can use the properties of the normal distribution. We first convert the raw scores to Z-scores using the formula Z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.

The probability statement is P(64 < X < 69). To calculate this probability, we find the Z-scores for 64 and 69 inches and then find the area under the normal curve between these two Z-scores using a standard normal distribution table or a calculator with normal distribution functions.

Answer:

HCF(91,39) = 13 and HCF(73,21) = 1

Step-by-step explanation:

As per euclidian algorithm, a = bq + r, where a is dividend, b is divisor, q is quotient and r is remainder.

We can use euclidian algorithm to find the HCF of numbers.

To find: HCF ( 91, 39 ):

On dividing 91 by 39, we get

91=39×2+13

Here, remainder = 13 [tex]\neq 0[/tex]

So, again applying division algorithm on 39 and 13, we get

[tex]39=13\times 3+0[/tex]

As remainder = 0 and divisor at this step is equal to 13, HCF = 13 .

To find: HCF ( 73, 21 )

On dividing 73 by 21, we get

[tex]73=21\times 3+10[/tex]

Here, remainder = 10 [tex]\neq 0[/tex]

On applying division algorithm on 21 and 10, we get

[tex]21=10\times 2+1[/tex]

Here, remainder = 1 [tex]\neq 0[/tex]

On applying division algorithm on 10 and 1, we get

[tex]10=1\times 10+0[/tex]

As remainder = 0 and divisor at this step is 1, HCF = 1

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:

The height of tower is 322.7 feet.

Step-by-step explanation:

Given

Distance between a building and tower= 250 feet

BCDE is a rectangle .Therefore, we have  BC=ED and CD=BE=250 feet

In triangle ABE

[tex]tan\theta=\frac{perpendicula \; side }{hypotenuse}[/tex]

[tex]\theta=38^{\circ}[/tex]

[tex] tan38^{\circ}=\frac{AB}{BE}[/tex]

[tex]\frac{AB}{250}=0.781[/tex]

[tex] AB=0.781\times250[/tex]

AB=195.25 feet

In triangle EDC

[tex]\theta=27^{\circ}[/tex]

[tex]tan27^{\circ}=\frac{ED}{CD}[/tex]

[tex]\frac{ED}{250}=0.509[/tex]

[tex]ED=250\times0.509[/tex]

ED=127.25 feet

ED=BC=127.25 feet

The height of tower=AB+BC

The height of tower=195.25+127.25=322.5 feet

Final answer:

The repeating decimal 0.342 can be expressed as the fraction 38/111 in simplest form, by using algebraic methods to eliminate the repeating pattern and then simplifying the resulting fraction.

Explanation:

Expressing the repeating decimal 0.342 as a fraction involves using algebra to convert the decimal into a form that can be simplified into a fraction. First, we set the repeating decimal equal to a variable:

Let x = 0.342

To remove the repeating part, we multiply x by a power of 10 that corresponds to the number of digits in the repeating sequence. Since there are three digits repeating (342), we multiply by 1000:

1000x = 342.342

We now subtract the original equation (1x) from this new equation (1000x) to get rid of the decimal:

1000x - x = 342.342 - 0.342

999x = 342

Now we simplify by dividing both sides by 999:

x = 342 / 999

This fraction can be further simplified by dividing both the numerator and denominator by their greatest common divisor:

x = 38 / 111. Therefore, the fractional form of 0.342 is 38/111 in its simplest form.

To convert the repeating decimal 0.342 (with 342 repeating indefinitely) to a fraction, let's follow these steps:
1. Let x equal the repeating decimal:
  \( x = 0.342342342... \)
2. Recognize the pattern: the digits 342 repeat every three places. To isolate the repeating sequence, we need to multiply x by a power of 10 that has the same number of digits as the repeating sequence.
3. The repeating sequence has three digits, so we'll multiply x by \( 10^3 = 1000 \):
  \( 1000x = 342.342342... \)
4. Now, subtract the original number x from this new number to remove the repeating part:
  \( 1000x - x = 342.342342... - 0.342342342... \)
5. Notice that on the right-hand side, the repeating decimals cancel each other out, leaving whole numbers:
  \( 999x = 342.000000... \)
6. Now solve for x:
  \( x = \frac{342}{999} \)
7. To simplify the fraction, we find the greatest common divisor (GCD) of the numerator (342) and the denominator (999).
To find the GCD, we can list the factors of each:
Factors of 342 include: \( 1, 2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342 \).
Factors of 999 include: \( 1, 3, 9, 27, 37, 111, 333, 999 \).
The largest factor that appears in both lists is 9.
8. Divide both the numerator and the denominator by their GCD:
  \( x = \frac{342 ÷ 9}{999 ÷ 9} \)
9. Doing the division:
  \( x = \frac{38}{111} \)
Thus, the repeating decimal 0.342 (with a line over the entire decimal) can be expressed as the fraction \( \frac{38}{111} \) in its simplest form.

Answer: 1.49

Step-by-step explanation:

We know that the standard deviation in a binomial distribution is given by :-

[tex]\sigma=\sqrt{n\cdot p\cdot (1-p)}[/tex], where n is the number of trials and p is the probability of success.

Given : The number of trials : [tex]n=10[/tex]

The probability of success = [tex]p=\dfrac{1}{3}[/tex]

Then , the standard deviation will be :-

[tex]\sigma=\sqrt{10\cdot \dfrac{1}{3}\cdot (1-\dfrac{1}{3})}\\\\\Rightarrow\ \sigma=1.490711985\approx1.49[/tex]

The standard deviation of a given binomial distribution with 10 trials and a probability of success of 1/3 is calculated using the formula σ = √npq, resulting in approximately 1.49. This value helps us understand the spread of successes in the experiment.

To compute the standard deviation of a given binomial distribution with 10 trials and a probability of success p=1/3, we first identify the probability of failure, which is q=1-p=2/3. The formula for the standard deviation (σ) of a binomial distribution is σ = √npq, where n is the number of trials, p is the probability of success, and q is the probability of failure.

By substituting the given values into the formula:

We get σ = √(10)(1/3)(2/3) = √(20/9) ≈ 1.49. Therefore, the standard deviation of the binomial distribution is approximately 1.49.

This calculation helps us understand the dispersion or spread of the distribution of successes in our binomial experiment, indicating how far the number of successes can deviate from the expected value on average.

Answer: (C) 0.9104

Step-by-step explanation:

Given : The hours studied follows the normal distribution

Mean : [tex]\mu=\text{20 hours}[/tex]

Standard deviation : [tex]\sigma=6\text{ hours}[/tex]

Sample size : [tex]n=144[/tex]

The formula to calculate the z-score is given by :-

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

Let x be the number hours taken by randomly selected undergraduated student.

Then for x = 19.25 , we have

[tex]z=\dfrac{19.25-20}{\dfrac{6}{\sqrt{144}}}=-1.5[/tex]

for x = 21.0  , we have

[tex]z=\dfrac{20-21}{\dfrac{6}{\sqrt{144}}}=2[/tex]

The p-value : [tex]P(19.25<x<21)=P(-1.5<z<2)[/tex]

[tex]P(2)-P(-1.5)= 0.9772498- 0.0668072=0.9104426\approx0.9104[/tex]

Thus, the probability that the mean of this sample is between 19.25 hours and 21.0 hours = 0.9104.

Answer: [tex]\$\ 3600[/tex]

Step-by-step explanation:

Given : A bank charges a 1% fee to process a credit card cash advance which is taken out of the cash advance amount.

The rate of fee can be also written as 0.01.

The amount of money a customer wants as a cash advance = $4,000

The amount of bank fee = [tex]0.01\times4000=40[/tex]

The amount which customer receive after cash advance has been processed = [tex]4000-40=\$3600[/tex]

Hence, the amount of money will he receive from the bank after the cash advance has been processed = [tex]\$\ 3600[/tex]

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]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.

Answer:

1,460 hours per year.

113,880 hours in a lifetime.

Step-by-step explanation:

Great question, it is always good to ask away and get rid of any doubts that you may be having.

Let's assume that on an average day Americans drive 4 hours a day. This includes going and coming from work as well as any groceries they might need to do. Since there are 365 days in a year we can multiply this by the hours a day driven.

[tex]365*4 = 1,460[/tex]

We can "guesstimate" that on average an American drives 1,460 hours per year.

Based on my research the average lifetime of an American is 78 years old. We can now multiply this by the number of hours driven in a year to have an idea of the number of hours driven in a lifetime.

[tex]1,460*78 = 113,880[/tex]

On average we "guesstimate" that an American drives 113,880 hours in a lifetime.

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Answer:

TT means pi and e means Euler

Step-by-step explanation:

Answer:

The given function is

  f(x)=cos 3x-7 x²+ 4x

f'(x)=-3 sin 3 x-14 x+4

When you will draw the graph of the function , you will find that root of the function lie between (-1,0).

Consider initial root as,

 [tex]x_{0}=0[/tex]

Using Newton method to find the roots of the equation

 [tex]x_{n+1}=x_{n}-\frac{f{x_n}}{f'{x_{n}}}\\\\x_{1}=x_{0} - \frac{cos 3x_{0}-7 x_{0}^2+ 4x_{0}}{-3 sin 3 x_{0}-14 x_{0}+4}\\\\x_{1}=-\frac{\cos 0^{\circ}-0+0}{-3 \times 0-0+4}\\\\x_{1}=\frac{-1}{4}\\\\x_{1}= -0.25\\\\x_{2}=x_{1} - \frac{cos 3x_{1}-7 x_{1}^2+ 4x_{1}}{-3 sin 3 x_{1}-14 x_{1}+4}\\\\x_{2}=-0.25 -\frac{cos (-0.75)-7\times (0.0625)- 1}{-3 sin (-0.75)+3.50+4}\\\\x_{2}= -0.176054[/tex]

[tex]x_{3}=x_{2} - \frac{cos 3x_{2}-7 x_{2}^2+ 4x_{2}}{-3 sin 3 x_{2}-14 x_{2}+4}\\\\x_{3}=-0.176054 -\frac{cos (3\times -0.176054)-7\times (-0.176054)^2+4 \times -0.176054}{-3 sin (-0.176054)-14 \times (-0.176054)+4}\\\\x_{3}= -0.1689[/tex]

So, root of the equation is

         =0.1688878

       =0.1689(approx)

Answer:

none

Step-by-step explanation:

We have three sides

Adding the two smallest sides together must be bigger than the third side

3.1+5.2 > 9.8

8.3> 9.8

This is false, so we cannot make a triangle

Answer:

[tex]\boxed{\text{(a)}A = 100 - 99e^{-t/25}; \,\text{(b) 15 s}}[/tex]

Step-by-step explanation:

(a) Expression for mass of salt as function of time

[tex]\text{Let A = mass of salt after t min}\\\text{and }r_{i} = \text{rate of salt coming into talk}\\\text{and }r_{o}$ =\text{rate of salt going out of tank}[/tex]

i. Set up an expression for the rate of change of salt concentration.

[tex]\dfrac{\text{d}A}{\text{d}t} = r_{i} - r_{o}\\\\r_{i} = \dfrac{\text{4 gal}}{\text{1 min}} \times \dfrac{\text{1 lb}}{\text{1 gal}} = \text{4 lb/min}\\\\r_{o} = \dfrac{\text{4 gal}}{\text{1 min}} \times \dfrac {A\text{ lb}}{\text{100 gal}} =\dfrac{A}{25}\text{ lb/min}\\\\\dfrac{\text{d}A}{\text{d}t} = 4 - \dfrac{x}{25}[/tex]

ii. Integrate the expression

[tex]\dfrac{\text{d}A}{\text{d}t} = \dfrac{100 - A}{25}\\\\\dfrac{\text{d}A}{100 - A} = \dfrac{\text{d}t}{25}\\\\\int \frac{\text{d}A}{100 - A} = \int \frac{\text{d}t}{25}\\\\-\ln(100 - A) = \dfrac{t}{25} + C[/tex]

iii. Find the constant of integration

[tex]-\ln (100 - A) = \dfrac{t}{25} + C\\\\\text{At $t$ = 0, $A$ = 1, so}\\\\-\ln (100 - 1) = \dfrac{0}{25} + C\\\\C = -\ln 99[/tex]

iv. Solve for A as a function of time.

[tex]\text{The integrated rate expression is}\\\\-\ln (100-A) = \dfrac{t}{25} - \ln 99\\\\\text{Solve for } A\\\\\ln(100 - A) = \ln 99 - \dfrac{t}{25}\\\\100 - A = 99e^{-t/25}\\\\A = \boxed{\mathbf{100 - 99e^{-t/25}}}[/tex]

The diagram shows A as a function of time. The mass of salt in the tank starts at 1 lb and increases asymptotically to 100 lb.

(b) Time to 2 lb salt

[tex]A = 100 - 99e^{-t/25}\\\\2 = 100 - 99e^{-t/25}\\\\99e^{-t/25} = 98\\\\e^{-t/25} = 0.989899\\\\-t/25 = -0.01015\\\\t = 25\times 0.01015 =\text{0.25 min = 15 s}\\\\\text{The tank will contain 2 lb of salt after } \boxed{\textbf{15 s}}[/tex]

To find the amount of salt in the tank at any time t, we need to consider the rate at which the brine solution is being poured in and leaving the tank. Initially, there is 1 pound of salt in the tank. The rate at which salt is added is 1 pound per gallon, and the rate at which the mixture is being removed is 4 gallons per minute. Therefore, the amount of salt in the tank at time t can be calculated using the equation. To find the time at which the mixture in the tank contains 2 pounds of salt, we can set up the equation. Solving this equation for t will give us the time at which the mixture contains 2 pounds of salt.

To find the amount of salt in the tank at any time t, we need to consider the rate at which the brine solution is being poured in and leaving the tank. Initially, there is 1 pound of salt in the tank. The rate at which salt is added is 1 pound per gallon, and the rate at which the mixture is being removed is 4 gallons per minute. Therefore, the amount of salt in the tank at time t can be calculated using the equation:

Amount of salt at time t = 1 + (1 pound/gallon)(4 gallons/minute)(t minutes)

To find the time at which the mixture in the tank contains 2 pounds of salt, we can set up the equation:

2 = 1 + (1 pound/gallon)(4 gallons/minute)(t minutes)

Solving this equation for t will give us the time at which the mixture contains 2 pounds of salt.

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Answer:

open fb and search ravi verma

Step-by-step explanation:

send friend request

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:

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.