At noon, ship A is 30 nautical miles due west of ship B. Ship A is sailing west at 20 knots and ship B is sailing north at 21 knots. How fast (in knots) is the distance between the ships changing at 4 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)

Answer:[tex]-8\hat{i}-6\hat{k}[/tex]

[tex]\theta =\tan^{-1}\left ( \frac{3}{4} \right )[/tex]

Step-by-step explanation:

Given

Velocity of hailstones fall[tex]\left ( V_h\right )=2\hat{i}-6\hat{k}[/tex] m/s

Velocity of cyclist [tex]\left ( V_c\right )=10\hat{i}[/tex] m/s

Therefore

Velocity of hail with respect to cyclist[tex]\left ( V_{hc}\right )[/tex]

[tex]V_{hc}=V_h-V_c[/tex]

[tex]V_{hc}=2\hat{i}-6\hat{k}-10\hat{i}[/tex]

[tex]V_{hc}=-8\hat{i}-6\hat{k}[/tex]

and angle of hails falling relative to the cyclist is given by

[tex]\theta =\tan^{-1}\left ( \frac{3}{4}\right )[/tex]

[tex]\theta [/tex] is the angle made with the vertical

The answer is:

Center: (1,-2)

Radius: 4 units.

To solve the problem, using the given formula of a circle, we need to find its standard equation form which is equal to:

[tex](x-h)^{2}+(y-k)^{2}=r^{2}[/tex]

Where,

"h" and "k"are the coordinates of the center of the circle and "r" is its radius.

So, we need to complete the square for both variable "x" and "y".

The given equation is:

[tex]x^2+y^2-2x+4y-11=0[/tex]

So, solving we have:

[tex]x^2+y^2-2x+4y=11[/tex]

[tex](x^2-2x+(\frac{2}{2})^{2} )+(y^2+4y+(\frac{4}{2})^{2})=11+(\frac{2}{2})^{2} +(\frac{4}{2})^{2}\\\\(x^2-2x+1)+(y^2+4y+4)=11+1+4\\\\(x^2-1)+(y^2+2)=16[/tex]

[tex](x^2-1)+(y^2-(-2))=16[/tex]

Now, we have that:

[tex]h=1\\k=-2\\r=\sqrt{16}=4[/tex]

So,

Center: (1,-2)

Radius: 4 units.

Have a nice day!

Note: I have attached a picture for better understanding.

Answer:

The correct option is 4.

Step-by-step explanation:

It is given that 10 men and 12 women will be seated in a row of 22 chairs.

Total possible ways to arrange n terms is n!.

Similarly,

Total possible ways to place 22 people on 22 chairs = 22!

[tex]\text{Total outcomes}=22![/tex]

It is given that all men will be seated side by side in 10 consecutive positions.

Total possible ways to place 10 people on 10 chairs = 10!

Let 10 men = 1 unit because all men will be seated side by side in 10 consecutive positions. 12 women = 12 units because women can any where.

Total number of units = 12 + 1 = 13.

Total possible ways to place 13 units = 13!

Total possible ways to place 10 men and 12 women, when all men will be seated side by side in 10 consecutive positions is

[tex]\text{Favorable outcomes}=10!\cdot 13![/tex]

The probability that all men will be seated side by side in 10 consecutive positions

[tex]P=\frac{\text{Favorable outcomes}}{\text{Total outcomes}}=\frac{10!\cdot 13!}{22!}[/tex]

Therefore the correct option is 4.

Answer:

See below.

Step-by-step explanation:

This will be a parabola with axis of symmetry x = 0 and will open downwards.

The vertex will be at the point (0 , 5). The graph will intersect the x axis  at

(-√5, 0) and (√5, 0).

Answer:

its a

Step-by-step explanation:

Answer:

5005

Step-by-step explanation:

In the question we have to form a committee of six from a club of consisting of 15 people.

This a simple case of selection of six people from a group of 15 people.

which can be done in

[tex]^{15}C_6= \frac{15!}{6!\times9!}=5005[/tex]

hence, the number of ways of forming committees of size six from a club of 15 members= 5005

Answer:

Not a probability distribution

Step-by-step explanation:

The given table doesn't describe a probability distribution as in order for the given distribution to be a probability distribution the sum of probabilities is required to be equal to one.

Here,

Sum of probabilities = 0.001+0.025+0.101+0.246+0.503 = 0.876

The sum of probabilities is not equal to one.

Therefore, the given distribution is not a probability distribution ..

Answer:

Here x represents the number of square feet to be refinished and y represents the cost of refinishing the floor,

Given,

The cost of a tile floor for up to 1000 square feet is $1.83 per square,

So, the cost of x square feet of tile = 1.83x for x ≤ 1000

⇒ y = 1.83x for x ≤ 1000

Since, there is an additional charge of $350 for toxic waste disposal for any job which includes more than 150 square feet of tile.

That is, y = 1.83x + 350, for x > 150

So, y must be 1.83x for x ≤ 150.

A) Hence, the function that express the cost, y, of refinishing a floor as a function of the number of square feet, x, to be refinished, is,

[tex]y=\begin{cases}1.83x & \text{ if } 0\leq x\leq 150 \\ 1.83x+350 & \text{ if } 150< x\leq 1000\end{cases}-----(1)[/tex]

B) The domain of the function =  all possible value of x

⇒ Domain = 0 ≤ x ≤ 1000

Range = All possible value of y,

Since, the range of function y=1.83x, 0≤ x ≤ 150 is [0, 274.5]

While the range of function y = 1.83x + 350, for x > 150 is (624.5, 2180]

Hence, the range of the function (1) = [0, 274.5]∪(624.5, 2180]

The cost of refinishing a floor can be expressed as a piecewise function based on the number of square feet to be refinished. The domain of the function is all real numbers, and the range is all real numbers greater than or equal to 0.

Let x represent the number of square feet to be refinished.

For x ≤ 150, the cost of refinishing a floor is simply $1.83 per square foot. So, the cost function, y, for x ≤ 150 is y = 1.83x.

For x > 150, there is an additional charge of $350 for toxic waste disposal. So, the cost function, y, for x > 150 is y = 350 + 1.83x.

The overall cost function, y, is given by:

y = 1.83x, for x ≤ 150

y = 350 + 1.83x, for x > 150

The domain of the function is all real numbers, since any positive number of square feet can be refinished. The range of the function is all real numbers greater than or equal to 0, since the cost cannot be negative.

Answer:

  (a) 120 square units (underestimate)

  (b) 248 square units

Step-by-step explanation:

(a) left sum

See the attachment for a diagram of the areas being summed (in orange). This is the sum of the first 4 table values for f(x), each multiplied by 2 (the width of the rectangle). Quite clearly, the curve is above the rectangle for the entire interval, so the rectangle area underestimates the area under the curve.

  left sum = 2(1 + 5 + 17 + 37) = 2(60) = 120 . . . . square units

(b) right sum

The right sum is the sum of the last 4 table values for f(x), each multiplied by 2 (the width of the rectangle). This sum is ...

  right sum = 2(5 +17 + 37 +65) = 2(124) = 248 . . . . square units

Answer:

q(r(4)) = -61

Step-by-step explanation:

q(x) = -2x +1

r(x) = 2x^2 - 1

q(r(4))

First find r(4)

f(4) = 2 (4)^2 -1

      = 2 *16 -1

      = 32-1

     = 31

Then put this value in for x in q(x)

q(r(4)) = q(31) = -2(31)+1

                     = -62+1

                     = -61

Answer:

The value of q( r(4) ) = -61

Step-by-step explanation:

It is given that,

q(x) = - 2x +1

r(x) = 2x^2 - 1

To find the value of q(r(4))

r(x) = 2x^2 - 1

r(4) = 2( 4^2) - 1  [Substitute 4 instead of x]

 = 2(16) - 1

 = 32 - 1 = 31

q( x ) = -2x +1

q( r(4) ) = q(31)        [Substitute 31 instead of x)

 =  (-2*31) +1

 = -62 + 1 = -61

Therefore the  value of q(r(4)) = -61

Answer:

3 students

Step-by-step explanation:

If everyone in the class has either a pierced nose or ear, we just simply have to add up the total number of hands raised and minus the number of students in the class.

35+8=43

43-40=3

3 students have both a pierced nose and pierced ear.

Answer:

A) w > -7

B) 1.5 < t

C) -3.5 makes A) true and B) false

    10 makes both inequalities true

Step-by-step explanation:

The idea of these exercises is to clear our variable, we need it to be alone on one side of the inequality

A) 2w + 17 ˃ -4w -25

First, we will put together on one side the terms with a w and on the other the terms without w.

For that, we have to add 4w - 17 on both sides

2w + 17 + 4w - 17 ˃ -4w -25 + 4w - 17 (Notice that 17-17=0 and -4w+4w=0, so we don't have to write them below)

2w + 4w > -25 - 17

Now we can sum the terms (we didn't do it before because we can't sum a term with a w with one without it)

6w > -42

We divide by 6 on both sides and we have

6/6w > -42/6

w > -7

B) 2.3 + 0.6t ˂ 2 + 0.8t

We start as before; in this case we have to put together the terms with a t (our variable changes name but the idea is the same)

We will add -2 - 0.6t on both sides

2.3 + 0.6t -2 - 0.6t ˂ 2 + 0.8t -2 - 0.6t  

2.3 - 2 < 0.8t - 0.6t

Now we sum the terms  

0.3 < 0.2t

We divide by 0.2 on both sides and we have

0.3/0.2 < 0.2/0.2t

1.5 < t

C) Let's check -3.5 on both inequalities:

We have to replace the variable by -3.5:

2*(-3.5) + 17 ˃ -4*(-3.5) -25 (remember that if there is no sign between a number and a variable, it means that is a multiplication)

Now we just solve the calculation

-7 + 17 > 14 -25

10 > -11

That's true, so -3.5 makes the inequality true.

Now, in the other inequality, we replace the t by -3.5 and solve as before

2.3 + 0.6*(-3.5) ˂ 2 + 0.8*(-3.5)

2.3 - 2.1 < 2 - 2.8

0.2 < -0.8

That's false because we are saying that a negative number is bigger than a positive one, so -3.5 makes the inequality not true.

  Now we do the same with 10 in both inequalities:

   2*10 + 17 ˃ -4*10 -25

   20 + 17 > -40 -25

   37 > - 65

   It's true!

   2.3 + 0.6*10 ˂ 2 + 0.8*10

   2.3 + 6 < 2 + 8

   8.3 < 10

   It's true!

Answer:

Step-by-step explanation:

1. Greater density means the sphere has more mass in the same volume. The volume of water that must be displaced to equal that increased mass must be increased, causing the water level to rise.

__

2. Less mass means less water must be displaced to equal the mass of the new sphere, causing the water level to fall.

__

3. More mass is the same as higher density (see 1). The water level will rise.

Archimedes' principle states that the upward force acting on a body floating or immersed in a fluid is equal to the weight of the displaced fluid

The level of the water in the three situations are as follows;

Situation 1; Falls or stays the same, E: f or s

Situation 2; Falls, B: f

Situation 3, Rises A: r

The reason for the above selection is as follows;

The given details of the arrangements are;

The mass of the solid homogeneous sphere = m₀

The radius of the sphere = r₀

The density of the sphere = ρ₀

The location the sphere is placed = Floating in a container of water

The required parameter;

The provision of an estimate of the water level when the sphere is replaced with a new sphere with different physical parameters

Notation;

r = The water level rises

f = The water level falls

s = The water level stays the same

Situation 1; The mass of the new sphere, m = m₀

The density of the new sphere, ρ > ρ₀

Here, the denser sphere of equal mass = Smaller sphere, r < r₀

if the sphere floats, then the volume of the water displaced is equal to the

mass of the sphere, which is therefore, equal to the volume of the water

displaced by the original sphere

Therefore, the water level remains the same, s

However, if the sphere sinks, then the water displaced is less than the

mass m = m₀, of the sphere and therefore, the level falls, f

Therefore, the correct option is E: f or s

Situation 2: The mass of the new sphere, m < m₀

The radius of the new sphere, r = r₀

Here, we have equal radius and therefore equal volume and lesser density

Given that the volume of the water displaced for a floating body is equal to

the weight of body, and that the mass of the new sphere is less than the

mass of the original sphere, the mass of the water displaced and therefore,

the volume of water displaced is less and therefore, the water level falls

The correct option is therefore B: f falls

Situation 3: The mass of the new sphere, m > m₀, and the radius r = r₀

therefore the new sphere is denser than the original sphere and the

therefore, the mass of the water displaced where the sphere floats is m >

m₀, which is more than the water displaced for the original sphere and the

level of water rises, r, and the correct option is A: r

Therefore;

In situation 1, we have option E: f or s

In situation 2, the correct option is B: f

In situation 3, the correct option is A: r

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The probability of guessing the correct seven-digit pin code on the first try is very low, approximately 0.000024%.

Given that,

A thief steals an ATM card.

The thief must guess the correct seven-digit pin code.

The pin code is entered using a 4-key keypad.

The probability of guessing the correct seven-digit pin code on the first try depends on a few factors.

To break it down,

if the thief has a 4-key keypad and repetition of digits is allowed, that means there are four options for each digit.

So, there are a total of 4⁷ (4 raised to the power of 7) possible combinations.

Since the thief is trying to guess the correct pin code on the first try, there is only one correct combination out of the total possible combinations.

Therefore,

The probability of guessing the correct pin code on the first try would be 1 out of 4⁷, or approximately 0.00000024, or 0.000024%.

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The probability of randomly guessing a 7-digit PIN from a 4-key keypad is 1 in 16,384. This equals approximately 0.000061 or 0.0061%. Each digit has 4 possible options, and there are 7 digits in total.

The probability of guessing a seven-digit PIN code correctly from a 4-key keypad (where repetition of digits is allowed) can be calculated as follows:

Since each of the 7 digits in the PIN can be any of 4 possible digits (0 through 3), the total number of possible combinations is calculated by raising the number of choices per digit to the power of the number of digits:

→ Total possible combinations = 4^7

= 4⁷

= 16384

Therefore, the probability of guessing the correct PIN on the first try is the reciprocal of the total number of possible combinations:

→ Probability of a correct guess = 1 / 16384

Hence, the probability is approximately 0.000061 or 0.0061%.

Answer:

Step-by-step explanation:

First, finding the horizontal asymptote:

[tex]\lim_{t \to \infty} = \frac{200+40t}{1+0.05t} = \frac{\frac{200}{t} 40 }{\frac{1}{t} 0.05} = 800[/tex]

In the context of the problem, the horizontal asymptote speaks about where the population of the fish is headed and capped.

Answer:

last installment is $540

Step-by-step explanation:

principal amount (p) = $7500

rate (r) = 16.5 %

installment = $300

to find out

full payment is increased to pay off the loan and the last smaller payment is made one month after the last full payment

solution

we know monthly installment is $300 so amount will be paid i.e.

amount = $300×12×N ..............1

here N is no of installment

and we know amount formula i.e.

amount = principal ( 1+r/100)^N

put amount value and principal rate

300×12×N = 7500 ( 1+16.5/100)^N

(3600 ×N ) / 7500 = 1.165^N

0.48N = 1.165^N

by the graphical we will get N = 3.65

so 3.65 year

so as that put N in equation 1 we get

amount = $300×12× 3.65

amount = $13140

we can say there are 43 installment so remaining money is  $13140 - ($300 × 43 installment )

i.e. = $240 and last installment will be $300 + $240 = $540

so last installment is $540

Answer:

The correct answer is first option

24

Step-by-step explanation:

From the figure we get, mAXM = 72° and  m<AMR = 38°

Also it is given that, all triangles are isosceles triangles and

m<FXA = 96°

To find the measure of <FXM

From the figure we get,

m<FXA =  m<AXM + m<FXM

m<FXM = m<FXA - m<AXM

 = 96 - 72

 = 24

Therefore the  correct answer is first option

24

Answer:

The equation 2.5x -10.5 = 64(0.5^x) is true when x=5

Step-by-step explanation:

we need to solve the equation 2.5x -10.5 = 64(0.5^x)

We have to put the given values of x in the functions f(x) and g(x) and find their values

x     f(x) = 2.5x -10.5            g(x) = 64(0.5^x)

2     2.5(2)-10.5 = -5.5          64(0.5^2) = 16      

3     2.5(3) - 10.5 = -3           64(0.5^3) = 8

4     2.5(4) - 10.5 = -0.5        64(0.5^4) = 4

5     2.5(5) - 10.5 = 2            64(0.5^5) = 2

6     2.5(6) - 10.5 = 4.5         64(0.5^6) = 1

So, we need to solve the equation 2.5x -10.5 = 64(0.5^x)

This holds when x = 5 as shown in the table above.

I suspect there's a typo in the question, because [tex]y_1=x[/tex] is *not* a solution to the corresponding homogeneous equation. We have [tex]{y_1}'=1[/tex] and [tex]{y_1}''=0[/tex], so the ODE reduces to

[tex]0+7x+5x=12x\neq0[/tex]

Let [tex]y=x^m[/tex], then [tex]y'=mx^{m-1}[/tex] and [tex]y''=m(m-1)x^{m-2}[/tex], and substituting these into the (homogeneous) ODE gives

[tex]m(m-1)x^m+7mx^m+5x^m=0\implies m(m-1)+7m+5=m^2+6m+5=(m+5)(m+1)=0[/tex]

which then admits the characteristic solutions [tex]y_1=\dfrac1x[/tex] and [tex]y_2=\dfrac1{x^5}[/tex].

Now to find a solution to the non-homogeneous ODE. We look for a solution of the form [tex]y(x)=v(x)y_1(x)[/tex] or [tex]y(x)=v(x)y_2(x)[/tex].

It doesn't matter which one we start with, so let's use the first case. We get derivatives [tex]y'=x^{-1}v'-x^{-2}v[/tex] and [tex]y''=x^{-1}v''-2x^{-2}v'+2x^{-3}v[/tex]. Substituting into the ODE yields

[tex]x^2(x^{-1}v''-2x^{-2}v'+2x^{-3}v)+7x(x^{-1}v'-x^{-2}v)+5x^{-1}v=x[/tex]

[tex]xv''+5v'=x[/tex]

Substitute [tex]w=v'[/tex], so that [tex]w'=v''[/tex] and

[tex]xw'+5w=x[/tex]

which is linear in [tex]w[/tex], and we can condense the left side as the derivative of a product after multiplying both sides by [tex]x^4[/tex]:

[tex]x^5w'+5x^4=x^5\implies(x^5w)'=x^5\implies x^5w=\dfrac{x^6}6+C\implies w=\dfrac x6+\dfrac C{x^5}[/tex]

Integrate to solve for [tex]v[/tex]:

[tex]v=\dfrac{x^2}{12}+\dfrac{C_1}{x^4}+C_2[/tex]

Then multiply both sides by [tex]y_1=\dfrac1x[/tex] to solve for [tex]y[/tex]:

[tex]y=\dfrac x{12}+\dfrac{C_1}{x^5}+\dfrac{C_2}x[/tex]

so we found another fundamental solution [tex]y_3=x[/tex] that satisifes this ODE.

Answer:

C

Step-by-step explanation:

A. First two rectangles are dilations because

[tex]\dfrac{2}{4}=\dfrac{4}{8}=0.5[/tex]

B. Second two rectangles are dilations because

[tex]\dfrac{2}{4}=\dfrac{3}{6}=0.5[/tex]

C. Third two rectangles are not dilations because

[tex]\dfrac{3}{4}\neq \dfrac{2}{3}[/tex]

D. Fourth two rectangles are dilations because

[tex]\dfrac{3}{4}=\dfrac{1.5}{2}=0.75[/tex]

Answer:

c please correct me if im wrong

Step-by-step explanation:

Answer: 0.0775

Step-by-step explanation:

Given : Mean : [tex]\mu = 25[/tex]

Standard deviation : [tex]\sigma =15[/tex]

Sample size : [tex]n=36[/tex]

Since its normal distribution , then the formula to calculate the z-score is given by :-

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

For x= 28.2 hours

[tex]z=\dfrac{28.2-25}{\dfrac{15}{\sqrt{36}}}=1.28[/tex]

For x= 30 hours

[tex]z=\dfrac{30-25}{\dfrac{15}{\sqrt{36}}}=2[/tex]

The P- value = [tex]P(1.28<z<2)[/tex]

[tex]=P(z<2)-P(z<1.28)= 0.9772498-0.8997274=0.0775224\approx0.0775[/tex]

Hence, the probabiliy that the average time spent stydying for the sampe was between 28.2 and 30 hours = 0.0775

The probability that the average time spent studying for the sample was between 28.2 and 30 hours is calculated as 0.0775 or 75 %.

To calculate the probability that the average time spent studying for the sample was between 28.2 and 30 hours, we use the normal distribution and standardize the sample means to a z-score.

Given the population mean (μ) is 25 hours, the population standard deviation (σ) is 15 hours, and the sample size (n) is 36, the standard error of the mean (SEM) is σ/√n which is 15/6 = 2.5 hours.

The z-scores for 28.2 and 30 hours are calculated as (X - μ)/(SEM).

Z for 28.2 hours = (28.2 - 25)/2.5 = 1.28
Z for 30 hours = (30 - 25)/2.5 = 2

Now we can look up these z-scores in the standard normal distribution table (or use calculator/software) to find the probabilities for these z-scores and then find the probability that lies between them by subtracting the two.

Example: Let's assume the probability corresponding to z=1.28 is 0.8997 and to z=2 is 0.9772.

The probability that the sample mean lies between 28.2 and 30 hours is:

P(1.28 < Z < 2) = P(Z < 2) - P(Z < 1.28)

= 0.9772 - 0.8997
= 0.0775

Hence, there is a 7.75% probability that the sample mean is between 28.2 and 30 hours.

Audrey can visit the six cities in which she plans to conduct book signings and return to her starting point in 720 different ways. This is because of the mathematical principle of permutations.

Audrey's problem deals with permutations because the order of the places she visits matters. In general, the number of ways to arrange 'n' items (in Audrey's case, 'n' cities) in a specific order is given by 'n things taken n at a time' which is mathematically represented as n! (n factorial). In this case, Audrey is visiting 6 cities (Knoxville, Chattanooga, Chapel Hill, Charlotte, Raleigh, and Richmond), and then returning to her original city, Wilmington. So, the number of ways she can visit these cities can be represented as 6!, which equals 720.

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Answer:

The number of dosage is 0.13.

Step-by-step explanation:

Here, the given function that represents the blood pressure,

[tex]B(x)=0.06x^2 - 0.3x^3[/tex]

Where, x is the number of dosage in cubic centimeters​,

Differentiating the above function with respect to x,

[tex]B'(x)=0.12x-0.9x^2[/tex]

For maximum or minimum blood pressure,

[tex]B'(x)=0[/tex]

[tex]0.12x-0.9x^2=0[/tex]

[tex]-0.9x^2=-0.12x[/tex]

[tex]x=\frac{0.12}{0.9}=\frac{2}{15}[/tex]

Again differentiating B'(x) with respect to x,

[tex]B''(x)=0.12-1.8x[/tex]

Since, at x = 2/15,

[tex]B''(\frac{2}{15})=0.12-1.8(\frac{2}{15})=0.12-0.24=-0.12=\text{Negative value}[/tex]

So, at x = 2/15 the value of B(x) is maximum,

Hence, the number of dosage at which the resulting blood pressure is maximized = 2/15 = 0.133333333333 ≈ 0.13

The maximum blood pressure results from a dosage of approximately 0.13 cubic centimeters, based on the mathematical model given in the problem.

To find the maximum blood pressure using the formula B (x) = 0.06 x^2 - 0.3 x^3, we need to first find the derivative of this equation, as the maximum point on any curve happens when its derivative equals zero.

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Answer:

They should be able to sell their home for $149706.9

Step-by-step explanation:

Let's first understand the situation.

There is an initial value for the house which is $189000. However, this value varies every 6 years because of a 11% depreciation of the total value.

Because the depreciation is not executed during the 6 years in a constant way, but instead after the whole 6 years have passed, then we can calculate how many depreciations will be applied within the next 15 years:

total years/years needed for depreciation=15years/6years=2.5

The above means that only 2 depreciations are going to be applied. Remember that depreciation is only applied if the whole 6 years have passed.  

Now, after the first 6 years the depreciation (D) is:

D = 0.11 * $189000 = $20790,

which means that the value of the house will be:

(initial value) - D = $189000 - $20790 = $168210

Now, after the following 6 years, first 12 years, the depreciation (D) is:

D = 0.11 * $168210 = $18503.1,

which means that the value of the house will be:

(initial value) - D = $168210 - $18503.1 = $149706.9

In conclusion, in 15 years from now, they should be able to sell their home for $149706.9

Answer: (a) $264.375 ⇒ Amount of Interest

(b) Future Value = $3264.375

Step-by-step explanation:

(a) Principal amount = $3000

Time period = 9 months

Interest rate = 11.75%

Simple interest(SI) = principal amount × rate of interest (i) × time period

                              = 3000 × [tex]\frac{11.75}{100}[/tex] × [tex]\frac{9}{12}[/tex]

                              = 3000 × 0.1175 × 0.75

                               = $264.375 ⇒ Amount of Interest

(b) Future value of loan = principal amount + interest amount

                                        = 3000 + 264.375

                                        = 3264.375

Answer:

The expression [tex]m^3-n^3[/tex] is even if both variables (m and n) are even or both are odd

Step-by-step explanation:

Let's remember the logical operations with even and odd numbers

odd*odd=odd

even*even=even

odd*even=even

odd-odd=even

even-even=even

even-odd=odd

Now, the original expression is:

[tex]m^3-n^3[/tex] which can be expressed as:

[tex](m*(m*m))-(n*(n*n))[/tex]

If m and n are both odd, then:

[tex](m*(m*m))=odd*(odd*odd)=odd*(odd)=odd[/tex]

[tex](n*(n*n))=odd*(odd*odd)=odd*(odd)=odd[/tex]

Then, [tex](m*(m*m))-(n*(n*n))=odd-odd=even[/tex]

If m and n are both even, then:

[tex](m*(m*m))=even*(even*even)=odd*(even)=even[/tex]

[tex](m*(m*m))=even*(even*even)=odd*(even)=even[/tex]

Then, [tex](m*(m*m))-(n*(n*n))=even-even=even[/tex]

Finally if one of them is even, for example m, and the other is odd, for example n, then:

[tex](m*(m*m))=even*(even*even)=odd*(even)=even[/tex]

[tex](n*(n*n))=odd*(odd*odd)=odd*(odd)=odd[/tex]

Then, [tex](m*(m*m))-(n*(n*n))=even-odd=odd[/tex]

In conclusion, the expression [tex]m^3-n^3[/tex] is even if both variables (m and n) are even or both are odd. If one of them is even and the other one is odd, then the expression is odd.

To solve the problem, first calculate all possible combinations of selecting 3 apples from 11. Then calculate the favorable combinations, which include selecting 1 red apple (from 6 available) and 2 yellow apples (from 5 available). Divide these values to get the probability.

The topic at hand is one of probability, more specifically, it's a problem of combinations in probability. The bag contains a total of 11 apples (6 red and 5 yellow). When 3 apples are chosen, we want to find the probability that 1 is red and 2 are yellow.

First, calculate the total number of ways to choose 3 apples from 11, which is denoted as '11 choose 3', using combination formula C(n,r) = n! / [r!(n - r)!]. Then, consider the number of favorable outcomes: choosing 1 red apple from 6 (denoted as '6 choose 1') and 2 yellow apples from 5 (denoted as '5 choose 2'). Multiply these two results because we choose '1 red' and '2 yellow', using the rule of product. Calculate these individual results and then divide the favorable outcomes by the total outcomes to get the required probability.

https://brainly.com/question/32117953

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Answer:

P=0.3698  or 36.98%

Step-by-step explanation:

Complete the table by adding the totals to each column and row.

                                Pedestrian     Pedestrian

                                Intoxicated    Not intoxicated Totals

Driver Intoxicated           56                     71               127

Driver Not intoxicated   292                   522             814

Totals                            348                    593             941

The probability that the pedestrian was intoxicated or the driver was intoxicated is the opposite event of neither of them was intoxicated.  The total of cases when neither of them was intoxicated is 593. So the probability is:

P1=593/941=0.6302

The probability of the opposite event is one minus the probability calculated:

P=1-0.6302=0.3698

And this is the probability that the pedestrian was intoxicated or the driver was intoxicated.

a.

[tex]x^{11}=13\pmod{35}\implies\begin{cases}x^{11}\equiv13\equiv3\pmod5\\x^{11}\equiv13\equiv6\pmod7\end{cases}[/tex]

By Fermat's little theorem, we have

[tex]x^{11}\equiv (x^5)^2x\equiv x^3\equiv3\pmod5[/tex]

[tex]x^{11}\equiv x^7x^4\equiv x^5\equiv6\pmod 7[/tex]

5 and 7 are both prime, so [tex]\varphi(5)=4[/tex] and [tex]\varphi(7)=6[/tex]. By Euler's theorem, we get

[tex]x^4\equiv1\pmod5\implies x\equiv3^{-1}\equiv2\pmod5[/tex]

[tex]x^6\equiv1\pmod7\impleis x\equiv6^{-1}\equiv6\pmod7[/tex]

Now we can use the Chinese remainder theorem to solve for [tex]x[/tex]. Start with

[tex]x=2\cdot7+5\cdot6[/tex]

[tex]x=2\cdot7\cdot4\cdot2+5\cdot6[/tex]

[tex]x=2\cdot7\cdot4\cdot2+5\cdot6\cdot4\cdot6[/tex]

[tex]\implies x\equiv832\pmod{5\cdot7}\implies\boxed{x\equiv27\pmod{35}}[/tex]

b.

[tex]x^5\equiv3\pmod{64}[/tex]

We have [tex]\varphi(64)=32[/tex], so by Euler's theorem,

[tex]x^{32}\equiv1\pmod{64}[/tex]

Now, raising both sides of the original congruence to the power of 6 gives

[tex]x^{30}\equiv3^6\equiv729\equiv25\pmod{64}[/tex]

Then multiplying both sides by [tex]x^2[/tex] gives

[tex]x^{32}\equiv25x^2\equiv1\pmod{64}[/tex]

so that [tex]x^2[/tex] is the inverse of 25 mod 64. To find this inverse, solve for [tex]y[/tex] in [tex]25y\equiv1\pmod{64}[/tex]. Using the Euclidean algorithm, we have

64 = 2*25 + 14

25 = 1*14 + 11

14 = 1*11 + 3

11 = 3*3 + 2

3 = 1*2 + 1

=> 1 = 9*64 - 23*25

so that [tex](-23)\cdot25\equiv1\pmod{64}\implies y=25^{-1}\equiv-23\equiv41\pmod{64}[/tex].

So we know

[tex]25x^2\equiv1\pmod{64}\implies x^2\equiv41\pmod{64}[/tex]

Squaring both sides of this gives

[tex]x^4\equiv1681\equiv17\pmod{64}[/tex]

and multiplying both sides by [tex]x[/tex] tells us

[tex]x^5\equiv17x\equiv3\pmod{64}[/tex]

Use the Euclidean algorithm to solve for [tex]x[/tex].

64 = 3*17 + 13

17 = 1*13 + 4

13 = 3*4 + 1

=> 1 = 4*64 - 15*17

so that [tex](-15)\cdot17\equiv1\pmod{64}\implies17^{-1}\equiv-15\equiv49\pmod{64}[/tex], and so [tex]x\equiv147\pmod{64}\implies\boxed{x\equiv19\pmod{64}}[/tex]

I suppose you mean

[tex]f(x)=\dfrac{x^3}{x^2+1}[/tex]

Recall that for [tex]|x|<1[/tex], we have

[tex]\dfrac1{1-x}=\displaystyle\sum_{n=0}^\infty x^n[/tex]

Then

[tex]\dfrac1{1+x^2}=\dfrac1{1-(-x^2)}=\displaystyle\sum_{n=0}^\infty(-x^2)^n=\sum_{n=0}^\infty(-1)^nx^{2n}[/tex]

which is valid for [tex]|-x^2|=|x|^2<1[/tex], or more simply [tex]|x|<1[/tex].

Finally,

[tex]f(x)=\displaystyle\frac{x^3}{x^2+1}=\sum_{n=0}^\infty(-1)^nx^{2n+3}[/tex]