ABCD is a parallelogram AC = 15, m∠ BAC = 22° , m∠DAC = 27° Find: AB and BC

Value of (x + y + z) = 8

Suppose p and q are the positive numbers for which

[tex]log_(9)p=log_(12)q=log_(16)(p+q)[/tex] from the given expression,

[tex]log_(9)p=log_(12)q[/tex]

[tex](logp)/(log9)=(log(q))/(log12)[/tex]

log(p).log(12) = log(q).log(9)

log(q).2log(3) = log(p).log(12) ------(1)

[tex]Now log_(12)q=log_(16)(p+q)[/tex]

[tex](logq)/(log12)=(log(p+q))/(log(16))[/tex]

log(q).log(16) = log(p + q).log(12)

2log(4).log(q) = log(p + q).log12 -------(2)

By adding both the equations (1) and (2),

2log(3).log(q) + 2log(4).log(q) = log(12).log(p) + log(12).log(p + q)

log(q)[2log(3) + 2log(4)] = log(12)[logp + log(p + q)]

2log(q).log(12) = log(12).log[p.(p + q)]

2log(q) = log[p.(p+q)]

q² = p(p + q)

(q)/(p)=(p+q)/(q)

(q)/(p)=(p)/(q)+1

Let (q)/(p)=a

a = (1)/(a)+1

a² - a - 1 = 0 from quadratic formula,

a = [tex]\frac{1\pm \sqrt{(-1)^(2)-4 \times 1 \times (-1)}}{2}[/tex]

a = [tex](1\pm √((1+4)))/(2)[/tex]

a = [tex](1\pm √((5)))/(2)[/tex]

If the solution is represented by (x+√(y))/(z) then  it will be equal to

(1+√((5)))/(2) then x = 1, y = 5 and z = 2.

Now we have to find the value of (x + y + z).

By placing the values of x, y and z,

(x + y + z) = (1 + 5 + 2) = 8

Therefore, value of (x + y + z) = 8

The helicopter uses 11 gallons of fuel per hour, so for 4 hours, it would use 44 gallons of fuel.

The question provides that the number of gallons of fuel used by a helicopter is directly proportional to the number of hours it spends flying. This is a proportionality problem, which can be solved using basic algebraic principles.

Firstly, we know from the question that the helicopter uses 33 gallons of fuel for 3 hours of flight. So, the proportionality constant, or the rate of fuel consumption, is 33 gallons/3 hours = 11 gallons/hour.

To find out how many gallons of fuel the helicopter would consume in 4 hours, we simply multiply the rate of fuel consumption by 4:

11 gallons/hour x 4 hours = 44 gallons.

Therefore, the helicopter would use 44 gallons of fuel to fly for 4 hours.

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Answer:

  6

Step-by-step explanation:

Each ice cream cone costs 4 quarters or 10 dimes, so Simon has ...

  20/4 + 12/10 = 5 + 1.2 = 6.2

times the price of an ice cream cone. He can buy 6 cones for his friends.

Answer:

The statement is true only when both (1) and (2) are valid. If only one of (1) and (2) is valid, them the statement is not true.

Step-by-step explanation:

(1) alone is not sufficient, 27 is 3³ but is not a 12th power

(2) alone is not sufficient either, 81 is 3³ but it is not a 12th power

If both (1) and (2) are valid, then for each prime p that divides x, p should divide y and z, with y³ = x and z⁴=x.

Lets suppose that k is the highest power of p that divides y and m is the highest power that divides z, then (p^k)³ = (p^m)⁴. Therefore

p^3k = p^4m

This means that the power of p that appears on x is a multiple of both 3 and 4. Since those numbers are coprime, then that power is a multiple of 12.

This ensures that every prime dividing x has at least a power of 12 in the prime factirization, hence x is a 12th power.

[tex]\boxed{\vec{v}=\frac{2}{5}i+4j}[/tex]

In this exercise, we have the following facts for the vector [tex]\vec{u}[/tex]:

Since the vector [tex]\vec{u}[/tex] goes from point [tex]P_{1}[/tex] to [tex]P_{2}[/tex], then:

[tex]\vec{u}=(19,-8)-(21,12) \\ \\ \vec{u}=(19-21,-8-12) \\ \\ \vec{u}=(-2,-20)[/tex]

On the other hand, we have the following facts for the vector [tex]\vec{v}[/tex]:

So we can write this relationship as follows:

[tex]5\vec{v}=-\vec{u} \\ \\ \vec{v}=-\frac{1}{5}\vec{u} \\ \\ \vec{v}=-\frac{1}{5}(-2,-20) \\ \\ \vec{v}=(\frac{2}{5},4) \\ \\ \\ Finally: \\ \\ \boxed{\vec{v}=\frac{2}{5}i+4j}[/tex]

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Answer:

See proof below

Step-by-step explanation:

Let [tex]r=\frac{w+z}{1+wz}[/tex]. If w=-z, then r=0 and r is real. Suppose that w≠-z, that is, r≠0.

Remember this useful identity: if x is a complex number then [tex]x\bar{x}=|x|^2[/tex] where [tex]\bar{x}[/tex] is the conjugate of x.

Now, using the properties of the conjugate (the conjugate of the sum(product) of two numbers is the sum(product) of the conjugates):

[tex]\frac{r}{\bar{r}}=\frac{w+z}{1+wz} \left(\frac{1+\bar{w}\bar{z}}{\bar{w}+\bar{z}}{\right)[/tex]

=[tex]\frac{(w+z)(1+\bar{w}\bar{z})}{(1+wz)(\bar{w}+\bar{z})}=\frac{w+z+w\bar{w}\bar{z}+z\bar{z}\bar{w}}{\bar{w}+\bar{z}+\bar{w}wz+\bar{z}zw}=\frac{w+z+w+|w|^2\bar{z}+|z|^2\bar{w}}{\bar{w}+\bar{z}+|w|^2z+|z|^2w}=\frac{w+z+\bar{z}+\bar{w}}{\bar{w}+\bar{z}+z+w}=1[/tex]

Thus [tex]\frac{r}{\bar{r}}=1[/tex]. From this, [tex]r=\bar{r}[/tex]. A complex number is real if and only if it is equal to its conjugate, therefore r is real.

Answer:

EF = (d − b) / 2

Step-by-step explanation:

Let's say that G is the intersection of the trapezoid's diagonals.

Triangle GBC is similar to triangle GDA, so we can write a proportion:

GD / GB = AD / BC

GD / GB = d / b

GD = (d / b) GB

Next, F is the midpoint of BD, so BF equals FD.

BF = FD

GB + GF = GD − GF

2GF = GD − GB

2GF = (d / b) GB − GB

2GF = ((d − b) / b) GB

GF / GB = (d − b) / (2b)

Finally, triangle GBC is similar to triangle GFE, so we can write another proportion:

GF / GB = EF / BC

(d − b) / (2b) = EF / b

EF = (d − b) / 2

Answer: b. 2x = 30

Step-by-step explanation:

Given : 30 states joined the United States between 1776 and 1849

and x states joined between 1850 and 1900 .

If the number of states that joined the United States between 1776 and 1849 is twice the number of states that joined between 1850 and 1900.

i.e. No. of states joined the United States between 1776 and 1849= 2 (No. of states that joined between 1850 and 1900)

i.e . 30=  2(x)          [Substitute the values]

i.e . 2x=30

Hence, the true equation : 2x=30

Answer:

A.102%

Step-by-step explanation:

Let cost price of painting=$100

In first year price increased 20%

Then , the price=[tex]100+100(0.20)=[/tex]$120

In second year

Price decreased 15%

Then , the price of painting=[tex]120-120(0.15)[/tex]

The price of painting=$102

Percent =[tex]\frac{final\;price}{Initial\;price}\times 100[/tex]

By using this formula

Then, we get

Percent of the original price=[tex]\frac{102}{100}\times 100[/tex]

Percent of the original price=102%

Option A is true.

The ratio of sixth-grade students to fifth-grade students is 7/8 or 7:8

Step-by-step explanation:

Let us define ratio first.

"Ratio is the quantitative relationship between two quantities which tells that one quantity is how many times of another quantity"

Given

Number of fifth grade students = 8

Number of sixth grade students = 7

We have to find the ratio of sixth grade students to fifth grade students

So,

[tex]r = \frac{sixth-grade\ students}{fifth-grade\ students}\\r = \frac{7}{8}\\r = 7:8[/tex]

Hence,

The ratio of sixth-grade students to fifth-grade students is 7/8 or 7:8

Keywords: ratio, fraction

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Answer:

The probability that  Smith will lose his first 5 bets is 0.15

The probability that  his first win will occur on his fourth bet is 0.1012

Step-by-step explanation:

Consider the provided information.

A roulette wheel consisting of 38 numbers 1 through 36, 0, and double 0. Smith always bets that the outcome will be one of the numbers 1 through 12,

It is given that Smith always bets on the numbers 1 through 12.

There are 12 numbers from 1 to 12.

Thus, the probability of success (winning) is=  [tex]\frac{12}{38}[/tex]

The probability of not success (loses) is=  [tex]1-\frac{12}{38}=\frac{26}{38}[/tex]

Part (A) Smith will lose his first 5 bets.

The probability  that Smith loses his first 5 bets is,

[tex]\frac{26}{38}\times\frac{26}{38}\times\frac{26}{38}\times\frac{26}{38}\times\frac{26}{38}=(\frac{26}{38})^5\approx0.15[/tex]

Hence, the probability that  Smith will lose his first 5 bets is 0.15

Part (B)  His first win will occur on his fourth bet?

Smith’s first win occurring on the fourth bet means that he loses the first 3 bets and wins on the fourth bet. That is,

[tex]\frac{26}{38}\times\frac{26}{38}\times\frac{26}{38}\times\frac{12}{38}=(\frac{26}{38})^3\times\frac{12}{38}\approx0.1012[/tex]

Hence, the probability that  his first win will occur on his fourth bet is 0.1012

Answer:

[tex]\displaystyle y=-2sin(6t)[/tex]

Step-by-step explanation:

Sinusoid

The shape of a sinusoid is well-know because it describes a curve with a smooth and periodic oscillation. The sine and cosine are the two trigonometric functions in which the graph matches the description of a sinusoid. The sine can be identified because its value is zero at time zero.

The graph shown in the figure corresponds to a sine. The other characteristics of the sine function are

* It completes a cycle in[tex]2\pi[/tex] radians

* It has a maximum of 1 and a minimum of -1

* It's increasing for a quarter of the cycle, decreasing for half of the cycle, and increasing for the remaining quarter of the cycle

* The equation is

[tex]y=Asin(wt)[/tex]

The function starts decreasing for the first quarter, which only is possible if the amplitude A is negative. We can also see the maximum and minimum values are 2 and -2 respectively. This means the amplitude is A=-2

We can also see the function completes 3 cycles in [tex]t=\pi[/tex] radians or 6 cycles in [tex]2\pi[/tex] radians. Or, equivalently

[tex]wt=12\pi[/tex]

[tex]w(2\pi)=12\pi[/tex]

[tex]\displaystyle w=\frac{12}{2}=6\ rad/sec[/tex]

Thus, the function can be expressed as

[tex]\boxed{\displaystyle y=-2sin(6t)}[/tex]

Answer:

y=-2sin6x

Step-by-step explanation:

Answer: D 205

Step-by-step explanation:

Let,

Number of all trouts = N

Number of speckled trouts = Ns = 645

Number of rainbow trouts = Nr

Number of male speckled trouts = Ms

Number of female speckled trouts = Fs

Number of male rainbow trouts = Mr

Number of female rainbow trouts = Fr

Since, Ms = 2Fs + 45

Also, Ms + Fs = 645

Therefore, 2Fs + 45 + Fs = 645

Fs = (645-45)/3 = 200

Female speckled trouts = 200

Since

Ms + Fs = 645

Ms = 645- 200 = 445

Since, Fs/Mr = 4/3

Mr = 3x200/4 = 150

Since,

Mr/N = 3/20

N = 20x 150/3 = 1000

Recall that,

N = Ms+Fs+Mr+Fr

Fr = N-Ms-Fs-Mr

Fr = 1000-445-200-150

Fr = 205

Therefore, the number of rainbow female trouts = 205

The number of female rainbow trout is calculated using given ratios and the total number of speckled trout. After solving a series of equations, the answer is determined to be 205 female rainbow trout, which is option D.

To find the number of female rainbow trout, we first need to unpack the information provided in the question and express it in equations.

Let n be the number of female speckled trout. We know that the number of male speckled trout is 45 + 2n. Since there are 645 speckled trout in total, we can express this as:

n + (45 + 2n) = 645

Solving for n, we get:

3n + 45 = 645

3n = 600

n = 200

Now, we have the ratio of female speckled trout to male rainbow trout as 4:3, and since we've found there are 200 female speckled trout (n = 200), we can figure out the number of male rainbow trout. Let's call this number m. We have:

200/4 = m/3

50 = m/3

m = 150

The ratio of male rainbow trout to all trout is given as 3:20. If the total number of trout is T, then:

150/T = 3/20

We can solve for T:

20 × 150 = 3T

3000 = 3T

T = 1000

So there are 1000 trout in total, of which 645 are speckled. This means there must be 1000 - 645 = 355 rainbow trout. As we have already found there are 150 male rainbow trout, the remainder must be female. So:

355 - 150 = 205 female rainbow trout

Therefore, the correct answer is D. 205.

Answer:

The answer is 1240

Step-by-step explanation:

We know that tuxedo shop rents different number of tuxedos each day and at max they rented 55 tuxedos in May. To maximize the number of tuxedos that rented, rented number of tuxedos need to be drop down by 1 for each day. So minimum 25 tuxedos were rented in May. To find the maximum number of tuxedos that rented in May, we need to add all numbers with using the formula for adding consecutive numbers.

[tex]Total = ((55+25)/2)*31=1240[/tex]

Answer:

c. Will produce a slope field with rows of parallel tangents

Step-by-step explanation:

We can write a differential equation that is a function of y only as:

[tex]y'=f(y)[/tex]

So the derivative, in this particular case, of any function will be a function of the dependent variable y only, it means that the curves you will get should all be pointing in the same direction for each value of x. Therefore the sketch of the slopes field would have parallel curves for each value of x, in other words, it will produce a slope field with rows of parallel tangents.

I hope it helps you!

Final answer:

The standard deviation for the given data is approximately 4.47.

Explanation:

The standard deviation is a measure of how spread out the data is from the mean. To calculate the standard deviation, we follow a formula that involves finding the difference between each data point and the mean, squaring those differences, and taking the average of the squared differences. Finally, we take the square root of the average to get the standard deviation.

For the given data, which is 13, 17, 9, and 21, we first find the mean (average) which is (13 + 17 + 9 + 21) / 4 = 60 / 4 = 15. Next, we calculate the squared differences from the mean for each data point:
(13 - 15)^2 = 4
(17 - 15)^2 = 4
(9 - 15)^2 = 36
(21 - 15)^2 = 36

Then, we find the average of the squared differences:
(4 + 4 + 36 + 36) / 4 = 80 / 4 = 20

Finally, we take the square root of the average:
sqrt(20) ≈ 4.47

So, the standard deviation for the given data is approximately 4.47.

Answer:

Step-by-step explanation:

Last year, sales at a book store increased from $5,000 to $10,000. The amount by which it increased would be 10000 - 5000 = $5000

The percentage by which the sales increased would be

5000/5000 × 100 = 100%

This year, sales decreased to $5,000 from $10,000.The amount by which it decreased would be 5000 - 10000 = - $5000

The percentage by which the sales increased would be

5000/10000 × 100 = 50%

Answer:

11 slips

Step-by-step explanation:

A perfect square is a positive integer that is the square of another integer. For example, 25 is a perfect square of 5. See the calculation below

.

[tex]5^{2} = 5*5\\5^{2} = 25[/tex]

There is a rule that should be kept in mind

1. When two perfect squares are multiplied by each other (e.g. 4 * 9), the result is a perfect square ([tex]36 = 6^{2}[/tex])

Let's identify the combination of numbers that result in perfect square, when multiplied with each other. These combinations are as follows,

• 1*4, 1*9, 1*16  

• 2*8  

• 3*12  

• 4*9, 4*16  

• 9*16  

From the list of numbers 1, 4, 9 and 16 are already perfect square e.g. [tex]2^{2} = 4, 3^{2} = 9[/tex]. If they are multiplied by each other, the result will also be a perfect square. Let’s assume that our first number is 1. Now we can't have any of the three numbers (except for 1), mentioned above. This rule out these three numbers.  

Next, from 2, 8, 3 and 12 we can only draw two numbers. e.g. if we take 2, we can’t take 8 as it will give a perfect square. Same goes with 3 and 12. Hence from these four numbers we can discard two of them.

We discarded three numbers initially and two now. Therefore, out of 16 slipds we can draw a maximum of 11 slips without obtaining a product that is a perfect square.

Answer:

(√2 − √6) / 4

Step-by-step explanation:

Rewrite using special angles.

sin(-11π/12)

sin((4−15)π/12)

sin(4π/12 − 15π/12)

sin(π/3 − 5π/4)

Use angle difference formula:

sin(π/3) cos(5π/4) − sin(5π/4) cos(π/3)

Evaluate:

(√3/2) (-√2/2) − (-√2/2) (1/2)

-√6/4 + √2/4

(√2 − √6) / 4

explanation:

The sides of a cube are squares, and they are covered by the respective sides of the cubes covering that side of the big cube. If we can show that a sqaure cannot be descomposed in squares of different sides, then we are done.

We cover the bottom side of that square with the bottom side of smaller squares. Above each square there is at least one square. Those squares have different heights, and they can have more or less (but not equal) height than the square they have below.

There is one square, lets call it A, that has minimum height among the squares that cover the bottom line, a bigger sqaure cannot fit above A because it would overlap with A's neighbours, so the selected square, lets call it B, should have less height than A itself.

There should be a 'hole' between B and at least one of A's neighbours, this hole is a rectangle with height equal to B's height. Since we cant use squares of similar sizes, we need at least 2 squares covering the 'hole', or a big sqaure that will form another hole above B, making this problem inifnite. If we use 2 or more squares, those sqaures height's combined should be at least equal than the height of B. Lets call C the small square that is next to B and above A in the 'hole'. C has even less height than B (otherwise, C would form the 'hole' above B as we described before). There are 2 possibilities:

If the second case would be true, next to C and above A there should be another 'hole', making this problem infinite. Assuming the first case is true, then C would fit perfectly above A and between B and A's neighborhood.  Leaving a small rectangle above it that was part of the original hole.

That small rectangle has base length similar than the sides of C, so it cant be covered by a single square. The small sqaure you would use to cover that rectangle that is above to C and next to B, lets call it D, would leave another 'hole' above C and between D and A's neighborhood.

As you can see, this problem recursively forces you to use smaller and smaller squares, to a never end. You cant cover a sqaure with a finite number of squares and, as a result, you cant cover a cube with finite cubes.

The solution to the compound inequality is 20 < x < 50. Malcolm should consume more than 20 but less than 50 grams of carbohydrates to adhere to his low-carbohydrate diet.

To solve the compound inequality 50 < 2x + 10 and 2x + 10 < 110 for Malcolm's diet, we'll deal with each part of the inequality separately and then combine the results to find the range of values for x that satisfies both conditions.

Combining these inequalities gives us 20 < x < 50. This means that Malcolm should consume more than 20 but less than 50 grams of carbohydrates to stay within his targeted dietary range on his low-carbohydrate diet.

Answer:

The answer is 8

Step-by-step explanation:

Lets consider that Sam's call will last x minutes in total. Then, equations for cost will be as follows:

A Pay Phone Cost: [tex]p(x)=1+0,5*x[/tex]

Operator Cost: [tex]o(x)=3+0,25*x[/tex]

If we make both costs equal, the equation will be:

[tex]1+0,5*x=3+0,25*x\\0,25*x=2\\x=8[/tex]

Sam's call need to last 8 minutes to make payphone cost and operator cost same.

Answer:

  8x^2 + 34; 45.52 centimeters

Step-by-step explanation:

The perimeter is twice the sum of adjacent edges of a rectangle:

  P = 2((2x^2 +9) +(2x^2 +8)) = 2(4x^2 +17)

  P = 8x^2 +34

Then when x=1.2, the perimeter is ...

  P = 8(1.2^2) +34 = 8·1.44 +34 = 45.52 . . . . cm

Answer:

The Answer is: B  8x^2 + 34; 45.52 centimeters

Step-by-step explanation:

Took it on edg

Answer:

8 seconds

15 quarts

Step-by-step explanation:

Givens

6 seconds emits 9 quarts of water.

x seconds emits 12 quarts of water.

Formula

6/x = 9/12

Solution

Cross multiply

9x = 12*6         Simplify the right

9x = 72            Divide by 9

9x/9 = 72/9

x = 8 seconds  Answer.

In 8 seconds, the hose will emit 12 quarts

=========================

Givens

9 quarts of water are emitted in 6 second

x quarts of water are emitted in 10 seconds

Formula

9/x = 6/10            Cross multiply

Solution    

9*10 = 6x              Combine the left

90 = 6x                 Divide by 6

90/6=6x/6            

15 = x

In 10 seconds the hose emits 15 quarts.

Answer:

Step-by-step explanation:

The garden hose emits 9 quarts of water in 6 seconds. It means that the number of seconds that it takes the garden hose to emit 1 quart of water would be 6/9 = 2/3 seconds

Therefore, the number of seconds that it will take the garden hose to emit 12 quarts of water would be

12 × 2/3 = 8 seconds.

Again, the number of quarts that the garden hose will emit in one second is 9/6 = 3/2 quarts.

Therefore, the number of quarts of water that the hose emits in 10 seconds would be

10 × 3/2 = 15 quarts.

Carter has 25 dimes

Solution:

Let "n" be the number of nickels

Let "d" be the number of dimes

Given that Carter has 37 coins, all nickels and dimes in his piggy bank

number of nickels + number of dimes = 37

n + d = 37 -------- eqn 1

Given that value of coins is $ 3.10

Also, value of nickel is 0.05 dollar and value of dime is 0.10 (in dollars) and total value of these 37 coins is 3.10, so we can write:

number of nickels x value of 1 nickel + number of dimes x value of 1 dime = 3.10

[tex]n \times 0.05 + d \times 0.10 = 3.10[/tex]

0.05n + 0.10d = 3.10 ------- eqn 2

Let us solve eqn 1 and eqn 2 to find values of "n" and "d"

From eqn 1,

n = 37 - d ---- eqn 3

Substitute eqn 3 in eqn 2

0.05(37 - d) + 0.10d = 3.10

1.85 - 0.05d + 0.10d = 3.10

0.05d = 3.10 - 1.85

0.05d = 1.25

Thus carter has 25 dimes

By setting up and solving a system of equations, we find that Carter has 25 dimes in his piggy bank.

To solve how many dimes Carter has, we need to set up two equations based on the given information.

Carter has 25 dimes in his piggy bank.

Answer:

We can expect 8255 numbers of students are liberal arts majors.

Step-by-step explanation:

Given:

Total Number of students in the college = 8943

Now According to Mr. Jones Survey;

60 out of 65 students are liberal arts majors.

We need to find the number of students who are liberal arts majors out of total number of students in college.

Solution:

First we will find the Percentage number of students who are liberal arts majors according to survey.

Percentage number of students who are liberal arts majors can be calculated by dividing 60 from 65 then multiplying by 100.

framing above quote in equation form we get;

Percentage number of students who are liberal arts majors = [tex]\frac{60}{65}\times 100 = 92.31\%[/tex]

Now we will find the Total number of students who are liberal arts major.

Total number of students who are liberal arts major can be calculated by Multiplying Percentage number of students who are liberal arts majors with total number of students in the college and then dividing by 100.

framing above quote in equation form we get;

Total number of students who are liberal arts major = [tex]\frac{92.31}{100}\times 8493 \approx8255.28[/tex]

Since number of students cannot be in point, so we will round the value.

Hence We can expect 8255 numbers of students are liberal arts majors.

Answer:

Step-by-step explanation:

Let x represent the fuel efficiency of the car that used the most gas. Since the sum of the fuel efficiencies is 40 mpg, the other car had a fuel efficiency of (40-x). Then the combined miles driven is ...

  40x +25(40-x) = 1225

  15x = 225

  x = 15

  (40-x) = 25

The first car got 25 miles per gallon; the second car got 15 miles per gallon.

_____

We made use of the relation ...

  gallons × (miles/gallon) = miles

Answer:

(1, 10)

Step-by-step explanation:

Use midpoint formula.

Mₓ = (x₁ + x₂) / 2

3 = (5 + x) / 2

6 = 5 + x

x = 1

Mᵧ = (y₁ + y₂) / 2

9 = (8 + y) / 2

18 = 8 + y

y = 10

The coordinates of point D are (1, 10).

The coordinates of point D are (1, 10).

Midpoint, as the word suggests, means the point which lies in the middle of something.

Midpoint of a line segment means a point which lies in the mid of the given line segment.

Given; CD has endpoints C and D, with C at coordinates (5,8). CD has midpoint M at (3,9).

Using midpoint formula;

Mₓ = (x₁ + x₂) / 2

3 = (5 + x) / 2

6 = 5 + x

x = 1

Now, Mᵧ = (y₁ + y₂) / 2

9 = (8 + y) / 2

18 = 8 + y

y = 10

Hence, The coordinates of point D are (1, 10).

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The inequality is used to solve how many hours of television Julia can still watch this week is [tex]x + 1.5 \leq 5[/tex]

The remaining hours of TV Julia can watch this week can be expressed is 3.5 hours

Given that Julia is allowed to watch no more than 5 hours of television a week

So far this week, she has watched 1.5 hours

To find: number of hours Julia can still watch this week

Let "x" be the number of hours Julia can still watch television this week

"no more than 5" means less than or equal to 5 ( ≤  5 )

Juila has already watched 1.5 hours. So we can add 1.5 hours and number of hours Julia can still watch television this week which is less than or equal to 5 hours

number of hours Julia can still watch television this week + already watched ≤ Total hours Juila can watch

[tex]x + 1.5 \leq5[/tex]

Thus the above inequality is used to solve how many hours of television Julia can still watch this week.

Solving the inequality,

[tex]x + 1.5 \leq5\\\\x \leq 5 - 1.5\\\\x \leq 3.5[/tex]

Thus Julia still can watch Television for 3.5 hours

Answer:

a) [tex]P(X>3.0)=1-P(X \leq 3.0)= 1- [1- e^{-\frac{1}{2.4} 3.0}]=e^{-\frac{1}{2.4} 3.0}=0.287[/tex]

b) [tex]P(2<X<3)= 1- e^{-\frac{1}{2.4} 3} -[ 1- e^{-\frac{1}{2.4} 2}][/tex]

[tex]P(2<X<3)=e^{-\frac{1}{2.4} 2}- e^{-\frac{1}{2.4} 3}=0.435-0.287=0.148[/tex]

Step-by-step explanation:

Definitions and concepts

The Poisson process is useful when we want to analyze the probability of ocurrence of an event in a time specified. The probability distribution for a random variable X following the Poisson distribution is given by:

[tex]P(X=x) =\lambda^x \frac{e^{-\lambda}}{x!}[/tex]

And the parameter [tex]\lambda[/tex] represent the average ocurrence rate per unit of time.

The exponential distribution is useful when we want to describ the waiting time between Poisson occurrences. If we assume that the random variable T represent the waiting time btween two consecutive event, we can define the probability that 0 events occurs between the start and a time t, like this:

[tex]P(X>x)= e^{-\lambda x}[/tex]

We can express in terms of the mean [tex]\mu =\frac{1}{\lambda}[/tex]

[tex]P(X>x)= e^{-\frac{1}{\mu} x}[/tex]

And the cumulative function would be given by the complement rule like this:

[tex]P(X\leq x)=1- e^{-\frac{1}{\mu} x}[/tex]

Solution for the problem

For this case we have that X the random variable that represent the magnitude of earthquakes recorded in a region of North America, we know that the distribution is given by:

[tex]X\sim Expon(\mu =2.4)[/tex]

a) Find theprobability that an earthquake striking this region will a exceed 3.0 on the Richter scale

So for this case we want this probability:

[tex]P(X>3.0)=1-P(X \leq 3.0)= 1- [1- e^{-\frac{1}{2.4} 3.0}]=e^{-\frac{1}{2.4} 3.0}=0.287[/tex]

b) fall between 2.0 and 3.0 on the Richter scale.

For this case we want this probability:

[tex]P(2 < X <3) = P(X<3) -P(X<2)[/tex]

And replacing we have this:

[tex]P(2<X<3)= 1- e^{-\frac{1}{2.4} 3} -[ 1- e^{-\frac{1}{2.4} 2}][/tex]

[tex]P(2<X<3)=e^{-\frac{1}{2.4} 2}- e^{-\frac{1}{2.4} 3}=0.435-0.287=0.148[/tex]