11. Arrange the following number in order of magnitude starting from the smallest: ⅔, ¾, ⁵/⁷, ⅖ [A] ⅔, ¾, ⅖, ⁵/7 [B]⅖, ¾, ⅔, ⁵/7 [C]⁵/7, ⅖, ¾, ⅔ [D]⅖, ⅔, ⁵/7, ¾​

The measures of the central and circumscribed angles are :

m ∠EFG=83∘, m ∠GCE=97∘

The correct option is (b)

There are two tangents on the circle C at the point E and G.

m ∠EFG=(3x+11)∘, and m ∠GCE=(5x−23)∘

Now, We have to find the measures of the central and circumscribed angles.

The line joining the center of circle to the point on circle on which there is a tangent, make an angle of 90° with the tangent itself.

∠CGF = ∠CEF = 90° ( G and F are the points on circle's tangent drawn from point F.)

Now, we can see that CGFE is a quadrilateral.

And sum of all internal angles of a quadrilateral is equal to 360°

∠C + ∠G + ∠F + ∠E = 360°

(5x - 23) + 90 + 3x + 11 + 90 = 360

=> 8x - 12 + 180 = 360

8x - 12 = 360 - 180

8x = 180 + 12

=> x = 24°

m ∠EFG = (3x + 11)°

m ∠EFG = (3× 24 + 11)° = 83°

m ∠GCE=(5x−23)∘

m ∠GCE = (5 × 24 −23)∘

m ∠GCE = 97°

The correct option is (b)

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For complete question , to see the attachment.

The following features of the circle are true:

Center: (h, k) = (- 1, 1) (Right choice: D)

Domain: - 4 ≤ x ≤ 2 (Right choice: A)

Range: - 2 ≤ y ≤ 4 (Right choice: E)

Herein we find a circle represented by a equation in general form:

x² + y² + C · x + D · y + E = 0

Where A, B, C, D, E are real coefficients.

To derive the features of the circle, we need to find the standard form from the previous expression:

(x - h)² + (y - k)² = r²

Where:

The domain and range of the equation of the circle are, respectively:

Domain: h - R ≤ x ≤ h + R

Range: k - R ≤ y ≤ k + R

First, we find the standard form by completing the square:

x² + y² + 2 · x - 2 · y - 7 = 0

(x² + 2 · x) + (y² - 2 · y) = 7

(x² + 2 · x + 1) + (y² - 2 · y + 1) = 9

(x + 1)² + (y - 1)² = 3²

Center: (h, k) = (- 1, 1)

Domain: - 4 ≤ x ≤ 2

Range: - 2 ≤ y ≤ 4

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Using the given volume and the length of the cylinder we know that the radius is 1.89 cm approximately.

A cylinder is one of the most basic curvilinear geometric shapes and has traditionally been solid in three dimensions.

In elementary geometry, it is regarded as a prism with a circle as its basis.

A cylinder can instead be described as an infinitely curved surface in a number of modern domains of geometry and topology.

So, the volume of the cylinder is 45 cm³.

The length is 4 cm.

Now, the formula for volume is: V=πr²h

Insert values and calculate r as follows:

V=πr²h

45=3.14r²4

45=12.56r²

45/12.56=r²

3.58 = r²

1.89 = r

Therefore, using the given volume and the length of the cylinder we know that the radius is 1.89 cm approximately.

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Complete question:

Find the radius of a cylinder or volume of 45 cm³ and length of 4 cm.

Answer:

The vertex is: [tex](\frac{5}{2} - \frac{17}{4} )[/tex]

The zero is: [tex]\frac{5+-\sqrt{17} }{2}[/tex]

Hope this helps :)

Pls brainliest...

The corresponding to Vector R is 4.8 units 47 East of South 4.8 units 47circ East of South

Vector C comprises a magnitude of -3.5i

Vector D is established as –3.3j (South bearing is thought to be negative along the y-axis).

Vector R = Vector D - Vector C

= (-3.3j) - (-3.5i)

= 3.5i - 3.3j

To evaluate the strength of Vector R, we must first compute its magnitude:

Magnitude of R = √((3.5)^2 + (-3.3)^2) ≈ 4.8 units.

determine the direction,

we shall need to calculate the angle θ with respect to the South direction (the negative y-axis):

tan(θ) = (3.5) / (3.3);

θ = arctan(3.5 / 3.3) ≈ 47°

Hence the answer is option 2

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The value of x is 55.6

In order to find the value of x we use sine,

sin ∅ = opposite / hypotenuse

From the question, x is the hypotenuse, the opposite is 19

So, we have

sin 20 = 19/x

x = 19/sin 20

x = 55.55

We have the answer as x = 55.6 to the nearest tenth

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Answer:

  34 m

Step-by-step explanation:

You want the width of a river as found using similar triangles.

In the attached, we have labeled the vertices of the figure and drawn it to scale. Triangles ABC and ADE are similar, so corresponding sides have the same ratio:

  BE/BA = DE/DA

  20.1/35 = ?/59

  ? = 59(20.1/35) ≈ 33.88 ≈ 34 . . . . . . multiply both sides by 59

The width of the river is about 34 meters.

__

Additional comment

The triangles are similar by the AA similarity postulate. The vertical angles are congruent, as are the right angles. You can learn to rapidly identify similar triangles and the sides that correspond. (One way to write the similarity statement is the way we did: name the vertices of the congruent angles in the same order: ∆ABC ~ ∆ADE)

The width of the river is 39m , we found by forming proportional equation because it is similar triangle.

The triangles are similar by the AA similarity postulate. The vertical angles are congruent, as are the right angles

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion .

We have to find the width of the river

Let us form a proportional equation

x/20.1 = 59/35

Apply cross multiplication

35x=59×20.1

35x=1185.9

Divide both sides by 35

x=33.8

x=39

Hence, the width of the river is 39m , we found by forming proportional equation

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the probability that all 6 prizes are gift certificates to restaurants is: 0.083.

There are a total of 9 prizes, out of which 7 are gift certificates to restaurants. If 6 prizes are randomly raffled away in the first hour, there are a total of 9 choose 6 possible outcomes, or 84 possible sets of 6 prizes.

The number of outcomes in which all 6 prizes are gift certificates to restaurants is 7 choose 6, or 7 possible sets of 6 gift certificates.

Therefore, the probability that all 6 prizes are gift certificates to restaurants is:

7/84 = 0.0833 or approximately 8.33%

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In the given problem, the furniture company should produce 300 bar stools and 50 chairs to maximize their profit, and should not produce any tables.

To maximize profit, the furniture company needs to determine how many units of each product (bar stools, chairs, and tables) they should produce. Let's use x, y, and z to represent the number of bar stools, chairs, and tables respectively.

The objective function to maximize profit is:

Profit = 10x + 25y + 35z

The company has constraints on the amount of labor, wood, and stain available. These constraints can be expressed as follows:

Labor: 1x + 4y + 2z ≤ 700 (the company has 700 hours of labor available each week)

Wood: 8x + 16y + 32z ≤ 3360 (the company has 3360 feet of wood available each week)

Stain: 0.2x + 0.4y + 0.5z ≤ 60 (the company has 60 gallons of stain available each week)

Also, since we cannot produce negative units of any product, we must add the constraint that x, y, and z are all non-negative.

x ≥ 0, y ≥ 0, z ≥ 0

We can now use linear programming techniques to solve this problem. One possible method is the simplex method, which involves converting the problem into an augmented matrix and applying row operations until we obtain the optimal solution.

The optimal solution is:

x = 300 (produce 300 bar stools)

y = 50 (produce 50 chairs)

z = 0 (do not produce any tables)

The maximum profit that the company can earn is:

Profit = 10x + 25y + 35z = 10(300) + 25(50) + 35(0) = $3,250

Therefore, the furniture company should produce 300 bar stools and 50 chairs to maximize their profit, and should not produce any tables.

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1) 45°

2) 45°

3) BC= 3 AB= sqrt(18)=3sqrt(2)

4) BC= 4 AB= sqrt(32)=4sqrt(2)

5) 9sqrt(2)

6) 7

7) 2

sqrt means square root

AC = AB because it is isosceles

pythagore theorem is used to solve 3 to 7

AB²= AC²+CB²

In 1 and 2 it is the angle in an isosceles triangle

Answer:

[tex]375 \times .18 = 67.5[/tex]

The probability of selecting a lemon-filled donut followed by a custard-filled donut is given as follows:

p = 15/92

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The probability of each event is given as follows:

Hence the probability is obtained as follows:

p = 9/24 x 10/23

p = 9/12 x 5/23

p = 3/4 x 5/23

p = 15/92

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The total budget is $1.200

The percentage of drinks is 15%

The percentage of food is 30%

We find out how much the drinks cost.

[tex] \bf 15\% \: of \: \$1200 = \\ \\ \bf = \frac{15}{1 \cancel 0 \cancel 0} \times 12 \cancel 0 \cancel 0 = \\ \\ \bf = 15 \times 12 = \green{\$180}[/tex]

We find out how much the food cost.

[tex] \bf 30\% \: of \: \$1200 = \\ \\ \bf = \frac{3 \cancel 0}{10 \cancel 0} \times 1200 = \\ \\ \bf = \frac{3}{1 \cancel 0} \times 120 \cancel 0 = \\ \\ \bf = 3 \times 120 = \green{\$360}[/tex]

How many dollars is the food and drinks?

[tex] \bf \$180 + \$360 = \red{\boxed{\bf \$540}} [/tex]

The answer is C $540.

Good luck! :)

Answer:

1 min = 60s

1 h = 60 mins

14277s/60s = 237 mins and 57s = 237.95 mins

237.95mins/60mins = 3 hours 57 mins 57s

= 3.97 h （cor.ti 3 sig fig.）

Answer: 237.95 minutes and 3.97 hours

Step-by-step explanation:

If you have 14,277 seconds and need to convert them into minutes, you can set up the equation 1(minute)/60(seconds) x 14,277(seconds), which equals 237. 95 minutes. To convert into hours, you just take the minutes and set up the equation 1(hour)/60(minutes) x 237.95, which equals approximately 3.966 hours. I assume your asking to convert to seconds was an accident because you started with seconds to begin with.

7) The probability of not being dealt a queen is: 12/13

8) The probability of not being dealt a 9 is: 12/13

9) The probability of not being dealt a heart is: 3/4

7) We know that in a standard deck of cards, that we have:

(4 Aces, 4 Kings, 4 Queens, 4 jacks)

Thus, probability of not selecting a queen is:

48/52 = 12/13

8) There are 4 nines in a standard deck of 52 cards. The probability of selecting the first nine is thus: 4/52.

Probability of not being dealt a 9 is: 48/52 = 12/13

9) In a standard deck of 52 cards, we know that there are 4 suits (Clubs, Hearts, Diamonds, and Spades) and there are 13 cards in each suit (Clubs/Spades are black, Hearts/Diamonds are red)

Thus, probability of not being dealt a heart = (52 - 13)/52

= 39/52

= 3/4

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Applying the Pythagorean Theorem and the triangle area formula, the area is calculated as: b. 24 cm².

The Pythagorean Theorem states that the square of the longest side (hypotenuse) of a right triangle is equal to the sum of the squares of the other shorter sides or legs.

Thus, applying the Pythagorean Theorem, we have:

a = √(10² - 8²)

a = 6 cm

Base (b) = 8 cm

Height (h) = a = 6 cm

Plug in the values:

Area of triangle = 1/2 * 8 * 6

Area of triangle = 24 cm²

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The z-scores are given as follows:

The z-score of a measure X of a normally distributed variable that has mean represented by [tex]\mu[/tex] and standard deviation represented by [tex]\sigma[/tex] is obtained by the equation presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

Josh's z-score is given as follows:

Z = (185.16 - 185.81)/0.363

Z = -1.79.

Rita's z-score is given as follows:

Z = (109.89 - 110.1)/0.133

Z = -1.58.

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Mean number of births per day ≈ 11.41.The probability of having exactly 13 births in a day,is approximately 11.41, is about 11.79%.. No, it doesnot appear likely that on any given day,there will be exactly 13 births.

Probability can be defined as the ratio of favourable outcome to the total number of outcome.

A Poisson distribution is a discrete probability distribution. The chance of an event occurring a specific number of times (k) during a specific time or space period is provided by the Poisson distribution. The mean number of occurrences, denoted by the letter "lambda," is the single parameter of the Poisson distribution.

To find the mean number of births per day, we divide the total number of births in a year (4164) by the number of days in a year. Assuming a year has 365 days, the mean number of births per day would be:

Mean number of births per day = Total number of births in a year / Number of days in a year

Mean number of births per day = 4164 / 365

Mean number of births per day ≈ 11.41

Now, we can use the Poisson distribution to find the probability of having exactly 13 births in a day, given that the mean number of births per day is approximately 11.41.

The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space, when the events are rare and random, and the average rate of occurrence is known.

The probability mass function (PMF) of the Poisson distribution is given by the formula:

P(X=k) = ( [tex]lambda^{k}[/tex]×e^(-λ)) / k!)

Where:

λ is the average rate of occurrence (mean) of events in the given interval

e=2.71828

k is the number of events for which we want to find the probability

k! is the factorial of k (k factorial)

In this case, the average rate of occurrence (mean) of births per day is approximately 11.41 (calculated in the previous step). So, we can plug in the values into the Poisson PMF formula:

P(X=13) = (λ¹³ ×e^(-λ)) / 13!

P(X=13) = (11.41¹³ × e^(-11.41)) / 13!

Calculating this value using a calculator or software, we can find that:

P(X=13) ≈ 0.1179 or 11.79%

So, the probability of having exactly 13 births in a day, given the mean number of births per day is approximately 11.41, is about 11.79%.

Based on this probability, it appears unlikely that on any given day there will be exactly 13 births, as the probability is relatively low. However, it is important to note that the Poisson distribution assumes that the events are rare and random, and there may be other factors that can affect the actual number of births in a day, such as seasonality, day of the week, and other external factors. Therefore, further analysis and consideration of other factors may be needed to make a more accurate assessment of the likelihood of exactly 13 births occurring in a day at a specific hospital.

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a) The minimum sample size required is 45.

b) The population mean could be within 10% of the sample mean, but less likely that it could be within 11% of the sample mean.

(a) How to determine the minimum sample size?

To determine the minimum sample size required to construct a 90% confidence interval for the population mean with a margin of error of 1.5 years, we use the formula n = (z ×σ / E)²

where n is the sample size, z is the z-score for the desired confidence level (in this case, 1.645 for a 90% confidence level), σ is the population standard deviation (1.8 years), and E is the margin of error (1.5 years).

Substituting the given values,

n = (1.645 × 1.8 / 1.5)² = 6.67² = 44.5

Rounding up to the nearest whole number, the minimum sample size required is 45.

(b) Using the minimum sample size of 45 with a 90% confidence level, the margin of error is 1.5 years. Therefore, the 90% confidence interval for the population mean is:

20 - 1.5 ≤ μ ≤ 20 + 1.5

18.5 ≤ μ ≤ 21.5

To determine whether it is likely that the population mean could be within 10% or 11% of the sample mean, we need to calculate the ranges of values that correspond to these percentages of the sample mean,

10% of 20 = 2

11% of 20 = 2.2

The range of values that are 10% of the sample mean is 20 ± 2 = 18 to 22

The range of values that are 11% of the sample mean is

20 ± 2.2 = 17.8 to 22.2

Comparing these ranges to the 90% confidence interval for the population mean, we see that:

The range of values that are 10% of the sample mean (18 to 22) is completely within the 90% confidence interval (18.5 to 21.5), so it is likely that the population mean could be within 10% of the sample mean.

The range of values that are 11% of the sample mean (17.8 to 22.2) extends slightly beyond the 90% confidence interval, so it is less likely that the population mean could be within 11% of the sample mean.

Therefore, we can conclude that it is likely that the population mean could be within 10% of the sample mean, but less likely that it could be within 11% of the sample mean.

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the limit of (x² - 1)/(x³ - 1) as x approaches 1 is equal to 2/3. got answer by  try substituting x = 1 directly into the expression

what is limit of equation ?

In calculus, the limit of a function is the value that the function approaches as the input (usually denoted by x) approaches a certain value (usually denoted by a). In other words, it describes the behavior of a function as the input value gets closer and closer to a certain point.

In the given question,

To evaluate the limit of (x² - 1)/(x³ - 1) as x approaches 1, we can try substituting x = 1 directly into the expression:

(1² - 1)/(1³ - 1) = 0/0

We cannot determine the limit using direct substitution because we get an indeterminate form of 0/0.

One way to evaluate this limit is to factor the numerator and denominator and cancel out any common factors. We can factor the numerator using the difference of squares formula:

x² - 1 = (x + 1)(x - 1)

We can factor the denominator using the difference of cubes formula:

x³ - 1 = (x - 1)(x² + x + 1)

Canceling out the common factor of (x - 1) in the numerator and denominator, we get:

(x + 1)/(x² + x + 1)

Now we can substitute x = 1 directly into this expression:

(1 + 1)/(1² + 1 + 1) = 2/3

Therefore, the limit of (x² - 1)/(x³ - 1) as x approaches 1 is equal to 2/3.

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The probability that a student had blue eyes and wore glasses is 4% or 0.04.

B. The probability that a student's eye color is brown given that the student wears glasses is 42.9% or 0.429.

To find the probability of blue-eyed glasses-wearers, when you look where "Blue" intersects "Yes" in the table. The relative frequency is 0.04, or 4%, hence the probability of a student having blue eyes and wearing glasses 4%.

To find the probability that a student's eye color is brown since the fact is: that the student wears glasses, it can be calculated as:

P(A|B) = P(A and B)/P(B)

P(Brown and Yes) = 0.12

P(Yes) = 0.04 + 0.12 + 0.08 + 0.04

          = 0.28

Hence, the probability that a student's eye color is brown since they wear glasses are:

P(Brown | Yes) = P(Brown and Yes) / P(Yes)

 = 0.12 / 0.28

= 0.429 or 42.9%

So we can say that, Blank 1 is "0.04" while Blank 2 is "42.9%".

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See text below



Eye Color

Question 15 (2 points)

A high school class conducts a survey where students are asked about their eye color and whether or not they wear glasses. The two-way table below shows the results of the survey as relative frequencies.

                             Glasses?

                        Yes                     No

Blue                 0.04                 0.16

Brown             0.12                 0.24

Green             0.08                  0.12

Hazel             0.04                  0.20

A. Based on thee results of the survey, what is the probability, rounded to the nearest tenth, that a student had blue eyes and wore glasses?

B. Based on the results of the survey, what is the probability, rounded to the nearest tenth, that a student's eye color is brown given that the student wears glasses?

Word Bank:

8% 4% 16% 20% 42.9%

24% 12%

Blank 1:

Blank 2:

Step-by-step explanation:

To solve this problem, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:

A = the amount of money Billy will have when the investment ends

P = the principal amount he invested, which is $18,000

r = the annual interest rate, which is 1.5%

n = the number of times the interest is compounded per year, which is monthly or 12 times per year

t = the time period in years, which is 10 years

Plugging these values into the formula, we get:

A = 18000(1 + 0.015/12)^(12*10)

A ≈ $24,134.44

Therefore, Billy will have approximately $24,134.44 when the investment ends.

Billy can calculate his investment with the compound interest formula, using his initial investment amount, the monthly interest rate, and the number of compounding periods in 10 years. Doing so gives an end balance of approximately $20,448.24.

Billy's investment in the CD can be calculated using the compound interest formula, which is A = P (1 + r/n)^(nt). In this formula, A represents the amount of money accumulated after n years, including interest. P is the principal amount (the initial amount of money), r is the annual interest rate (in decimal format), n is the number of times that interest is compounded per year, and t is the time in years that the money is invested for.

Plugging the given values into the formula, we have A = 18000 (1 + 0.015/12)^(12*10). Calculating this expression gives a result of approximately $20,448.24. Therefore, after 10 years, Billy will have approximately $20,448.24 in his CD.

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Answer:

61.69924423

Step-by-step explanation:

This is an example of arc tangent. You can use arctan(13/7) on a calculator to get your answer!

Answer:

Step-by-step explanation:

If the order you select them is important

nmber of ways = 9! / (9-7)!

= (9*8*7*6*5*4*3*2*1) / (2*1)

= 9*8*7*6*5*4*3

= 181,440.

If the order does not matter:

nmber of ways = 9! / ((9-7)! * 7!)

= 9*8 / 2*1

= 36.

The perimeter of the triangle is

To ascertain the perimeter of a triangle whose verticies reside at points P(-3, -2),Q(0, 2), and R(1, -2),calculate the distances between these points and then add them.

Using the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Distance for PQ =√((0 - (-3))^2 + (2 - (-2))^2)

Distance for PQ =√(3^2 + 4^2)

Distance for PQ = √(9 + 16)

Distance for PQ = √25

Distance for PQ = 5

Distance for PR = (1 - (-3))

Distance for PR = 4

Distance for QR =√((1 - 0)^2 + ((-2) - 2)^2)

Distance for QR =√(1^2 + 4^2)

Distance for QR =√17

Perimeter

Perimeter = Distance PQ  +  Distance PR  +  Distance QR

Perimeter =  5  +   4   +   √17

Perimeter = 9 +√17 units    

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The perimeter of the triangle is

To ascertain the perimeter of a triangle whose verticies reside at points P(-3, -2),Q(0, 2), and R(1, -2),calculate the distances between these points and then add them.

Using the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Distance for PQ =√((0 - (-3))^2 + (2 - (-2))^2)

Distance for PQ =√(3^2 + 4^2)

Distance for PQ = √(9 + 16)

Distance for PQ = √25

Distance for PQ = 5

Distance for PR = (1 - (-3))

Distance for PR = 4

Distance for QR =√((1 - 0)^2 + ((-2) - 2)^2)

Distance for QR =√(1^2 + 4^2)

Distance for QR =√17

Perimeter

Perimeter = Distance PQ  +  Distance PR  +  Distance QR

Perimeter =  5  +   4   +   √17

Perimeter = 9 +√17 units    

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After 153 years, 38.18 mg of Polonium-209 would remain from the initial sample of 108 mg.

When we are given that half-life of Polonium-209 is 102 years, this means that after 102 years, half of the initial sample would have decayed.

We will use the below formula to calculate the amount of Polonium-209 remaining after 153 years:

= Initial amount × (1/2)^(t/half-life)

Substituting the values given in the problem, we get:

= 108 mg × (1/2)^(153/102)

= 108 mg × 0.35355339059

= 38.1837662 mg

= 38.18 mg.

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Logarithm [tex]log(xy^{4})[/tex]  is equal to  36.4.

We can use the properties of logarithms to simplify [tex]log(xy^{4})[/tex]  :

[tex]log(xy^{4})[/tex]   = [tex]log(x)[/tex] +  [tex]log(y^{4} )[/tex] (by the product rule)

= [tex]log(x)+4log(y)[/tex]   (by the power rule)

Substituting the given values:

[tex]log(xy^{4})[/tex]   =  [tex]log(x)+4log(y)[/tex]

= 14.11 + 4(5.43)

= 36.4

Therefore, [tex]log(xy^{4})[/tex]  = 36.4. This means that [tex]xy^{4}[/tex] equals 10 to the power of 36.4. Using the inverse property of logarithms, we can find that:

[tex]xy^{4}[/tex]  = [tex]10^{36.4}[/tex]

= 4.17 x  [tex]10^{36}[/tex]

In summary, [tex]log(xy^{4})[/tex]   equals 36.4 and [tex]xy^{4}[/tex]  equals 4.17 x  [tex]10^{36}[/tex].

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After 12 years, the environment can no longer support the population growth. (option a).

Population growth is a fundamental concept in ecology and biology. It refers to the increase in the number of organisms in a population over time.

The carrying capacity is the maximum number of individuals of a particular species that can be supported by the environment without degrading it. In this case, the carrying capacity of the environment is 600 organisms. When the population reaches this limit, the environment can no longer support the population, and the growth rate slows down until it reaches equilibrium.

The table provides the approximate number of organisms after y years. To determine when the environment can no longer support the population, we need to find the year when the population exceeds the carrying capacity of the environment. In other words, we need to find the year when the population reaches or exceeds 600 organisms.

Looking at the table, we can see that the population reaches 600 organisms in the year 12.

Hence the correct option is (a).

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Answer:  The environment will no longer be able to support the population after 24 years

Step-by-step explanation:

The environment will no longer be able to support the population after 24 years

How to determine the number of years?

The proper representation of the table is given as:

y   1     2     3    4

n  55  60  67  75

An exponential function is represented as:

                                [tex]n=ab^y[/tex]                

Where:

Next, we determine the function equation using a statistical calculator.

From the statistical calculator, we have:

[tex]a=49.19[/tex] [tex]and[/tex] [tex]b=1.11[/tex]

Substitute these values in [tex]n=ab^y[/tex] .

So, we have:

     [tex]n=49.19*1.11^y[/tex]

From the question, the maximum is 600.

So, we have:

      [tex]49.19*1.11^y=600[/tex]

Divide both sides by 49.19

        [tex]1.11^y=12.20[/tex]

Take the logarithm of both sides

        [tex]y log(1.11) = log(12.20)[/tex]

Divide both sides by log(1.11)

     [tex]y=23.97[/tex]

Approximate

    [tex]y=24[/tex]

Hence, the environment will no longer be able to support the population after a period of 24 years.

We can say that it is true that the function g(x)=(e^x)sinb is an antiderivative of the function f(x)=(e^x)sinb.

We can see here that to prove this, we need to show that g'(x) = f(x), where g'(x) is the derivative of g(x).

Using the product rule of differentiation, we have that:

g'(x) = (e^x)(cosb) + (sinb)(e^x).

A further simplification will give us:

g'(x) = (e^x)(cosb + sinb)

Thus, comparing g'(x) with f(x), we have: f(x) = (e^x)(sinb)

Therefore, comparing the two expressions, we can see that:

g'(x) = f(x) if and only if cosb + sinb = sinb.

This is true for all values of b, since:

We can then conclude that g(x) = (e^x)sinb is indeed an antiderivative of f(x) = (e^x)sinb.

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We can say that it is true that the function g(x)=(e^x)sinb is an antiderivative of the function f(x)=(e^x)sinb.

We can see here that to prove this, we need to show that g'(x) = f(x), where g'(x) is the derivative of g(x).

Using the product rule of differentiation, we have that:

g'(x) = (e^x)(cosb) + (sinb)(e^x).

A further simplification will give us:

g'(x) = (e^x)(cosb + sinb)

Thus, comparing g'(x) with f(x), we have: f(x) = (e^x)(sinb)

Therefore, comparing the two expressions, we can see that:

g'(x) = f(x) if and only if cosb + sinb = sinb.

This is true for all values of b, since:

We can then conclude that g(x) = (e^x)sinb is indeed an antiderivative of f(x) = (e^x)sinb.

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