a number b divided by 6 equals 15

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...

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

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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Answer:

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

Answer: 16

Step-by-step explanation:

1. Distribute the 12 to the x variable and the constant.

You will get 12x - 240 = -48

2. -48 + 240 = 192. You now have 12x = 192

3. solve for x. x = 192/12 -> x = 16

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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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!

Okay, here are the steps to solve this problem:

1) Savings start in year 3 at $10,000.

2) Savings increase by $1,000 each year for 5 years, so years 4-8 savings are $11,000, $12,000, $13,000, $14,000, $15,000.

3) There are 8 years of savings total.

4) Interest rate is 15%.

To calculate the present value at year 0 (today), we use the following formula:

PV = FV / (1 + r)^n

Where FV is the future value, r is the interest rate, and n is the number of years.

So for year 3 savings of $10,000:

PV = $10,000 / (1 + 0.15)^3 = $8,562

For year 4 savings of $11,000:

PV = $11,000 / (1 + 0.15)^4 = $8,948

And so on...

Adding up the present values for years 3 to 8:

$8,562 + $8,948 + $9,371 + $9,820 + $10,298 + $10,796 + $11,310 + $11,842 = $80,847

Therefore, the total present value of the 8 years of savings is $80,847.

So the software is worth $80,847 today at a 15% interest rate.

Let me know if you have any other questions!

The cost of renting the vehicle per mile is $0.524.

To get the cost per mile of renting a vehicle, multiply the total cost of renting the vehicle by the total number of miles driven.

The rental cost:

10 days multiplied by $29.95 per day equals $299.50.

The total rental cost is $299.50.

The mileage cost:

865 miles minus 500 free miles equals 365 miles.

365 miles multiplied by $0.25 each mile equals $91.25.

The total cost of miles is $91.25.

Finally, let's figure up the overall cost:

Total rental expense + total mileage expense + gasoline expense = $299.50 + $91.25 + $62 = $452.75

Let us now compute the cost per mile:

Cost per mile = total cost / total distance traveled = $452.75 / 865 miles = $0.524 per mile

Hence, the cost of renting the vehicle per mile is $0.524.

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Since, P(male)*P(fail)= 0.33 and P(male and fail) = 0.33 , the two results are equal and independent.

What is Probability?

The possibility of outcome of any event is found by probability. as for an example whenever we toss a coin in the air, then what is the possibility that we get a head? Only based on possible outcomes we can answer this question. It is that part of events which deals with the results of random events. We can basically call it as prediction of any event that is based on study of previous record or the type and no of possible outcome.

Probability of happening of an event= Total no of favorable outcomes/                          Total no of outcomes

Here in this question;

P(male/fail)= P(male and fail)/P(fail)

                 = [tex]\frac{8/123}{(8+16)/123}[/tex]

                 =[tex]\frac{8}{24} = \frac{1}{3} = 0.33[/tex]

P(male) = [tex]\frac{33+8}{123} = \frac{41}{123} =\frac{1}{3} = 0.33[/tex]

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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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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! :)

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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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.

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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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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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:

In order to determine a system of inequalities from a graph, these steps must be undertaken:

Initially, locate the boundary lines that form either solid or dashed sections of the graph. These lines serve as an illustration of the equations defining the bounds of the specified solution.

Next, it is important to identify which inequality symbol exists for each line on this graph. One can accomplish this by examining the shaded portion located within the region bounded by the particular line being considered.

Finally, once all relevant information has been gathered when assessing the inequality pairs against their respective graphical boundaries, you should study the resulting solution set closely. This collection represents the specific region depicted in the graph satisfying every provided inequality.

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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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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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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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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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The measure of angle or arc FDG is 285°.

A circle's circumference may be divided up into arcs. It is described as the curved line that divides a circle into segments by joining two points on its circumference. An arc's length is inversely proportional to the angle it forms at the circle's centre.

The length of the arc increases with increasing central angle, and vice versa. In geometry, trigonometry, and physics, arcs are frequently employed to represent and compute measurements of circles and circular objects.

For given problem,

mGC = 105° and

mFDG= 360° - mGF  (∵ Total arc or angle of circle is 360°)

           (As, mGF=mFC-mGC)

          = 360°-(180°-mGC) (Assuming FC is a diameter making mFC=180°)

          =  360°-(180°-105°)  (∵mGC = 105°)

          = 360°-75° = 285°

Hence, arc or angle FDG= 285 degrees.

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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.

Answer:

Euro

Step-by-step explanation:

1 EUR = 1.07 USD

Answer:

hello i d k the answer but im pretty sure the other guy is correct

Step-by-step explanation:

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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The probability that Anya will not be late is 0.7955.

a) Probability tree diagram:

     Car (0.35)       Bicycle (0.65)

     /                \

    /                  \

 Late (0.12)          Late (0.25)

/       \            /       \

Not Late (0.88)    Not Late (0.75)

b. To determine the likelihood that Anya will not be late:

P(No Tardiness) = P(No Tardiness | Car) * P(Car) + P(No Tardiness | Bicycle) * P(Bicycle)

P(No Tardiness) = (0.88 * 0.35) + (0.75 * 0.65) = 0.308 + 0.4875 = 0.7955

The probability that Anya will not be late is 0.7955.

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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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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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a. The boat is currently 60.5 miles from its pier.

b. The boat could have originally taken a bearing of either N45°E or S45°W to arrive at the current spot.

a.How to determine the distance from the boat's pier?

To determine the distance from the boat's pier, we can use the Pythagorean theorem since we have a right triangle formed by the boat's original position, the current position, and the distance traveled to reach the current position.

Let's call the distance from the boat's pier to the current position "d", and let's call the distance traveled after the turn "x".

Using trigonometry, we can see that:

sin(35°) = x/d

Rearranging this equation, we get:

x = d×sin(35°)

Now we can use the Pythagorean theorem to solve for "d":

d² = 29²+ (d×sin(35°))²

d² = 841 + 0.77d²

0.23d² = 841

d² = 3661.74

d = 60.5 miles (rounded to one decimal place)

Therefore, the boat is currently 60.5 miles from its pier.

b. How to determine the original bearing?

To determine the original bearing, we can use trigonometry again. Let's call the original bearing "θ". Then we have,

cos(θ) = 29/d

Rearranging this equation,

d = 29/cos(θ)

Now we can substitute this expression for "d" into the equation we used to solve for "x" earlier:

x = d×sin(35°)

x = (29/cos(θ))×sin(35°)

We know that after traveling this distance, the boat is currently 14.6 miles away from its original position. So we can set up another equation:

cos(θ) = 14.6/(29/cos(θ))

Simplifying, we get:

cos²(θ) = 0.5

Taking the inverse cosine of both sides,

θ = 45° or θ = 315°

Therefore, the boat could have originally taken a bearing of either N45°E or S45°W to arrive at the current spot.

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a. The boat is currently 60.5 miles from its pier.

b. The boat could have originally taken a bearing of either N45°E or S45°W to arrive at the current spot.

a.How to determine the distance from the boat's pier?

To determine the distance from the boat's pier, we can use the Pythagorean theorem since we have a right triangle formed by the boat's original position, the current position, and the distance traveled to reach the current position.

Let's call the distance from the boat's pier to the current position "d", and let's call the distance traveled after the turn "x".

Using trigonometry, we can see that:

sin(35°) = x/d

Rearranging this equation, we get:

x = d×sin(35°)

Now we can use the Pythagorean theorem to solve for "d":

d² = 29²+ (d×sin(35°))²

d² = 841 + 0.77d²

0.23d² = 841

d² = 3661.74

d = 60.5 miles (rounded to one decimal place)

Therefore, the boat is currently 60.5 miles from its pier.

b. How to determine the original bearing?

To determine the original bearing, we can use trigonometry again. Let's call the original bearing "θ". Then we have,

cos(θ) = 29/d

Rearranging this equation,

d = 29/cos(θ)

Now we can substitute this expression for "d" into the equation we used to solve for "x" earlier:

x = d×sin(35°)

x = (29/cos(θ))×sin(35°)

We know that after traveling this distance, the boat is currently 14.6 miles away from its original position. So we can set up another equation:

cos(θ) = 14.6/(29/cos(θ))

Simplifying, we get:

cos²(θ) = 0.5

Taking the inverse cosine of both sides,

θ = 45° or θ = 315°

Therefore, the boat could have originally taken a bearing of either N45°E or S45°W to arrive at the current spot.

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