Please i give 25 points

Answer:

Step-by-step explanation:

a) To solve the equation tan θ = 1/√3, we can find the angle whose tangent is 1/√3 by taking the inverse tangent (arctan) of 1/√3.

θ = arctan(1/√3)

θ ≈ 30.0°

Therefore, the angle in standard position that satisfies tan θ = 1/√3 is approximately 30.0°.

b) To solve the equation 2cos θ = √3, we can isolate the cosine term by dividing both sides of the equation by 2.

cos θ = √3 / 2

Now, we can find the angle whose cosine is √3/2 by taking the inverse cosine (arccos) of √3/2.

θ = arccos(√3/2)

θ ≈ 30.0°

Therefore, the angle in standard position that satisfies 2cos θ = √3 is approximately 30.0°.

To find the area of a rectangle, we need to know its length and width. Let's solve the problem step by step:

Let's assume that the width of the rectangle is represented by "w" (in feet).

According to the given information, the length of the rectangle is 4 feet longer than its width, which means the length can be represented as "w + 4" (in feet).

The perimeter of a rectangle is calculated by adding up all the sides. In this case, the perimeter is given as 32 feet.

Since a rectangle has two pairs of equal sides (length and width), we can express the perimeter equation as the following:

2(length + width) = perimeter

Substituting the values into the equation, we get:

2(w + (w + 4)) = 32

Simplifying the equation, we have:

2(2w + 4) = 32

4w + 8 = 32

4w = 24

w = 6

Now we know that the width of the rectangle is 6 feet. To find the length, we can substitute this value back into the equation for the length:

Length = w + 4 = 6 + 4 = 10 feet

The width is 6 feet, and the length is 10 feet. Now we can calculate the area of the rectangle:

Area = Length × Width = 10 × 6 = 60 square feet

Answer: The area of the rectangle is 60 square feet.

The graph of the function and its inverse is added as an attachment

From the question, we have the following parameters that can be used in our computation:

f(x) = 3 + 2ln(x)

Express as an equation

So, we have

y = 3 + 2ln(x)

Swap x and  y in the above equation

x = 3 + 2ln(y)

Next, we have

2ln(y) = x - 3

Divide by 2

ln(y) = (x - 3)/2

Take the exponent of both sides

[tex]y = e^{\frac{x - 3}{2}}[/tex]

Next, we plot the graphs

The graph of the functions is added as an attachment

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The estimated population of the state in the year 2022 is 19,280,700 people.

The given equation represents the population of a certain state as a function of time, where p is the population in thousands of people and t is the number of years since 2010.

The equation is given as p = 80.7t + 18,312.3.

To estimate the population of the state, we substitute the value of t into the equation. For example, if we want to estimate the population in the year 2022 (12 years since 2010), we substitute t = 12 into the equation:

p = 80.7(12) + 18,312.3

= 968.4 + 18,312.3

= 19,280.7.

The estimated population of the state in the year 2022 is 19,280,700 people.

We can estimate the population for any given year by substituting the corresponding value of t into the equation.

It's important to note that the population is given in thousands of people, so we multiply the final result by 1,000 to obtain the population in actual numbers.

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Answer:

When p, the probability of success, is zero in a binomial distribution, the probability of getting exactly k successes in n trials is also zero for all values of k except when k is zero (i.e., when there are no successes).

So, in the case of (0.3)^0, the result would be 1, because any number raised to the power of 0 is equal to 1. Therefore, the probability of getting zero successes in a binomial distribution when the probability of success is 0.3 is 1.

Answer:

BC = 24 units

Step-by-step explanation:

This is an isosceles triangle which always has:

In this triangle, the legs CA and BA are congruent so CA = BA and the angles C and B are congruent to each other so angle C = angle B.

Thus, we can find x by setting CA and BA equal to each other:

(3x - 15 = x + 33) + 15

(3x = x + 48) - x

(2x = 48) / x

x = 24

Thus, x = 24

Since the length of BC is x and x = 24, BC is 24 units long.

The function for which f(x) is equal to f⁻¹(x) is: C. [tex]f(x) = \frac{x+1}{x-1}[/tex]

In this exercise, you are required to determine the inverse of the function f(x) with an equivalent inverse function f⁻¹(x). This ultimately implies that, we would have to swap (interchange) both the independent value (x-value) and dependent value (y-value) as follows;

[tex]f(x) = y = \frac{x+6}{x-6} \\\\x=\frac{y+6}{y-6}[/tex]

x(y - 6) = y + 6

y = xy - 6x - 6

f⁻¹(x) = (-6x - 6)/(x - 1) ⇒ Not equal.

Option B.

[tex]f(x) = y = \frac{x+2}{x-2} \\\\x=\frac{y+2}{y-2}[/tex]

x(y - 2) = y + 2

y = xy - 2x - 2

f⁻¹(x) = (-2x - 2)/(x - 1) ⇒ Not equal.

Option C.

[tex]f(x) = y = \frac{x+1}{x-1} \\\\x=\frac{y+1}{y-1}[/tex]

x(y - 1) = y + 1

y - xy = x + 1

f⁻¹(x) = (x + 1)/(x - 1) ⇒ equal.

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Answer:

$6.84

Step-by-step explanation:

This is quite a simple question, simply add the new deposited amount into the original balance to get your answer.

The equivalent ratio of the corresponding sides and the triangle proportionality theorem indicates that the similar triangles are;

1. ΔAJK ~ ΔSWY according to the SAS similarity postulate

2. ΔLMN ~ ΔLPQ according to the AA similarity postulate

3. ΔPQN ~ ΔLMN

LM = 12, QP = 8

4. ΔLMK~ΔLNJ

NL = 21, ML = 14

Similar triangles are triangles that have the same shape but may have different sizes.

1. The ratio of corresponding sides between the two triangles circumscribing the congruent included angle are;

24/16 = 3/2

18/12 = 3/2

The ratio of each of the two sides in the triangle ΔAJK to the corresponding sides in the triangle ΔSWY are equivalent and the included angle, therefore, the triangles ΔAJK and ΔSWY are similar according to the SAS similarity rule.

2. The ratio of the corresponding sides in each of the triangles are;

MN/LN = 8/10 = 4/5

PQ/LQ = 12/(10 + 5) = 12/15 = 4/5

The triangle proportionality theorem indicates that the side MN and PQ are parallel, therefore, the angles ∠LMN ≅ ∠LPQ and ∠LNM ≅ ∠LQP, which indicates that the triangles ΔLMN and ΔLPQ are similar according to the Angle-Angle AA similarity rule

3. The alternate interior angles theorem indicates;

Angles ∠PQN ≅ ∠LMN and ∠MLN ≅ ∠NPQ, therefore;

ΔPQN ~ ΔLMN by the AA similarity postulate

LM/QP = (x + 3)/(x - 1) = 18/12

12·x + 36 = 18·x - 18

18·x - 12·x = 36 + 18 = 54

6·x = 54

x = 54/6 = 9

LM = 9 + 3 = 12

QP = x - 1

QP = 9 - 1 = 8

4. The similar triangles are; ΔLMK and ΔLNJ

ΔLMK ~ ΔLNJ by AA similarity postulate

ML/NL = (6·x + 2)/(6·x + 2 + (x + 5)) = (6·x + 2)/((7·x + 7)

ML/NL = LK/LJ = (24 - 8)/24

(24 - 8)/24 = (6·x + 2)/((7·x + 7)

16/24 = (6·x + 2)/(7·x + 7)

16 × (7·x + 7) = 24 × (6·x + 2)

112·x + 112 = 144·x + 48

144·x - 112·x = 32·x = 112 - 48 = 64

x = 64/32 = 2

ML = 6 × 2 + 2 = 14

NL = 7 × 2 + 7 = 21

MN = 2 + 5 = 7

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Answer:

x=50

Step-by-step explanation:

Make this equal to 180.

x+3x-35+x-35 = 180

5x = 180 + 70

5x=250

x=50

Answer:

KL = 6

Step-by-step explanation:

We see that the length of IL includes IJ, JK, and Kl and is 26.

Since IL = 26 and IJ + JK + KL = IL, we can subtract the sum of the lengths of IJ and Jk from IL to find KL:

IL = IJ + JK + KL

26  = 9 + 11 + KL

26 = 20 + KL

6 = KL

Thus, the length of KL is 6.

We can confirm this fact by plugging in 6 for KL and checking that we get 26 on both sides of the equation when simplifying:

IL = IJ + JK + KL

26 = 9 + 11 + 6

26 = 20 + 6

26 = 26

Thus, our answer is correct.

The measure of the numbered angles in the rhombus is determined as angle 1 = 90⁰, angle 2 = 57⁰, angle 3 = 45⁰, and angle 4 = 45⁰.

The measure of the numbered angles is calculated by applying the following formula as follows;

Rhombus has equal sides and equal angles.

angle 2 = angle 57⁰ (alternate angles are equal)

angle 1 = 90⁰ (diagonals of rhombus intersects each other at 90⁰)

angle 3 = angle 4 (base angles of Isosceles triangle )

angle 3 = angle 4 = ¹/₂ x 90⁰

angle 3 = angle 4 = 45⁰

Thus, the measure of the numbered angles in the rhombus is determined as angle 1 = 90⁰, angle 2 = 57⁰, angle 3 = 45⁰, and angle 4 = 45⁰.

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Step-by-step explanation:

Common difference , d, is  9

Sn = n/2 ( a1 + a33)                a33 = a1 + 32d = 2 + 32(9) = 290

S33 = 33/2 ( 2+290) = 4818

The required answers of the given questions are answered below.

a), b), c),  d)

A quadrilateral is geometric structure enclosed in 4 sides.

A) Because,

 1) the angle A = Angle D

 2) its has a common side

 3) the angle D = Angle B

 ∵ ΔEAB ≅ EDC

Thus, Both the triangle is congruent by ASA.

B.)

Jane conclusion is also write for AAA

because by angle E is congruent to angle B, implies angle F is congruent to angle C.

So that is why Jane conclude AAA.

C.) George conclusion is also perfect

George thinks the following conditions-

1) the angle A = Angle D

 2) its has a common side

 3) the angle D = Angle B

 ∵ ΔEAB ≅ EDC

Thus, Both the triangle is congruent.

Hence, Both the triangles are congruent by SSS.

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The completed sequence would then be: 2, 4, 7, 9, 16, 19, 29.

To complete the given number sequence, let's analyze the pattern and identify the missing terms.

Looking at the given sequence 2, 4, 7, __, 16, __, 29, __, we can observe the following pattern:

The difference between consecutive terms in the sequence is increasing by 1. In other words, the sequence is formed by adding 2 to the previous term, then adding 3, then adding 4, and so on.

Using this pattern, we can determine the missing terms as follows:

To obtain the third term, we add 2 to the second term:

7 + 2 = 9

To find the fifth term, we add 3 to the fourth term:

16 + 3 = 19

To determine the seventh term, we add 4 to the sixth term:

__ + 4 = 23

Therefore, the missing terms in the sequence are 9, 19, and 23.

By identifying the pattern of increasing differences, we can extend the sequence and fill in the missing terms accordingly.

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Answer:

$4929

Step-by-step explanation:

I assume the cost is proportional to the area.

15 ft × 20 ft = 300 ft²

25 ft × 26 ft = 650 ft²

650/300 = x/$2275

300x = 650 × $2275

x = $4929

Answer: $4929

The limit of the trigonometric expression is equal to 0.

In this problem we find the case of a trigonometric expression, whose limit must be found. This can be done by means of algebra properties, trigonometric formula and known limits. First, write the entire expression below:

[tex]\lim_{\Delta x \to 0} \frac{\cos (\pi + \Delta x) + 1}{\Delta x}[/tex]

Second, use the trigonometric formula cos (π + Δx) = - cos Δx to simplify the resulting formula:

[tex]\lim_{\Delta x \to 0} \frac{1 - \cos \Delta x}{\Delta x}[/tex]

Third, use known limits to determine the result:

0

The limit of the trigonometric function [cos (π + Δx) + 1] / Δx evaluated at Δx → 0 is equal to 0.

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Answer:A' = (15, -15), B' = (9, -9), and C' = (15, -9)

Step-by-step explanation:

To dilate triangle ABC with a center of dilation at the origin (0,0) and a scale factor of 3, you need to multiply the coordinates of each vertex by the scale factor.

Let's calculate the coordinates of triangle A'B'C':

For point A:

x-coordinate of A' = scale factor * x-coordinate of A = 3 * 5 = 15

y-coordinate of A' = scale factor * y-coordinate of A = 3 * (-5) = -15

Therefore, A' = (15, -15)

For point B:

x-coordinate of B' = scale factor * x-coordinate of B = 3 * 3 = 9

y-coordinate of B' = scale factor * y-coordinate of B = 3 * (-3) = -9

Therefore, B' = (9, -9)

For point C:

x-coordinate of C' = scale factor * x-coordinate of C = 3 * 5 = 15

y-coordinate of C' = scale factor * y-coordinate of C = 3 * (-3) = -9

Therefore, C' = (15, -9)

Hence, the correct coordinates of triangle A'B'C' are A' = (15, -15), B' = (9, -9), and C' = (15, -9).

Answer:

Let's denote the height of the triangle as "h" inches.

According to the given information, the base of the triangle is 3 inches more than two times the height. Therefore, the base can be expressed as (2h + 3) inches.

The formula to calculate the area of a triangle is:

Area = (1/2) * base * height

Substituting the given values, we have:

7 = (1/2) * (2h + 3) * h

To simplify the equation, let's remove the fraction by multiplying both sides by 2:

14 = (2h + 3) * h

Expanding the right side of the equation:

14 = 2h^2 + 3h

Rearranging the equation to bring all terms to one side:

2h^2 + 3h - 14 = 0

Now, we can solve this quadratic equation. We can either factor it or use the quadratic formula. In this case, let's use the quadratic formula:

h = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, the values are:

a = 2

b = 3

c = -14

Substituting these values into the quadratic formula:

h = (-3 ± √(3^2 - 4 * 2 * -14)) / (2 * 2)

Simplifying:

h = (-3 ± √(9 + 112)) / 4

h = (-3 ± √121) / 4

Taking the square root:

h = (-3 ± 11) / 4

This gives us two possible solutions for the height: h = 2 or h = -14/4 = -3.5.

Since a negative height doesn't make sense in this context, we discard the negative solution.

Therefore, the height of the triangle is h = 2 inches.

To find the base, we substitute this value back into the expression for the base:

base = 2h + 3

base = 2(2) + 3

base = 4 + 3

base = 7 inches

Hence, the base of the triangle is 7 inches and the height is 2 inches.

Step-by-step explanation:

-The answer for the height is 5.5 units.

-The base of the triangle is aproximately 2.5454 units.

To answer this problem, you have to set an equation with the information you're given. If you do it correctly, it should look like this:

7=1/2(3+2h)

-Now, you have to solve for h:

7=1.5+h

7-1.5=h

5.5=h

-Now that you have the height, you plug it in into the triangle area formula to solve for the base:

7=1/2(b)5.5

7=2.75b

7/2.75=b

b≈2.5454

-To make sure that the corresponding values for the base and height are correct, we plug the values in and this time we are going to solve for a(AREA):

A(triangle)=1/2(2.5454)(5.5)

A=1/2(13.9997)

A=6.99985 square units

-We round the result to the nearest whole number and we get our 7, which is the given value they gave us.

Answer:

113.1 is the answer. I used the arbitory height of 4 so the volume of both are now 314.16 and 201.06

314.16-201.06=113.1

The net area of the curve represented by f(x) = ex - e on the interval [0, 2] is e2 - 1.

To find the net area of the curve represented by the function f(x) = ex - e on the interval [0, 2], we need to calculate the definite integral of the function over that interval. The net area can be determined by taking the absolute value of the integral.

The integral of f(x) = ex - e with respect to x can be computed as follows:

∫[0, 2] (ex - e) dx

Using the power rule of integration, the antiderivative of ex is ex, and the antiderivative of e is ex. Thus, the integral becomes:

∫[0, 2] (ex - e) dx = ∫[0, 2] ex dx - ∫[0, 2] e dx

Integrating each term separately:

= [ex] evaluated from 0 to 2 - [ex] evaluated from 0 to 2

= (e2 - e0) - (e0 - e0)

= e2 - 1

The net area of the curve represented by f(x) = ex - e on the interval [0, 2] is e2 - 1.

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Answer:

all real numbers

Step-by-step explanation:

Try a negative number, a positive number and zero for x.

All of them work.

Answer: all real numbers

We can solve for the missing elements as follows:

1. Radius - 10 inches

Diameter - 20

Circumference -  62.8

Area - 314

2.  Radius  - 6ft

Diameter - 12

Circumference - 37.68

Area - 113.04

3.  Radius - 18

Diameter - 36 yards

Circumference - 113.04

Area - 1017.36

4.  Radius 15

Diameter - 30 cm

Circumference 94.2

Area - 706.5

5.  Radius - 5 mm

Diameter 10

Circumference 31.4

Area -78.5

6. Radius 20

Diameter - 40 inches

Circumference  125.6

Area -1256

To solve for the given values, we will use the formulas for area, circumference. Also, we can obtain the radius by dividing the diameter by 2 and the diameter is 2r. So we will solve for the values this way:

1. radius = 10 inches

diameter = 20

circumference = 2pie*r 2 *3.14*10 = 62.8

Area = 314

2. radius = 6ft

diameter = 12

circumference = 37.68

Area = 113.04

3. radius = 18

diameter = 36 yards

circumference = 113.04

Area = 1017.36

4. radius = 15

diameter = 30 cm

circumference = 94.2

Area = 706.5

5. radius = 5 mm

diameter = 10

circumference = 31.4

Area = 78.5

6. radius = 20 inches

diameter = 40 inches

circumference = 125.6

area = 1256

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Answer:

180 meters

Step-by-step explanation:

To find the perimeter of the actual swimming pool, you need to first find the length and width of the actual swimming pool by multiplying the length and width of the pictured swimming pool by the scale factor of 1200 cm.

Now that we know that the perimeter of the actual pool is 18000 centimeters, we need to convert that to meters! Keep in mind that:

Now we can divide 18000 by 100:

18000 cm ÷ 100 = 180 m

Therefore, the perimeter of the actual swimming pool is 180 m.

Answer:x=-28

Step-by-step explanation:

Distribute the 4 on the left side of the equation:

32 - 8x = 256

Move the constant term to the right side of the equation:

-8x = 256 - 32

-8x = 224

Divide both sides of the equation by -8 to isolate x:

x = 224 / -8

x = -28

Answer:

In a 30°-60°-90° triangle, the length of the longer leg is √3 times the length of the shorter leg. So AC = 80√3.

In a 45°-45°-90° triangle, both legs are congruent. So BC = 80.

AB = AC - BC = (80√3 - 80) meters

= 80(√3 - 1) meters

= about 58.56 meters

The distance between points A and B is approximately 138.6 meters.

To find the distance between points A and B, we can use the concept of trigonometry and the given information.

Let's denote the distance between points A and B as x.

From point A, the angle of elevation to the top of the cliff at point D is 30°. This means that in the right triangle formed by points A, D, and the top of the cliff, the opposite side is the height of the cliff (80.0 m) and the adjacent side is x. We can use the tangent function to calculate the length of the adjacent side:

tan(30°) = opposite/adjacent

tan(30°) = 80.0/x

Simplifying the equation, we have:

x = 80.0 / tan(30°)

Using a calculator, we can find the value of tan(30°) ≈ 0.5774.

Substituting the value, we get:

x = 80.0 / 0.5774

Calculating the value, we find:

x ≈ 138.6 meters

In light of this, the separation between positions A and B is roughly 138.6 metres.

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