find the area that the curve encloses and then sketch it. r2 = 9 cos(2θ)

Answer:

Step-by-step explanation:

Let's factor the expression w^2(w+8)-5(w+8):

First, we can see that (w+8) is a common factor in both terms of the expression.

So, we can factor out (w+8) from both terms:

(w+8)(w^2 - 5)

Now, the expression is factored as (w+8)(w^2 - 5).

No, 34.05 is not greater than 34.59.

When comparing two numbers, we look at their digits from left to right.

In this case, both numbers start with 34, which means they have the same value in the tens place.

However, when we move to the decimal part, 0.05 is smaller than 0.59.

Therefore, 34.05 is smaller than 34.59.

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The area of the semicircle is 47.5 square centimeters

We have to find the area of the semi circle

The diameter of the semicircle is 11 cm

Radius is half of the diameter

Radius = 5.5 cm

Let us find the area of semicircle  by formula 1/2(πr²)

Radius = 1/2×3.14×(5.5)²

= 1/2×3.14×30.25

=47.4925 square centimeters

Hence,  the area of the semicircle is 47.5 square centimeters

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The angle measure of LJK is 27 degrees

Following the triangle sum theorem, we have that the sum of the interior angles of a triangle is 180 degrees

Also, we need to know that the sum of angles on a straight line is 180, then, we have;

<JLK = 180 - <LKM = 180 - (8x - 19)

Then, substitute the value, we have that;

<JKL + < JLK + < KJL = 180

Then,

3x - 16 + (180 - (8x - 19)) + 2x + 15 = 180

expand the bracket, we have;

3x - 16 - 8x - 19 + 2x + 15 = 0

add the like terms

-3x + 18 = 0

collect the terms

-3x = -18

x = 6

Then, the angle LJK = 2x + 15 = 2(6) + 15 = 27 degrees

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Answer: 187 apples

Step-by-step explanation:

Substitute "year 5" into the equation. y = 15.2(5) +111.

y = 76 + 111

y = 187

The length of fencing needed around her triangular garden is 41 feet.

A given shape which has three sides and three measures of internal angles which add up to 180^o is said to be a triangle.

For Mary to build a fence around her triangular garden, the amount of fencing needed can be determined by adding the length of each side of the garden.

So that;

A + B + C = 180^o

A + 49 + 40 = 180

A = 180 - 89

   = 91

A = 91^o

Applying the sine rule, we have;

a/Sin A = b/Sin B = c/Sin C

a/Sin A = b/Sin B

a/Sin 91 = 13/ Sin 49

aSin 49 = 13*Sin 91

             = 12.998

a = 12.998/ 0.7547

  = 17.22

a = 17 ft

Also,

b/Sin B = c/Sin C

13/Sin 49 = c/ Sin 40

cSin 49 = 13*Sin 40

             = 8.3562

c = 8.3562/ 0.7547

  = 11.0722

c = 11 feet

Thus the amount of fencing required = 13 + 17 + 11

                                                    = 41

The fencing required is 41 feet length.

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The required solution of the random variable X that follows an exponential distribution is shown below.

To find the expected value, standard deviation, and probabilities associated with the exponential distribution, we'll use the given parameter λ = 1/9.848.

Expected Value (Mean):

The expected value of an exponential distribution is equal to the reciprocal of the rate parameter λ. Therefore, the expected value is given by:

E(X) = 1 / λ

E(X) = 1 / (1/9.848)

E(X) = 9.848

So, the expected value of the random variable X is 9.848.

Standard Deviation:

The standard deviation of an exponential distribution is equal to the reciprocal of the rate parameter λ. Therefore, the standard deviation is given by:

σ = 1 / λ

σ = 1 / (1/9.848)

σ = 9.848

So, the standard deviation of the random variable X is also 9.848.

P(X < 12):

To find the probability that X is less than 12, we can use the cumulative distribution function (CDF) of the exponential distribution. The CDF is given by:

[tex]F(x) = 1 - e^{(-\lambda x)}[/tex]

Substituting the given λ = 1/9.848 and x = 12, we have:

[tex]P(X < 12) = F(12) = 1 - e^{-(1/9.848) * 12)}[/tex]

Calculating this expression, we find:

P(X < 12) ≈ 0.7043

Therefore, the probability that X is less than 12 is approximately 0.7737.

P(8 < X < 12):

To find the probability that X lies between 8 and 12, we subtract the cumulative probability at 8 from the cumulative probability at 12:

P(8 < X < 12) = F(12) - F(8)
                     = [tex]1 - e^{(-(1/9.848) * 12)]} - [1 - e^{(-(1/9.848) * 8)]}[/tex]

Calculating this expression, we find:

P(8 < X < 12) ≈ 0.2025

Therefore, the probability that X lies between 8 and 12 is approximately 0.2025.

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The actual length of the bus is given as follows:

45 feet.

The actual length of the bus is obtained applying the proportions in the context of the problem.

The scale drawing is of 0.5 inches to 5 feet, hence the length represented by each inch of drawing is given as follows:

5/0.5 = 10 feet.

The length of the drawing is given as follows:

4.5 inches.

Hence the actual length of the bus is given as follows:

4.5 x 10 = 45 feet.

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With a triangle, the sum of any two side lengths must be greater than the third side length. If this is not true, then the side lengths cannot make a triangle. Let's go through each set of side lengths and determine which would and wouldn't work.

a. 3, 4, 8 - will not make a triangle

3 + 4 = 7 > 8 = false

3 + 8 = 11 > 4 = true

4 + 8 = 12 > 3 = true

b. 7, 6, 12 - will make a triangle

7 + 6 = 13 > 12 = true

7 + 12 = 19 > 6 = true

6 + 12 = 18 > 7 = true

c. 5, 11, 13 - will make a triangle

5 + 11 = 16 > 13 = true

5 + 13 = 18 > 11 = true

11 + 13 = 24 > 5 = true

d. 4, 6, 12 - will not make a triangle

4 + 6 = 10 > 12 = false

4 + 12 = 16 > 6 = true

6 + 12 = 18 > 4 = true

e. 4, 6, 10 - will not make a triangle

4 + 6 = 10 > 10 = false

4 + 10 = 14 > 6 = true

6 + 10 = 16 > 4 = true

Hope this helps!

Answer:

[tex] \sin(x) + \sin(2x) = 0[/tex]

[tex] \sin(x) + 2 \sin(x) \cos(x) = 0[/tex]

[tex]( \sin(x) )(1 + 2 \cos(x) ) = 0[/tex]

sin(x) = 0 or 1 + 2cos(x) = 0

2cos(x) = -1

cos(x) = -1/2

x = kπ or x = 2π/3 + 2kπ or

x = 4π/3 + 2kπ

(k is an integer)

Answer:

The angle in the first problem measures 67°.

x=16, y=40

Step-by-step explanation:

1st problem: draw the line r in figure (red) parallel to the two given one and containing the vertex of the angle. The two green angles are congruent since alternate angles from the two parallels t and r when crossed with the upper side of the angle. Same with the two blue angles created by the parallels r and b. The measure of the angle you need is the sum of the two, or 36+31= 67°.

Second problem. The two angles in red are congruent since corresponding angles generated by the parallels m and b and the transversal r. we can then write [tex]4x-5=3x+11\\[/tex]. Solving for x we find x= 16. Now the two angles marked in blue are conjugates when looking at the parallels l and r cut by the transversal m, thus they add up to 180° We know how much the angle with double marks measures (since we found x earlier) so we can write

[tex](3y+1) + (4\times16-5) = 180\\3y+ 1 + 59 = 180\\3y=120 \implies y=40[/tex]

Answer:

Step-by-step explanation:

Total cost function, C = 15+9x-6x²+x

By differentiation,

dc/dx= 9-12x²+1

but

dc/dx= 0

9-12x²+1 = 0

-12x = -10

x=10/12

x=5/6

The complete statement should be If 3m = 7n, then m/n = 7/3. The proportion was obtained by solving for m, and then using the converse of the cross products property. Option B

The converse of the cross products property states that if two ratios are equal, then the product of the means is equal to the product of the extremes.

To justify the answer, you solve for m/n in the original equation by dividing each side by 3n, which gives you m/n = 7/3.

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Given statement solution is :- Outputs and values g(2/3) = 2/3.

To summarize:

g(0) = 0

g(-1) = undefined

g(2) = undefined

g(2/3) = 2/3

To find the values of g(x) for the given inputs, we substitute each input into the function g(x) = x / √([tex]1 - x^2[/tex]). Let's calculate the values:

g(0):

Substitute x = 0 into the function:

g(0) = 0 / √([tex]1 - 0^2[/tex])

= 0 / √(1 - 0)

= 0 / √1

= 0

Therefore, g(0) = 0.

g(-1):

Substitute x = -1 into the function:

g(-1) = (-1) / √(1 - [tex](-1)^2[/tex])

= (-1) / √(1 - 1)

= (-1) / √0

Since the square root of 0 is undefined, g(-1) is undefined.

g(2):

Substitute x = 2 into the function:

g(2) = 2 / √([tex]1 - 2^2[/tex])

= 2 / √(1 - 4)

= 2 / √(-3)

Since the square root of a negative number is undefined in the real number system, g(2) is undefined.

g(2/3):

Substitute x = 2/3 into the function:

g(2/3) = (2/3) / √(1 - [tex](2/3)^2[/tex])

= (2/3) / √(1 - 4/9)

= (2/3) / √(5/9)

= (2/3) / (√5/√9)

= (2/3) / (√5/3)

= (2/3) * (3/√5)

= 2√5 / 3√5

= 2/3

Therefore, outputs and values g(2/3) = 2/3.

To summarize:

g(0) = 0

g(-1) = undefined

g(2) = undefined

g(2/3) = 2/3

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Answer:

95 × 10^3 = 95000

95 × 10^4 = 950000

95 × 10^5 = 9500000

The average rate of change for this problem is given as follows:

2.4 inches per hour.

The average rate of change of a function is given by the change in the output of the function divided by the change in the input of the function.

The change in the output is given as follows:

8.1 - 3.3 = 4.8.

The change in the input is given as follows:

2 - 0 = 2.

Hence the average rate of change is given as follows:

4.8/2 = 2.4 inches per hour.

The table is given by the image presented at the end of the answer.

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For calculating the percentage of discount, we can use the formula:

Percentage of discount = (Discount amount / Original price) * 100

We have given that the discounted price is Birr 51,000 and the discount amount is Birr 9,000,we need to find the original price.

Original price = Discounted price + Discount amount

Original price = 51,000 + 9,000 = 60,000 Birr

Now, we can calculate the percentage of discount:

Percentage of discount = (9,000 / 60,000) * 100 = 15%

Hence, the percentage of discount for the four-door model is 15%.

Answer:9/41

SOH = opposite over hypotenuse

Step-by-step explanation:

We have that Linda's net and expression incorrect

With the Expression of the surface area of attic

45 (40 + 25 + 25) + 1/2 (40 x 15)

Hence, Surface area is,

X = 4050 + 600

X = 4650 feet²

Hence, The answer is No, Because Her expression for the triangles are incorrect;

1/2(40 x 15)

Instead of,

1/2 x 40 x 15

Hence, Linda's net and expression incorrect,

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Linda is adding padding to all of the surfaces inside her attic for extra warmth in the winter.

She needs to find the approximate surface area of the attic, including the walls, floor, and

ceiling. The attic is in the shape of a triangular prism. Linda draws the net and writes

the expression below to represent the surface area of the attic. Are Linda's net and

expression correct?

Answer:

To determine the sample size required to estimate the proportion of consumers who have seen the company's advertisement in the newspaper, we can use the formula for sample size calculation for proportions. The formula is as follows:

n = (Z^2 * p * q) / E^2

Where:

n = required sample size

Z = Z-score corresponding to the desired confidence level (in this case, 98% confidence corresponds to a Z-score of approximately 2.33)

p = estimated proportion (0.65)

q = 1 - p (the complement of the estimated proportion)

E = maximum error tolerance (+/- 0.05)

Let's plug in the values and calculate the sample size:

n = (2.33^2 * 0.65 * 0.35) / (0.05^2)

n = 339.28

Rounding up to the nearest whole number, the required sample size is 340.

Answer: x=25 degrees

Step-by-step explanation:

25 degrees (blue) and x degrees (orange) are vertical angles. Vertical angles always equal each other.

(80 degrees {green} is just there to throw you off)

14. The measure of angle ABC is 90⁰.

15. The measure of angle CED is 43⁰.

16. The measure of angle BDE is 47.5⁰.

17. The measure of angle CBD is 43⁰.

18. The measure of arc AD is 94⁰.

19. The measure of angle BCE is 47.5⁰.

20. The measure of angle ABD is 47⁰.

21.  The measure of arc ABC is 180⁰.

The measure of the missing angles is calculated by applying intersecting chord theorem, which states that the angle at tangent is half of the arc angle of the two intersecting chords.

The measure of angle ABC is calculated as follows;

m∠ABC = 90⁰ (since line AC is the diameter)

The measure of angle CED is calculated as follows;

m∠CED = ¹/₂ (arc CD) (interior angle of intersecting secants)

m∠CED = ¹/₂ (86⁰) = 43⁰

The measure of angle BDE is calculated as follows;

m∠BDE= ¹/₂ (arc BE) (interior angle of intersecting secants)

m∠BDE = ¹/₂ (95⁰) = 47.5⁰

The measure of angle CBD is calculated as follows;

m∠CBD = ¹/₂ (arc CD) (interior angle of intersecting secants)

m∠CBD = ¹/₂ (86⁰) = 43⁰

The measure of arc AD is calculated as follows;

arc ABC = 180 (sum of angles in a semi circle)

arc ADC = 180 (sum of angles in a semi circle)

arc ADC = arc AD + arc CD

180 = AD + 86

AD = 94⁰

The measure of angle BCE is calculated as follows;

m∠BCE= ¹/₂ (arc BE) (interior angle of intersecting secants)

m∠BCE = ¹/₂ (95⁰) = 47.5⁰

The measure of angle ABD is calculated as follows;

m∠ABD = ¹/₂ (arc AD) (interior angle of intersecting secants)

m∠ABD = ¹/₂ (94⁰) = 47⁰

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Answer:

8

Step-by-step explanation:

3x - 10 = x + 2

2x = 12

x = 6

3x - 10 = 3(6) - 10 = 8

The length of each leg of the triangle is 8.

The standard form (SF), center (C) and radius (R) are given as follow: (13) SF:  (x + 4)² + (y - 3)² = 50, Center: (-4, 3), R: √50 (14) SF: (x - 2)² + (y - 6)² = 169, C: (2, 6), R: 13 (15) SF: (x + 7)² + (y + 5)² = 1, C: (-7, -5), R: 1 (16) SF: (x - 8)² + y² = 225, C: (8,0), R: 15 (17) SF: (x - 12)² + (y - 2)² = 63, C: (12, 2), R: √63 (18) SF: (x - 5)² + (y + 4)² = 100 , C: (5, -4), R: 10

The general form of a circle is given as:

       (x - h)² + (y - k)² = r²

where:

        (h, k) represents the center of the circle

       r represents the radius.

Now we can use the above information to solve the following questions:

13. x² + y² + 8x - 6y - 25 = 0

Rearranging the equation:

x² + 8x + y² - 6y = 25

Completing the square for x terms:

(x² + 8x + 16) + y² - 6y = 25 + 16

Simplifying:

(x + 4)² + (y² - 6y) = 41

(x + 4)² + (y² - 6y + 9) = 41 + 9

(x + 4)² + (y - 3)² = 50

Center: (-4, 3)

Radius: √50

14. x² + y² - 4x - 12y - 129 = 0

Rearranging the equation:

x² - 4x + y² - 12y = 129

Completing the square for x terms:

(x² - 4x + 4) + y² - 12y = 129 + 4

Simplifying:

(x - 2)² + (y² - 12y) = 133

(x - 2)² + (y² - 12y + 36) = 133 + 36

(x - 2)² + (y - 6)² = 169

Center: (2, 6)

Radius: 13

15. x² + y² + 14x + 10y + 73 = 0

Rearranging the equation:

x² + 14x + y² + 10y = -73

Completing the square for x terms:

(x² + 14x + 49) + y² + 10y = -73 + 49

Simplifying:

(x + 7)² + (y² + 10y) = -24

(x + 7)² + (y² + 10y + 25) = -24 + 25

(x + 7)² + (y + 5)² = 1

Center: (-7, -5)

Radius: 1

16. x² + y² - 16x - 161 = 0

Rearranging the equation:

x² - 16x + y² = 161

Completing the square for x terms:

(x² - 16x + 64) + y² = 161 + 64

Simplifying:

(x - 8)² + y² = 225

Center: (8, 0)

Radius: 15

17. x² + y² = 24x + 4y - 85

Rearranging the equation:

x² - 24x + y² - 4y = -85

Completing the square for x and y terms:

(x² - 24x + 144) + (y² - 4y + 4) = -85 + 144 + 4

Simplifying:

(x - 12)² + (y - 2)² = 63

Center: (12, 2)

Radius: √63

18. x² + y² - 9x + 2y = x - 6y + 59

Rearranging the equation:

x² - 9x - x + y² + 2y + 6y = 59

Combining like terms:

x² - 10x + y² + 8y = 59

Completing the square for x and y terms:

(x² - 10x + 25) + (y² + 8y + 16) = 59 + 25 + 16

Simplifying:

(x - 5)² + (y + 4)² = 100

Center: (5, -4)

Radius: 10

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The regular-size granola bar is 1 1/3 inches longer than the snack-size granola bar.

The regular-size granola bar has a volume of 4 cubic inches, while the snack-size bar has a volume of 3 cubic inches.

Since both bars have the same width and height, we can use the formula for the volume of a rectangular prism to find the length of each bar:

Regular-size bar: V = lwh = 4 cubic inches, w = 1 inch, h = 3/4 inch

l = V/wh = 4/(1 × 3/4) = 16/3 inches

Snack-size bar: V = lwh = 3 cubic inches, w = 1 inch, h = 3/4 inch

l = V/wh = 3/(1 × 3/4) = 4 inches

Therefore, the regular-size granola bar is (16/3 - 4) = 4/3 inches longer than the snack-size granola bar.

This can also be written as the mixed number 1 1/3 inches.

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Answer:

1/12 gallons

Step-by-step explanation:

We can determine how much more hot chocolate than apple cider die Joe serve by subtracting 1/12 from 1/6.

Step 1:  Since 1/12 has the bigger denominator and 6 * 2 = 12, let's give 1/6 the same denominator as 1/12 by multiplying the entire fraction by 2/2

(1/6) * (2/2) = 2/12

Step 2:  Now we can subtract 1/12 from 2/12:

2/12 - 1/12 = 1/12

Thus, Joel served 1/12 more gallons of hot chocolate than apple cider.

The equation of the line is: x = 1 + 3t,, y = 10, z = 3 + 3t.

To determine the equation of the line passing through points M(1, 10, 3), N(4, 10, 6), and P(7, 10, 9), we need to find the direction vector of the line.

Let's denote the direction vector as D = (a, b, c). The line can be represented by the parametric equations:

x = 1 + at,

y = 10 + bt,

z = 3 + ct,

To find the direction vector, we can use two points on the line, for example, M and N. The direction vector can be obtained by subtracting the coordinates of M from the coordinates of N:

D = N - M = (4 - 1, 10 - 10, 6 - 3) = (3, 0, 3).

So, the equation of the line passing through M(1, 10, 3) and having the direction vector D = (3, 0, 3) is:

x = 1 + 3t,

y = 10 + 0t = 10,

z = 3 + 3t.

Therefore, the equation of the line is:

x = 1 + 3t,

y = 10,

z = 3 + 3t.

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The value of y from the given right angled triangle is 9.2 cm.

Given that, a right angled triangle has 8.2 cm, y cm and 12.3 cm.

Let, hypotenuse = 12.3 cm, perpendicular = 8.2 cm and base = y cm.

By using Pythagoras theorem, we get

8.2²+y²=12.3²

67.24+y²=151.29

y²=151.29-67.24

y²=84.05

y=9.2 cm

Therefore, the value of y from the given right angled triangle is 9.2 cm.

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The value of R and θ is 6.18 and 53.13 degrees, under the condition that a circle C has center at the origin and radius 5 .

In order to  evaluate the equation of the circle K. The diameter of K has endpoints at the origin and (0,15). Then, the center of K is at (0,7.5) and its radius is 7.5. Therefore, the evaluated equation of K is
x² + (y-7.5)² = 56.25.
The equation of circle C is x² + y² = 25.
The two circles intersect at two points. Now we have to evaluate the coordinates of these points.
Staging y = 5 - x² in the equation of K,
we get
x² + (5-x²-7.5)² = 56.25.
Applying simplification on this equation
x⁴ - 10x² + 31.25 = 0.
Calculating this quadratic equation gives us
x² = 5 ± √(10)/2.
If P lies in the first quadrant,
we choose x² = 5 + √(10)/2 and y = √(25-x²) to get P in Cartesian coordinates.
Converting P to polar coordinates gives us
r = √(x²+y²) and θ = arctan(y/x).
Staging x = √(5+√(10)/2) and y = √(25-x²) in these equations gives us
r ≈ 6.18 and θ ≈ 0.93 radians.
Using this formula to convert into degree
Rad × 180/π
= 0.93 × 180/π
≈ 53.13 degrees
Therefore, r ≈ 6.18 and θ ≈ 53.13 degrees.
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The value of a in the triangle, given that in the triangle ABC, b = 2, c = 6, and A = 35° is 7.7 (option A)

We can obtain the value of a in the triangle ABC as shown in the attached photo by using the cosine rule as illustrated below:

Cosine rule states as follow:

a² = b² + c² + 2bc Cos A

Inputting the given parameters, we can obtain the value of a as follow:

a² = 2² + 6² + (2 × 2 × 6 × Cos 35)

Clear the bracket

a² = 4 + 36 + 19.66

a² = 59.66

Take the square root of both sides

a = √59.66

a = 7.7

Thus, we can conclude from the above calculation that the value of a is 7.7 (option A)

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