12. What is the cosine of angle B

Answer:

√3/2

Step-by-step explanation:

Easy Method

The equation above is in the forms of sin(a)cos(b) - cos(a)sin(b), which is sin(a-b) according to the trig identities. sin(75-15) = sin(60) = √3/2

Harder Method

Find sin75 with equation sin(a+b) = sin(a)cos(b) + cos(a)sin(b)

sin75°

= sin(45° + 30°)

= [sin45°cos30° + cos45°sin30°]

= [√2/2 * √3/2 + √2/2 * 1/2]  <-- unit circle known values

= [(√6 + √2)/4]

Find cos75 with the equation: cos(a+b) = sin(a)sin(b) - cos(a)cos(b)

cos75°

= cos(45° + 30°)

= [cos45°cos30° - sin45°sin30°]

= [√2/2 * √3/2 - √2/2 * 1/2]  <-- unit circle known values

= [(√6 - √2)/4]

Find sin15 with the equation: sin(a-b) = sin(a)cos(b) - cos(a)sin(b)

sin15°

= sin(45° - 30°)

= [sin45°cos30° - cos45°sin30°]

= [√2/2 * √3/2 - √2/2 * 1/2]  <-- unit circle known values

= [(√6 - √2)/4]

Find cos15 with the equation: cos(a-b) = sin(a)sin(b) + cos(a)cos(b)

cos15°

= cos(45° - 30°)

= [cos45°cos30° + sin45°sin30°]

= [√2/2 * √3/2 + √2/2 * 1/2]  <-- unit circle known values

= [(√6 + √2)/4]

Now plug in all the solved values, we get: {[(√6 + √2)/4] * [(√6 + √2)/4]} - {[(√6 - √2)/4] * [(√6 - √2)/4]} = √3/2

The inverse of f(x) is f⁻¹(x) = (x - 3)/3.

To find the inverse of the function f(x), we need to switch the roles of x and y and solve for y.

Given:

f(x) = 3x + 3

Step 1: Replace f(x) with y:

y = 3x + 3

Step 2: Swap x and y:

x = 3y + 3

Step 3: Solve for y:

x - 3 = 3y

3y = x - 3

y = (x - 3)/3

Therefore, the inverse of f(x) is f⁻¹(x) = (x - 3)/3.

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The solution to the set {2x < 6} n {x - 5 > -4} is {x | 1 < x < 3 }

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

{2x < 6} n {x - 5 > -4}

Divide both sides of 2x < 6 by 2

So, we have

{x < 3} n {x - 5 > -4}

Add 5 to both sides of x - 5 > -4

So, we have

{x < 3} n {x > 1}

The above set when combined is

1 < x < 3

This means that the solution to the set is {x | 1 < x < 3 }

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Answer:

Step-by-step explanation:

D

Using a trigonometric relation we can see that the distance between the base and the wall is 26.76ft

In the image at the end you can see a diagram for this problem.

We have a right triangle where we want to find the value of d, which is the adjacent cathetus to the known angle.

Then we can use the trigonometric relation:

cos(a) = (adjacent cathetus)/hypotenuse

Where:

a = 48°

hypotenuse = 40ft

adjacent cathetus = d

Replacing that we will get:

cos(48°) = d/40ft

Solving that for d, we will get:

40ft*cos(48°) =d

26.76ft = d

That is the distance between the base and the wall.

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Answer:

22 = 2 × 11

40 = 2 × 2 × 2 × 5

LCM of 22 and 40 = 2^3 × 5 × 11 = 440

Answer:

x=5,1

Step-by-step explanation:

have a great day and thx for your inquiry :)

Answer: x = 5, 1

Step-by-step explanation:

Take the root of both sides and solve

Since 6 is less than 84, the statement is correct. The roof with dimensions 12 meters by 7 meters is sufficient to accommodate 6 solar panels, each with a width of 1 meter.

To verify Sam's statement, we can calculate whether a roof that is 12 meters long and 7 meters wide can accommodate 6 solar panels, with each panel having a width of 1 meter.

The total area required to install 6 solar panels with a width of 1 meter each is:

6 panels * 1 meter = 6 square meters

Now, let's calculate the area of the roof:

Area of the roof = length * width = 12 meters * 7 meters = 84 square meters

Since the area required for the solar panels is 6 square meters and the area of the roof is 84 square meters, we can determine if the statement is correct.

If 6 square meters is less than or equal to 84 square meters, then the roof is indeed enough to install the 6 solar panels.

6 square meters ≤ 84 square meters.

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Using a pattern we can find the missing values in the table, these are:

When you look at the table, you can see that for each increase of 1 unit in the value of x, we have an increase of 14 units in the value of y. So we have a really simple pattern:

f(x) = f(x - 1) + 14

Then to get the next values in the table, just add 14 to the previous ones.

A = 42 + 14 = 56

B = 56 + 14 = 70

C = 70 + 14 = 84

Thes are the 3 missing values in the table.

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Answer:

GJ ≈ 31.42 units

Step-by-step explanation:

the arc length GJ is calculated as

GJ = circumference of circle × fraction of circle

     = 2πr × [tex]\frac{30}{180}[/tex] ( r is the radius )

the radius HJ = 30 , then

GJ = 2π × 30 × [tex]\frac{1}{6}[/tex]

     = 60π × [tex]\frac{1}{6}[/tex]

     = 10π

     ≈ 31.42 units ( to the nearest hundredth )

The pH level of the pool is determined as 8.6.

The pH of a solution is a measure of its acidity or basicity. The pH scale ranges from 0 to 14, with a pH of 7 being neutral, less than 7 is acidic and above 7 is alkaline.

The pH of a solution can be calculated using the formula;

pH = -log[H₃O⁺]

where;

Using the measured [H₃O⁺] concentration of 2.40 x 10⁻⁹ moles/liter, we can calculate the pH of the swimming pool as follows;

pH = -log(2.40 x 10⁻⁹)

pH = 8.62

Therefore, the pH of the swimming pool is 8.6, rounded to the nearest tenth.

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The determinant of the given matrix is -43.

We are given that;

The matrix= 3 2 -2

1 2 4

-1 3 2

Now,

To evaluate the determinant of the matrix

[3  2 -2]

[1  2  4]

[-1 3  2]

you can use the rule of Sarrus. First, you need to copy the first two columns of the matrix and place them next to the third column:

[3  2 -2 | 3  2]

[1  2  4 | 1  2]

[-1 3  2 | -1 3]

Then, you can calculate the sum of the products of the three diagonals going from left to right:

(3 * 2 * 2) + (2 * 4 * -1) + (-2 * 1 * 3) = 12 -8 -6 = -2

and subtract from it the sum of the products of the three diagonals going from right to left:

(-1 * 2 * -1) + (3 * 4 * 3) + (2 * 1 * 2) = 1 +36 +4 =41

-2 -41 = -43.

Therefore, by the matrices the answer will be -43.

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Answer:

X=20

Step-by-step explanation:

khan academy :))

y=3(5)ˣ is the equation of the function represented by the table

We can see that as x increases, y increases exponentially.

To determine the specific form of the exponential function, we can take the ratio of successive y-values:

(3/5)/(3/25) = 5/1

(3)/(3/5) = 5

(15)/(3) = 5

(75)/(15) = 5

This shows that the ratio of successive y-values is constant at 5. Therefore, the function represented by the table is an exponential function of the form:

y = a(b)ˣ

To find a and b, we can use any two pairs of (x,y) values.

Let's use (0,3) and (1,15):

3 = a(b)⁰

15 = a(b)¹

3 = a

Now we get b =5

Hence, y=3(5)ˣ is the equation of the function represented by the table

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In the given diagram, the area of the shape is 2π units²

From the question, we are to calculate the area of the given shape.

In the given diagram, the shape is a quadrant.

The area of a quadrant is given by the formula

Area = 1/4 πr²

Where r is the radius

From the given information,

Radius = 8

Thus,

Area of the quadrant = 1/4 × π × 8

Area of the quadrant = 2π units²

OR

Area of the quadrant = 2 × 3.14 units²

Area of the quadrant = 6.28 units²

Hence,

The area of the shape is 2π units² OR 6.28 units².

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The probability that the teacher chooses a girl with glasses is 3/16, and the probability that the teacher chooses a boy with glasses is 1/4.

Let's calculate the probability that the teacher chooses a student with glasses.
a) A girl with glasses: There are 6 girls in the class, and 3 of them wear glasses. There are a total of 16 students (6 girls + 10 boys).
Probability = (Number of girls with glasses) / (Total number of students)
Probability = 3 girls with glasses / 16 total students
Probability = 3/16
b) A boy with glasses: There are 10 boys in the class, and 4 of them wear glasses.
Probability = (Number of boys with glasses) / (Total number of students)
Probability = 4 boys with glasses / 16 total students
Probability = 1/4
So, the probability that the teacher chooses a girl with glasses is 3/16, and the probability that the teacher chooses a boy with glasses is 1/4.

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Answer: The standard equation of a parabola is given by:

4p(x - h) = (y - k)^2

where (h, k) is the vertex of the parabola, p is the distance from the vertex to the focus and also the distance from the vertex to the directrix.

In this case, the focus is at (-2,1), so the vertex is halfway between the focus and the directrix, which is at (x, y) = (-2, (1+8)/2) = (-2, 4.5). The distance from the vertex to the focus (and to the directrix) is p = 3.5.

Substituting these values into the equation, we get:

4(3.5)(x + 2) = (y - 4.5)^2

Simplifying, we get:

14(x + 2) = (y - 4.5)^2

This is the standard equation of the parabola.

To sketch the graph, we can use the vertex (-2, 4.5) as a starting point, and use the distance from the vertex to the focus to plot some points on either side of the vertex. Since the focus is to the left of the vertex, we know that the parabola will open to the left.

Using the distance formula, we can find the coordinates of two points on the parabola that are equidistant from the focus and the directrix:

Point 1: (-5.5, 4.5)

Distance from focus: sqrt((-5.5 + 2)^2 + (4.5 - 1)^2) = sqrt(44.5) ≈ 6.67

Distance from directrix: |4.5 - 8| = 3.5

Point 2: (-5.5, 1.5)

Distance from focus: sqrt((-5.5 + 2)^2 + (1.5 - 1)^2) = sqrt(36.5) ≈ 6.04

Distance from directrix: |1.5 - 8| = 6.5

Parabola with a focus at (-2,1) and directrix y=8

The vertex is labeled as V, the focus as F, and the directrix as D. The distance from the vertex to the focus (and directrix) is labeled as p. The parabola opens to the left and is symmetric about the axis of symmetry, which is the vertical line passing through the vertex.

The value of missing side length is, 51.11

We have to given that;

A triangle is shown.

And, To find the missing side of triangle.

Since, An angle is a combination of two rays (half-lines) with a common endpoint. The latter is known as the vertex of the angle and the rays as the sides, sometimes as the legs and sometimes the arms of the angle.

And, A triangle is a three sided polygon, which has three vertices and three angles which has the sum 180 degrees.

Now, By using trigonometry formula, we get;

⇒ cos 63° = Hypotenuse / Base

⇒ cos 63° = 23 / x

⇒ 0.45 = 23 / x

⇒ x = 23 / 0.45

⇒ x = 51.11

Therefore, The value of missing side in the triangle is,

⇒ x = 51.11

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The sum of the equation 3y/y²+7y+10 + 2/y+2 is 5/(y + 5).

In order to find the sum, we have to first find a common denominator for the two fractions:

3y/y² + 7y + 10 + 2/y + 2 = (3y/(y + 5)(y + 2)) + (2(y + 5)/(y + 5)(y + 2))

Next, we join the two fractions by adding their numerators:

= (3y + 2(y + 5))/((y + 5)(y + 2))

Then, we simplify the numerator:

= (3y + 2y + 10)/((y + 5)(y + 2))

= (5y + 10)/((y + 5)(y + 2))

= 5(y + 2)/((y + 5)(y + 2))

= 5/(y + 5)

Therefore, the sum of 3y/y²+7y+10 + 2/y+2 = 5/(y + 5).

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The function that corresponds to the parabola Graph is f(x) = -3(x + 1)² + 2, which is option B.

The graph represents a downward open parabola that rises from (-2.4,-4) to (-1,2) and then declines through (0.4,-4) on the x-y coordinate plane.

To determine the function that corresponds to this graph, we need to consider the key features of a parabolic function, including its vertex and axis of symmetry.

The vertex of a parabola in the form f(x) = a(x-h)² + k is (h,k), and the axis of symmetry is x = h

From the graph, we can see that the vertex of the parabola is at (-1,2), which means that h = -1 and k = 2.

We also know that the parabola opens downward, which means that a < 0.

Therefore, we can eliminate options A and D since they both have a positive value of "a."

Next, we can test options B and C.

Option B has the form f(x) = -3(x + 1)² + 2, which means that the vertex is at (-1,2) and the parabola opens downward due to the negative coefficient of (x+1)².

To confirm if option B is correct, we can check if the point (0.4,-4) lies on the parabola:

f(0.4) = -3(0.4+1)² + 2 = -3(1.4)² + 2 = -5.88

Since the y-coordinate of the point (0.4,-4) is -4, which is equal to -5.88, we can see that the point lies on the parabola.

Therefore, the function that corresponds to the graph is f(x) = -3(x + 1)² + 2, which is option B.

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Total length of the two sides of the triangle is 2x² + 4x + 4 and  length of the third side of the triangle is 5x³ - 5x² + 3x - 12.

The total length of the two sides of the triangle is the sum of Side 1 and Side 2:

(3x² - 4x - 1) + (4x - x² + 5)

2x² + 4x + 4

The total length of the two sides of the triangle is 2x² + 4x + 4.

The length of the third side of the triangle can be found by subtracting the sum of Side 1 and Side 2 from the perimeter of the triangle:

Perimeter - (Side 1 + Side 2)

(5x³ - 2x² + 3x - 8) - (3x² - 4x - 1 + 4x - x² + 5)

Combining like terms

5x³ - 5x² + 3x - 12

The length of the third side of the triangle is 5x³ - 5x² + 3x - 12.

The polynomials are closed under addition and subtraction by part A and part B because when we added Side 1 and Side 2, and when we subtracted their sum from the perimeter of the triangle, the resulting expressions were still polynomials with real coefficients.

Therefore, the sum and difference of polynomials with real coefficients are also polynomials with real coefficients.

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Answer:

a) 2/25. b) 3/100. c) 1/20. d) 11/50.

Step-by-step explanation:

there are 3 + 6 + 4 + 12 = 25 markers

a) p(red) = 4/25. then p(blue) = 12/24.

4/25 X 12/24 = 2/25

b) p(yellow) = 6/25. then p(green) = 3/24.

6/25 X 3/24 = 3/100

c) p(yellow) = 6/25. p(2nd yellow) = 5/24

6/25 X 5/24 = 1/20

d) p(blue) = 12/25. p(2nd blue) = 11/24

12/25 X 11/24 = 11/50

Answer:

Step-by-step explanation:

You probably should check for silly mistakes.

Answer:

Step-by-step explanation:

We know that the cosine of an angle is the reciprocal of the secant of the same angle, and the cotangent of an angle is the reciprocal of the tangent of the same angle. Therefore, we can use these relationships to find the value of the secant of an angle when the cotangent of the same angle is given.We are given that cot(o) = 13/6. Using the definition of the cotangent function, we know that:cot(o) = adjacent side / opposite sideWe can use the Pythagorean theorem to find the hypotenuse of a right triangle with adjacent side 13 and opposite side 6:h^2 = 13^2 + 6^2

h^2 = 169 + 36

h^2 = 205

h = sqrt(205)Now we can use the definitions of the secant and cosine functions to find the value of sec(o):sec(o) = hypotenuse / adjacent side

sec(o) = sqrt(205) / 13Therefore, the value of sec(o) is:sec(o) = sqrt(205) / 13 ≈ 1.5276

The length FE is 6 units and the arc AE is 64 degrees

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

The circle

Given that the lengths from the center to either chords are equal

This means that

FE = 6 units

For the other circle, we have

BC = 58 degrees

AB = ED

The arc AE is calculated as

AE = 180 - BC - ED

Where

AB = ED = BC = 58 degrees

So, we have

AE = 180 - 58 - 58

Evaluate

AE = 64

Hence, the arc AE is 64 degrees

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Hello !

Answer:

198 apples

Step-by-step explanation:

12 apples <=> 7 pounds of dough

x apples <=> 115 pounds of dough

Let's use a cross product :

[tex]x = \frac{12 \times 115}{7} \approx197.14 [/tex]

198 apples are needed.

Have a nice day

The calculated value of the missing side length of the triangle is 13.96 units

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

The triangle

The missing side length of the triangle can be calculated using the law of sines

The law of sines states that

a/sin(A) = b/sin(B) = c/sin(C)

Using the above as a guide, we have the following equation

x/sin(61) = 15/sin(70)

Cross multiply

x = sin(61) * 15/sin(70)

Evaluate

x = 13.96 units

Hence, the missing side length of the triangle is 13.96 units

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To have an infinite number of solutions the value of a would be

D. 20

Information given in the problem includes

5(x - 3) + a = 5(x + 1)

To have an infinite number of solutions we can simplify as follows

5(x - 3) + a = 5(x + 1)

opening the parenthesis

5x - 15 + a = 5x + 5

collecting like terms

-15 + a = 5

Adding 15 to both sides, we get:

a = 20

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patterns a and d have a constant difference, pattern c has a constant difference, and patterns b, e, and f do not have a constant difference or a constant ratio.

a. The pattern has a constant difference. The first term is 1, the second term is 4, which is 3 more than 1, the third term is 10, which is 6 more than 4, and the fourth term is 19, which is 9 more than 10.

b. The pattern does not have a constant difference or a constant ratio. The differences between consecutive terms are 3, 2, and 4, respectively, which are not constant. The ratios between consecutive terms are 2.5/2, 1.4/2.5, and 11/1.4, respectively, which are also not constant.

c. The pattern has a constant difference. The first term is 3, the second term is 7, which is 4 more than 3, the third term is 13, which is 6 more than 7, and the fourth term is 21, which is 8 more than 13.

d. The pattern has a constant difference. The first term is 12, the second term is 10, which is 2 less than 12, the third term is 6, which is 4 less than 10, and the fourth term is 0, which is 6 less than 6.

e. The pattern does not have a constant difference or a constant ratio. The differences between consecutive terms are 4, 7, and 10, respectively, which are not constant. The ratios between consecutive terms are 2.5/2, 13/2.5, and 23/13, respectively, which are also not constant.

f. The pattern does not have a constant difference or a constant ratio. The differences between consecutive terms are 6, 12, and 18, respectively, which are not constant.

The ratios between consecutive terms are 13/7, 25/13, and 43/25, respectively, which are also not constant.

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