Classwork Topic: Solve problems 25 May 2023 1. Jessica and Melissa shared 12 pieces of dried pears. Jessica ate = of the dried pears! Melissa ate . How Pieces did they eat in all? What fraction of the dried eat altogether? many pears did they 2. There were is children playing in the park. One third of the children went home. How many children stayed in the park?​

Answer:

Step-by-step explanation:

-3≤ x = graph 6

-3 > x = graph 3

x≥ 3 = graph 2

x ≤ 3 = graph 4

3 > x = graph 1

x > 3 = graph 5

the closed circles means greater/less than or equal to

the open circle means greater/less than

the direction of the arrow tells you if the number is greater than x or less than x

Given statement solution is :- The volume of the resulting part is approximately 267 cubic inches.

To find the volume of the resulting part, we need to calculate the volume of the larger cone and subtract the volume of the smaller cone.

The formula for the volume of a cone is given by:

V = (1/3) * π * [tex]r^2[/tex] * h

where V is the volume, π is a mathematical constant approximately equal to 3.14159, r is the radius of the cone's base, and h is the height of the cone.

For the larger cone:

Radius (r) = 5 inches

Height (h) = 12 inches

V_large_cone = (1/3) * π * [tex](5^2)[/tex] * 12

V_large_cone = (1/3) * π * 25 * 12

V_large_cone = (1/3) * π * 300

V_large_cone = 100π

For the smaller cone:

Radius (r) = 3 inches

Height (h) = 5 inches

V_small_cone = (1/3) * π * [tex](3^2)[/tex] * 5

V_small_cone = (1/3) * π * 9 * 5

V_small_cone = (1/3) * π * 45

V_small_cone = 15π

Therefore, the volume of the resulting part is:

V_resulting_part = V_large_cone - V_small_cone

V_resulting_part = 100π - 15π

V_resulting_part = 85π

Now, we can approximate the value of π as 3.14159:

V_resulting_part ≈ 85 * 3.14159

V_resulting_part ≈ 267.10715

Rounded to the nearest whole number, the volume of the resulting part is approximately 267 cubic inches.

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Answer: 4



Step-by-step explanation:

The fisherman can expect to catch 120 unhealthy salmon during the next month.

We can use the proportion of unhealthy to healthy salmon that was seen in the fisherman's catch of 15 salmon (12 unhealthy, 3 healthy) and multiply that proportion by the total amount of salmon he plans to catch (150) to find our answer.

Unhealthy = 150 × (12/15)

Unhealthy = 120

Therefore, the fisherman can expect to catch 120 unhealthy salmon during the next month.

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The square inches of cardboard paper used is 56.8 square inches

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

The prism

The square inches of cardboard paper is the surface area of the pyramid

And this is calculated as

Area = bh + L(Sum of side lengths)

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

Area = 2 * 3.2 + 6 * (2 + 3.2 + 3.2)

Evaluate

Area = 56.8

Hence, the square inches of construction paper used to make the container is 56.8 square inches

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A triangular prism is a three-dimensional geometric shape that consists of two triangular bases and three rectangular faces connecting them. It is a polyhedron with six faces, nine edges, and six vertices.

The two triangular bases of a triangular prism are congruent and parallel to each other. The rectangular faces are perpendicular to the triangular bases, and their lateral edges connect the corresponding vertices of the triangular bases.

The triangular bases are identical and parallel to each other. Each base has three vertices, three edges, and one face.There are three rectangular lateral faces connecting the corresponding vertices of the triangular bases. Each lateral face has two edges and one face.

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Answer:

A

Step-by-step explanation:

To simplify the expression 3 11 5 ÷ 3 − 9 5, let's break it down step by step:

First, let's simplify the division 3 11 5 ÷ 3:

3 11 5 ÷ 3 = (3 × 115) ÷ 3 = 345 ÷ 3 = 115.

Next, let's subtract 9 5 from the result we obtained:

115 - 9 5 = 115 - (9 × 5) = 115 - 45 = 70.

Therefore, the simplified expression is 70.

The correct answer is A. 70.

Answer:

Step-by-step explanation:

9x-3y=-10 ...............(1)

3x-4y=1...............(2)

multiplying equation (2) by 3

9x-12y=3...................(3)

Using elimination method, then

9x-3y=-10 ...............(1)

9x-12y=3...................(3

9y= -13

y= -13/9

substituting y= -13/9 in equation (1) then

9x-3(-13/9)= -10

9x+13/3= -10

multiplying throughout by 3

27x+13= -30

27x= -30-13

27x= -43

x= -43/27

since x and y values are known, then

12x-7y = 12(-43/27) - 7(-13/9)

12x-7y = -516/27 + 91/9

12x-7y = -9

Answer:

40

Step-by-step explanation:

plugging in a and b will get us:

[tex]2^{2} + 6^{2} = c^{2}[/tex]

4 + 36 = [tex]c^{2}[/tex]

40 = [tex]c^{2}[/tex]

c= [tex]\sqrt{40\\[/tex]

The frequency distribution table would be:

Class ______ Freq ___ Midpoint __R/Freq__ C/Freq

0 - 9 _______ 4 ______ 4.5 ______ 0.2 _____ 4

10-19 _______ 5 ______ 14.5 _____0.25 _____9

20-29 ______ 4 ______ 24.5_____ 0.2______ 13

30-39_______3 ______ 34.5 _____0.15______ 16

40-49_______4 ______ 44.5 _____ 0.2 _____ 20

The data ranges from 0 - 49 . we had to use a class interval of 10 since we were interested in having just 5 classes.

The midpoint is the average value of the upper and lower class.

Relative frequency is the ratio of the class frequency to the total number of data in the distribution.

Cumulative Frequency is the running sum of frequencies in the distribution table.

Therefore, the frequency distribution captures all the information required.

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The area of the dining room table top is: 864 sq. in

The formula for the area of a triangle is:

Area = ¹/₂ * base * height

Formula for the area of a rectangle is:

Area = Length * Width

Area of trapezium = ¹/₂(sum of parallel sides) * height

Thus, if 1cm = 6 inches

Then: 9cm = 54 inches

3 cm = 18 inches

4 cm = 24 inches

Thus:

Area of trapezium = ¹/₂(54 + 18) * 24

= 864 sq. in

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The possible combinations of women to the total number of club members could be:

7 women out of 10 members.

14 women out of 20 members.

21 women out of 30 members.

28 women out of 40 members.

And so on, as long as "x" is a multiple of 10 that is divisible by 7.

The ratio of women to men in the local book club is 7 to 3. This means that for every 7 women, there are 3 men.

To find the possible combinations of women to the total number of club members, we can consider the total number of members as a multiple of 10 (since the ratio is given as 7 to 3). Let's assume the total number of club members is represented by the variable "x."

Based on the given ratio, we can calculate the number of women in terms of "x" by multiplying the ratio of women (7/10) by the total number of members:

Number of women = (7/10) * x

Since we're looking for possible combinations, the number of women must be a whole number. Therefore, "x" must be a multiple of 10 that is divisible by 7.

Let's explore some possible combinations:

If x = 10, the number of women = (7/10) * 10 = 7, which satisfies the ratio.

If x = 20, the number of women = (7/10) * 20 = 14, which satisfies the ratio.

If x = 30, the number of women = (7/10) * 30 = 21, which satisfies the ratio.

If x = 40, the number of women = (7/10) * 40 = 28, which satisfies the ratio.

As you can see, the possible combinations of women to the total number of club members could be:

7 women out of 10 members.

14 women out of 20 members.

21 women out of 30 members.

28 women out of 40 members.

And so on, as long as "x" is a multiple of 10 that is divisible by 7.

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Answer:

 consider the following data set 10 14 12 16 10 13 14 12 11 which of the relative frequency for the data value of 14

Step-by-step explanation:

The length of the arc RT is 11.72 units.

An arc is defined as a portion of the boundary of a circle or a curve.

To calculate the length of the arc RT, we use the formula below

Formula:

Where:

From the question,

Given:

Substitute these values into equation 1

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The cross-section that results from slicing a cube through the center vertically, perpendicular to the base, is both a rectangle and a square.

When a cube is sliced through the center vertically, perpendicular to the base, the resulting cross-section will be a rectangle and a square.

Let's consider the properties of a cube. A cube is a three-dimensional solid with six square faces. Each face of the cube is congruent and perpendicular to the adjacent faces. The edges of the cube are all equal in length, and the angles between the faces are all right angles (90 degrees).

When we slice the cube vertically through the center perpendicular to the base, we cut through the cube in such a way that the cross-section obtained is a plane figure. This cross-section will have the same shape as the face of the cube that was cut.

In this case, since the cube has square faces, the cross-section will also have the same shape. Therefore, a square is one of the shapes that apply to the resulting cross-section.

Additionally, since the slice is made through the centre of the cube, the resulting cross-section will pass through the midpoints of two opposite sides of the square face. As a result, the cross-section will have equal side lengths and parallel sides, making it a rectangle.

Final answer:

Therefore, the cross-section that results from slicing a cube through the center vertically, perpendicular to the base, is both a rectangle and a square. These shapes capture the properties of the cube's face and the nature of the cut.

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When solving a linear system of equations, you are looking for the points of intersection between the equations

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

Solving a system of linear equations

Also, we have the following from the options

The general rile is that

Slope = rate of change

x and y intercepts = when y and x equals 0

points of intersection = solution to the system

Hence, you are looking for the points of intersection between the equations

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Question

When solving a linear system of equations, you are looking for which of the following?

Slope

y-intercept

x-intercept

Points of intersection

A. In equation form: Revenue = 9x1 + 12x2 + 4x3 + 5x4 + 8x5

B. Non-negativity constraint: x1, x2, x3, x4, x5 ≥ 0

To solve this problem using E x c e l Solver, set up the revenue function and the constraints. Here's how you can do it:

a. Revenue Function:

Let's denote the number of cakes baked for each type as x1, x2, x3, x4, and x5 respectively.

The revenue function can be formulated as:

Revenue = (Selling Price of Cake 1 × Number of Cake 1) + (Selling Price of Cake 2 × Number of Cake 2) + (Selling Price of Cake 3 × Number of Cake 3) + (Selling Price of Cake 4 × Number of Cake 4) + (Selling Price of Cake 5 × Number of Cake 5)

In equation form:

Revenue = 9x1 + 12x2 + 4x3 + 5x4 + 8x5

b. Constraints:

The constraints for the availability of ingredients can be formulated as follows:

Milk constraint: 4x1 + 3x2 + x3 + 5x4 + 4x5 ≤ 280

Sugar constraint: 7x1 + 4x2 + 5x3 ≤ 300

Egg constraint: x1 + 2x2 + 3x3 + 4x4 + 5x5 ≤ 80

Flour constraint: 3x1 + 4x2 + 2x3 + 4x4 + 6x5 ≤ 250

Cream constraint: 4x1 + 5x3 + x4 + 3x5 ≤ 190

Non-negativity constraint: x1, x2, x3, x4, x5 ≥ 0

c. Solve the system of linear inequalities using E x c e l Solver:

To solve the system of linear inequalities using E x c e l Solver, follow these steps:

1. Open M i c r o s o f t E x c e l and enter the following data in a new sheet:

| | A | B |

|-----|-----------|-----------------|

| 1 | Cakes | Selling Price |

| 2 | Cake 1 | $9 |

| 3 | Cake 2 | $12 |

| 4 | Cake 3 | $4 |

| 5 | Cake 4 | $5 |

| 6 | Cake 5 | $8 |

| | | |

| | | Formula |

| 9 | Milk | 280 |

| 10 | Sugar | 300 |

| 11 | Eggs | 80 |

| 12 | Flour | 250 |

| 13 | Cream | 190 |

2. In cell B16, enter the formula for the revenue:

=B2×B7 + B3×B8 + B4×B9 + B5×B10 + B6×B11

3. In cell B18, enter the formula for the milk constraint:

=4×B2 + 3×B3 + B4 + 5×B5 + 4×B6

4. In cell B19, enter the formula for the sugar constraint:

=7×B2 + 4×B3 + 5×B4

5. In cell B20, enter the formula for the egg constraint:

=B2 + 2×B3 + 3×B4 + 4×B5 + 5×B6

6. In cell B21, enter the formula for the flour constraint:

=3×B2 + 4×B3 + 2×B4 + 4×B5 + 6×B6

7. In cell B22, enter the formula for the cream constraint:

=4×B2 + 5×B4 + B5 + 3×B6

8. In cell B24, enter the formula for the non-negativity constraint for Cake 1:

=B2

9. Repeat step 8 for the remaining cakes, entering the formulas in cells B25, B26, B27, and B28:

=B3

=B4

=B5

=B6

10. Now, select cells B16 to B28 and click on the "Solver" button in the "Data" tab.

11. In the Solver Parameters window, set the objective to maximize the cell B16 (Revenue).

12. Set the By Changing Variable Cells to B24:B28 (the number of cakes baked).

13. Click on the "Add" button in the "Subject to the Constraints" section.

14. In the Constraint window, select the range B18:B22 for the constraint cells.

15. In the Solver Parameters window, click on the "Add" button again and select the range B24:B28 for the non-negativity constraints.

16. Set the Solver options as desired, and click on the "Solve" button.

E x c e l Solver will calculate the optimal values for the number of cakes to be baked for each type that maximize the revenue, while satisfying the given constraints on ingredient availability. The solution will be displayed in cells B24:B28, indicating the number of cakes to be baked for each type.

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TanA value is 22/3 and TanB is 3/2 from the given triangle ABC.

We have to find the values of Tan A and TanB.

We know that tan function is a ratio of opposite side and adjacent side.

To find TanA, we have to take opposite side of vertex A has opposite side.

TanA=18/27

TanA=2/3

Now let us find TanB , which is 27/18

TanB =3/2

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If u =(1+i√3) and v=(1-i√3), product uv is: D. 4.

To find the product of u and v let us simply multiply them together:

u = 1 + i√3

v = 1 - i√3

uv = (1 + i√3)(1 - i√3)

Using the difference of squares formula (a² - b² = (a + b)(a - b)) we can simplify the expression:

uv = (1 + i√3)(1 - i√3)

uv= 1² - (i√3)²

uv= 1 - (-3)

uv= 1 + 3

uv= 4

Therefore the product uv is equal to 4.

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To express the value 6 × 10^(-10), you can use scientific notation or decimal notation. Here are some valid expressions that equal 6 × 10^(-10):

0.0000000006

6E-10

6 × 10^(-10)

These expressions represent the same value, which is 6 multiplied by 10 raised to the power of -10.

Decimal is a number system used in everyday arithmetic and mathematics. It is also known as the base-10 system because it uses ten digits (0-9) to represent numbers. Each digit's position in a decimal number represents a power of 10.

In a decimal number, the digits to the left of the decimal point represent whole numbers, while the digits to the right of the decimal point represent fractions or parts of a whole. For example, in the decimal number 123.45, the digit "1" represents 100, the digit "2" represents 10, the digit "3" represents 1, the digit "4" represents 0.1, and the digit "5" represents 0.01.

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The point B that partitions the segment AC into a 2:3 ratio is B (4.6, 9.6). To find the point B that partitions the segment AC into a 2:3 ratio, we can use the concept of section formulas or the concept of ratios of the coordinates.

Let's assume the coordinates of point B are (x, y). We can calculate the coordinates of point B using the following steps:

Step 1: Calculate the differences in the x and y coordinates between points A and C:

Δx = x-coordinate of C - x-coordinate of A = 5 - 3 = 2

Δy = y-coordinate of C - y-coordinate of A = 11 - 4 = 7

Step 2: Divide the differences by the sum of the ratios (2 + 3 = 5) to get the ratios of the segments AB and BC:

Ratio_AB = (2/5)[tex]\times[/tex] Δx = (2/5) [tex]\times[/tex]2 = 4/5

Ratio_BC = (3/5) [tex]\times[/tex] Δx = (3/5) [tex]\times[/tex] 2 = 6/5

Step 3: Apply the ratios to the coordinates of point A to find the coordinates of point B:

x-coordinate of B = x-coordinate of A + Ratio_AB [tex]\times[/tex] Δx = 3 + (4/5)[tex]\times[/tex]2 = 3 + 8/5 = 23/5

y-coordinate of B = y-coordinate of A + Ratio_AB [tex]\times[/tex]Δy = 4 + (4/5) [tex]\times[/tex] 7 = 4 + 28/5 = 48/5

Therefore, the coordinates of point B are (23/5, 48/5).

In decimal form, the coordinates of point B are approximately (4.6, 9.6).

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The rental car is getting approximately 37.56 miles per gallon.

To convert kilometers per liter (km/l) to miles per gallon (mpg), you can use the following conversion factor:

1 km/l ≈ 2.82475 mpg

Therefore, if your rental car is getting 13.3 km/l of gasoline, the equivalent value in miles per gallon would be:

13.3 km/l x 2.82475 mpg ≈ 37.56 mpg

So, your rental car is getting approximately 37.56 miles per gallon.

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Answer:

[tex]8 {x}^{2} - 6x - 5 = x[/tex]

[tex]8 {x}^{2} - 7x - 5 = 0[/tex]

x = (7 + √((-7)^2 - 4(8)(-5)))/(2×8)

= (7 + √(49 + 160))/16

= (7 + √209)/16

= -.4661, 1.3411 (to 4 decimal places)

John's z-score is 0.75.

John's score of 79 on the math test corresponds to a z-score of 0.75 and a percentile of 77.82%.

To get John's z-score, we will use the formula: z = (x - μ) / σ

Data

x = John's score (79)

μ = mean score (73)

σ = standard deviation (8)

z = (79 - 73) / 8

z = 6 / 8

z = 0.75.

To find John's percentile, we can use a standard normal distribution table or calculator.

The percentile represents the percentage of scores that fall below a given value. Using  standard normal distribution table, we find that a z-score of 0.75 equals to percentile of 77.82%.

Full question:

The scores on a math test have a mean of 73 points and a standard deviation of 8 points. John scores a 79 on the test. Convert this score to a z-score and a percentile..

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The item most likely to weigh 2 kilograms is a roast chicken.

The size, breed, and any other ingredients or stuffing used can all affect the weight of a roast chicken. Weights of roast chickens can range from petite ones weighing less than 1 kilogram to larger ones weighing more than 2 kilograms.

The other things that we have there would either weigh less than 2 Kg such as a tea bag or much more than 2 Kg such as a horse. The egg and the tea a very light and would be less than 2 Kg in weight while the bag and the horse would be above 2 Kg in weight.

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The output value of f(-1) include the following: -13.

In Mathematics and Geometry, a piecewise-defined function simply refers to a type of function that is defined by two (2) or more mathematical expressions over a specific domain.

Generally speaking, the domain of any piecewise-defined function simply refers to the union of all of its sub-domains.

Since the value of x is -1, the output value of this piecewise-defined function can be calculated by using the first function as follows;

f(x) = 3x - 10

f(-1) = 3(-1) - 10

f(-1) = -3 - 10

f(-1) = -13

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The volume of the cylinder is 98π cubic centimeters.

To find the volume of a cylinder, we use the formula:

Volume = π × [tex]r^2[/tex] × h

Given:

Radius (r) = 7 cm

Height (h) = 2 cm

Substituting the values into the formula, we have:

Volume = π × (7 [tex]cm)^2[/tex] × 2 cm

Calculating the values inside the parentheses:

Volume = π × 49 [tex]cm^2[/tex] × 2 cm

Multiplying the values:

Volume = 98π [tex]cm^3[/tex]

Therefore, the volume of the cylinder is 98π cubic centimeters.

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The roots of the function F(x) = x³ + 2x² + 4x + 8 are given as follows:

B. 1 real root and 2 complex roots.

The function in this problem is defined as follows:

F(x) = x³ + 2x² + 4x + 8.

The function is of the third degree, hence the total number of zeros of the function is of 3.

From the graph, the function has only one x-intercept, which is given as follows:

x = -2.

Hence the two remaining roots are the complex roots, meaning that option B is the correct option for this problem.

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a. The scalar product of vectors A = A₁i + A2₂j + A₃k and B = B₁i + B₂j + B₃k. is A ∙ B = A₁B₁ + A₂B₂ + A₃B₃.

b. The angle between the vectors A = 2i +2j - k and B = 7i + 24k is 97.67°

The scalar product of two vectors a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k is given by

a.b = abcosФ = a₁b₁ + a₂b₂ + a₃b₃ where

6a) Suppose A = A₁i + A2₂j + A₃k and B = B₁i + B₂j + B₃k. Prove that A ∙ B = A₁B₁ + A₂B₂ + A₃B₃.

We proceed as follows

We know that

A ∙ B = (A₁i + A₂j + A₃k). (B₁i + B₂j + B₃k)

= A₁i.(B₁i + B₂j + B₃k) + A₂j.(B₁i + B₂j + B₃k) + A₃k.(B₁i + B₂j + B₃k)

Expanding the brackets, we have

= A₁i.B₁i + A₁i.B₂j + A₁i.B₃k + A₂j.B₁i + A₂j.B₂j + A₂j.B₃k + A₃k.B₁i + A₃k.B₂j + A₃k.B₃k

= A₁B₁i.i + A₁B₂i.j + A₁B₃i.k + A₂B₁j.i + A₂B₂j.j + A₂B₃j.k + A₃B₁k.i + A₃B₂k.j + A₃B₃k.k

We know that

So, we have that

= A₁B₁i.i + A₁B₂i.j + A₁B₃i.k + A₂B₁j.i + A₂B₂j.j + A₂B₃j.k + A₃B₁k.i + A₃B₂k.j + A₃B₃k.k

= A₁B₁(1) + A₁B₂(0) + A₁B₃(0) + A₂B₁(0) + A₂B₂(1) + A₂B₃(0) + A₃B₁(0) + A₃B₂(0) + A₃B₃(1)

= A₁B₁ + 0 + 0 + 0 + A₂B₂ + 0 + 0 + 0 + A₃B₃

= A₁B₁ + A₂B₂ + A₃B₃

So, A ∙ B = A₁B₁ + A₂B₂ + A₃B₃.

6b) To find the angle between the vectors A = 2i +2j - k and B = 7i + 24k, we proceed as follows.

We know that the angle between two vectors is given by

Ф = cos⁻¹[(A.B)/AB​] where

Now, A.B =  (2i +2j - k).(7i + 24k) = (2 × 7 + 2 × 0 + (-1) × 24)

= (14 + 0 - 24)

= -10

Now, the magnitude of a vector C = C₁i + C₂j + C₃k is C = √(C₁² + C₂² + C₃²)

Since vector A = 2i +2j - k its magniude A = √(2² + 2² + (-1)²)

= √(4 + 4 + 1)

= √9

= 3

Also since vector B = 7i + 24k,  its magniude A = √(7² + 0² + 24²)

= √(49 + 0 + 576)

= √625

= 25

So, substituting the values of the variables into the equation, we have that

Ф = cos⁻¹[(A.B)/AB​]

Ф = cos⁻¹[(-10)/(3 × 25)]

Ф = cos⁻¹[(-2)/(3 × 5)]

Ф = cos⁻¹[-2/15]

Ф = -0.13333

Ф = 97.67°

So, the angle betwen the vectors is 97.67°

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The value of integral is 3.

Let's evaluate the integral over the positive half of the interval:

∫[0 to π] (cos(x) / √(4 + 3sin(x))) dx

Let u = 4 + 3sin(x), then du = 3cos(x) dx.

Substituting these expressions into the integral, we have:

∫[0 to π] (cos(x) / sqrt(4 + 3sin(x))) dx = ∫[0 to π] (1 / (3√u)) du

Using the power rule of integration, the integral becomes:

∫[0 to π] (1 / (3√u)) du = (2/3) . 2√u ∣[0 to π]

Evaluating the definite integral at the limits of integration:

(2/3)2√u ∣[0 to π] = (2/3) 2(√(4 + 3sin(π)) - √(4 + 3sin(0)))

(2/3) x 2(√(4) - √(4)) = (2/3) x 2(2 - 2) = (2/3) x 2(0) = 0

So, the value of integral is

= 3-0

= 3

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Answer:

[tex]3-\displaystyle \int^{\pi}_{-\pi} \dfrac{\cos x}{\sqrt{4+3 \sin x}}\; \text{d}x\approx 0.806\; \sf (3\;d.p.)[/tex]

Step-by-step explanation:

First, compute the indefinite integral:

[tex]\displaystyle \int \dfrac{\cos x}{\sqrt{4+3\sin x}}\; \text{d}x[/tex]

To evaluate the indefinite integral, use the method of substitution.

[tex]\textsf{Let} \;\;u = 4 + 3 \sin x[/tex]

Find du/dx and rewrite it so that dx is on its own:

[tex]\dfrac{\text{d}u}{\text{d}x}=3 \cos x \implies \text{d}x=\dfrac{1}{3 \cos x}\; \text{d}u[/tex]

Rewrite the original integral in terms of u and du, and evaluate:

[tex]\begin{aligned}\displaystyle \int \dfrac{\cos x}{\sqrt{4+3\sin x}}\; \text{d}x&=\int \dfrac{\cos x}{\sqrt{u}}\cdot \dfrac{1}{3 \cos x}\; \text{d}u\\\\&=\int \dfrac{1}{3\sqrt{u}}\; \text{d}u\\\\&=\int\dfrac{1}{3}u^{-\frac{1}{2}}\; \text{d}u\\\\&=\dfrac{1}{-\frac{1}{2}+1} \cdot \dfrac{1}{3}u^{-\frac{1}{2}+1}+C\\\\&=\dfrac{2}{3}\sqrt{u}+C\end{aligned}[/tex]

Substitute back u = 4 + 3 sin x:

                            [tex]= \dfrac{2}{3}\sqrt{4+3\sin x}+C[/tex]

Therefore:

[tex]\displaystyle \int \dfrac{\cos x}{\sqrt{4+3\sin x}}\; \text{d}x= \dfrac{2}{3}\sqrt{4+3\sin x}+C[/tex]

To evaluate the definite integral, we must first determine any intervals within the given interval -π ≤ x ≤ π where the curve lies below the x-axis. This is because when we integrate a function that lies below the x-axis, it will give a negative area value.

Find the x-intercepts by setting the function to zero and solving for x in the given interval -π ≤ x ≤ π.

[tex]\begin{aligned}\dfrac{\cos x}{\sqrt{4+3\sin x}}&=0\\\\\cos x&=0\\\\x&=\arccos0\\\\\implies x&=-\dfrac{\pi }{2}, \dfrac{\pi }{2}\end{aligned}[/tex]

Therefore, the curve of the function is:

So to calculate the total area, we need to calculate the positive and negative areas separately and then add them together, remembering that if you integrate a function to find an area that lies below the x-axis, it will give a negative value.

Integrate the function between -π and -π/2.

As the area is below the x-axis, we need to negate the integral so that the resulting area is positive:

[tex]\begin{aligned}A_1=-\displaystyle \int^{-\frac{\pi}{2}}_{-\pi} \dfrac{\cos x}{\sqrt{4+3\sin x}}\; \text{d}x&=- \left[\dfrac{2}{3}\sqrt{4+3\sin x}\right]^{-\frac{\pi}{2}}_{-\pi}\\\\&=-\dfrac{2}{3}\sqrt{4+3 \sin \left(-\frac{\pi}{2}\right)}+\dfrac{2}{3}\sqrt{4+3 \sin \left(-\pi\right)}\\\\&=-\dfrac{2}{3}\sqrt{4+3 (-1)}+\dfrac{2}{3}\sqrt{4+3 (0)}\\\\&=-\dfrac{2}{3}+\dfrac{4}{3}\\\\&=\dfrac{2}{3}\end{aligned}[/tex]

Integrate the function between -π/2 and π/2:

[tex]\begin{aligned}A_2=\displaystyle \int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \dfrac{\cos x}{\sqrt{4+3\sin x}}\; \text{d}x&= \left[\dfrac{2}{3}\sqrt{4+3\sin x}\right]^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\\\\&=\dfrac{2}{3}\sqrt{4+3 \sin \left(\frac{\pi}{2}\right)}-\dfrac{2}{3}\sqrt{4+3 \sin \left(-\frac{\pi}{2}\right)}\\\\&=\dfrac{2}{3}\sqrt{4+3 (1)}-\dfrac{2}{3}\sqrt{4+3 (-1)}\\\\&=\dfrac{2\sqrt{7}}{3}-\dfrac{2}{3}\\\\&=\dfrac{2\sqrt{7}-2}{3}\end{aligned}[/tex]

Integrate the function between π/2 and π.

As the area is below the x-axis, we need to negate the integral so that the resulting area is positive:

[tex]\begin{aligned}A_3=-\displaystyle \int^{\pi}_{\frac{\pi}{2}} \dfrac{\cos x}{\sqrt{4+3\sin x}}\; \text{d}x&= -\left[\dfrac{2}{3}\sqrt{4+3\sin x}\right]^{\pi}_{\frac{\pi}{2}}\\\\&=-\dfrac{2}{3}\sqrt{4+3 \sin \left(\pi\right)}+\dfrac{2}{3}\sqrt{4+3 \sin \left(\frac{\pi}{2}\right)}\\\\&=-\dfrac{2}{3}\sqrt{4+3 (0)}+\dfrac{2}{3}\sqrt{4+3 (1)}\\\\&=-\dfrac{4}{3}+\dfrac{2\sqrt{7}}{3}\\\\&=\dfrac{2\sqrt{7}-4}{3}\end{aligned}[/tex]

To evaluate the definite integral, sum A₁, A₂ and A₃:

[tex]\begin{aligned}\displaystyle \int^{\pi}_{-\pi} \dfrac{\cos x}{\sqrt{4+3\sin x}}\; \text{d}x&=\dfrac{2}{3}+\dfrac{2\sqrt{7}-2}{3}+\dfrac{2\sqrt{7}-4}{3}\\\\&=\dfrac{2+2\sqrt{7}-2+2\sqrt{7}-4}{3}\\\\&=\dfrac{4\sqrt{7}-4}{3}\\\\ &\approx2.194\; \sf (3\;d.p.)\end{aligned}[/tex]

Now we have evaluated the definite integral, we can subtract it from 3 to evaluate the given expression:

[tex]\begin{aligned}3-\displaystyle \int^{\pi}_{-\pi} \dfrac{\cos x}{\sqrt{4+3 \sin x}}\; \text{d}x&=3-\dfrac{4\sqrt{7}-4}{3}\\\\&=\dfrac{9}{3}-\dfrac{4\sqrt{7}-4}{3}\\\\&=\dfrac{9-(4\sqrt{7}-4)}{3}\\\\&=\dfrac{13-4\sqrt{7}}{3}\\\\&\approx 0.806\; \sf (3\;d.p.)\end{aligned}[/tex]

Therefore, the given expression cannot be zero.