Help pls i don't understand

The quotient of[tex](6x/(3x+15)) / ((x^2+2x)/ (x^2+7x+10))[/tex] is 2x.

To simplify the expression[tex](6x/(3x+15)) / ((x^2+2x)/ (x^2+7x+10)),[/tex] we can use the rule for dividing fractions, which states that dividing by a fraction is the same as multiplying by its reciprocal.

Let's simplify the expression :

Simplify the numerator and denominator of the first fraction.

The numerator is 6x, and the denominator is (3x+15).

We can factor out a common factor of 3 from the denominator, which gives us 3(x+5).

So the first fraction simplifies to 6x / 3(x+5).

Simplify the numerator and denominator of the second fraction.

The numerator is (x^2+2x), and the denominator is [tex](x^2+7x+10).[/tex]

We can factor both the numerator and denominator:

Numerator: x(x+2)

Denominator: (x+5)(x+2)

Canceling out the common factor of (x+2), we are left with x / (x+5).

Multiply the two simplified fractions.

Multiplying the fractions, we get:

[tex](6x / 3(x+5)) \times (x / (x+5))[/tex]

Simplify the resulting expression.

Canceling out the common factor of (x+5) in the numerator and denominator, we are left with:

2x / 1

So the quotient simplifies to 2x.

Therefore, the quotient of [tex](6x/(3x+15)) / ((x^2+2x)/ (x^2+7x+10))[/tex] is 2x.

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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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There were 600 downloads of the standard version.

An equation means the formula that expresses the equality of two expressions by connecting them with the equals sign.

Let x represents the number of downloads of the standard version.

Given:

Size of the standard version = 2.4 MB

Size of the high-quality version = 4.2 MB

Total download size = 4716 MB

Total downloads = 1380

We will set up the following equation:

2.4x + 4.2(1380 - x) = 4716

2.4x + 5796 - 4.2x = 4716

-1.8x + 5796 = 4716

-1.8x  = 4716 - 5796

-1.8x = -1080

x = 600

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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

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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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 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 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 factorization of[tex]8(p-q)^3-(p+q)^3 is 8(p-q)^3 - (p+q)^3 is [p - 3q][7p^2 - 8pq + 3q^2].[/tex]

The given expression is[tex]8(p-q)^{3}-(p-q)^{3}[/tex]

To factorize this expression, we can use the identity for the difference of cubes, which states that [tex]a^3 - b^3 = (a - b)(a^2 + ab + b^2).[/tex]

Let's apply this identity to the given expression:

[tex]8(p-q)^3 - (p+q)^3 = [2(p-q)]^3 - (p+q)^3.[/tex]

Now we can see that we have a difference of cubes:[tex](2(p-q))^3 - (p+q)^3.[/tex]

Using the identity, we can factorize this expression as follows:

[tex][(2(p-q)) - (p+q)][(2(p-q))^2 + (2(p-q))(p+q) + (p+q)^2].[/tex]

Simplifying further:

[tex][2p - 2q - p - q][(2(p-q))^2 + (2(p-q))(p+q) + (p+q)^2].[/tex]

Combining like terms:

[tex][p - 3q][(2(p-q))^2 + 2(p-q)(p+q) + (p+q)^2].[/tex]

Expanding the squared term:

[tex][p - 3q][4(p^2 - 2pq + q^2) + 2(p^2 - q^2) + (p^2 + 2pq + q^2)].[/tex]

Simplifying:

[tex][p - 3q][4p^2 - 8pq + 4q^2 + 2p^2 - 2q^2 + p^2 + 2pq + q^2].[/tex]

Combining like terms again:

[tex][p - 3q][7p^2 - 8pq + 3q^2].[/tex]

Therefore, the factorization of [tex]8(p-q)^3 - (p+q)^3 is [p - 3q][7p^2 - 8pq + 3q^2].[/tex]

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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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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 person will need approximately 28 square yards of material for the tent.

To calculate the amount of material needed, we first need to determine the total surface area of the tent.

The tent has six sides: a top, a bottom, a front, a back, and two sides. The top and bottom are both 10 feet by 5 feet, so their combined area is 10 feet x 5 feet = 50 square feet each.

The front and back are both 10 feet by 5 feet, so their combined area is also 10 feet x 5 feet = 50 square feet each.

The two sides are both 5 feet by 5 feet, so their combined area is 5 feet x 5 feet = 25 square feet each.

To find the total surface area, we add up all these areas:

Top and bottom: 50 square feet + 50 square feet = 100 square feet

Front and back: 50 square feet + 50 square feet = 100 square feet

Sides: 25 square feet + 25 square feet = 50 square feet

Total surface area: 100 square feet (top and bottom) + 100 square feet (front and back) + 50 square feet (sides) = 250 square feet

Since the material is sold in square yards, we need to convert square feet to square yards.

There are 9 square feet in 1 square yard.

So, 250 square feet / 9 = 27.78 square yards.

Rounding up to the nearest square yard, the person will need approximately 28 square yards of material for the tent.

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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: 4



Step-by-step explanation:

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

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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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.

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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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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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. 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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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 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 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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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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Experimental probability depends on Ted's frequency in the new set of names. The total number of names chosen also affects it. Theoretical probability is influenced by the total number of possible outcomes based on the number of names in the hat. If the number of names increases, the chance of choosing Ted decreases.

To find the experimental probability of choosing the name Ted, we divide the number of times Ted was chosen by the total number of names chosen.

Experimental probability = Number of times Ted was chosen / Total number of names chosen

Given that Ted was chosen 19 times out of 119 total selections, we can calculate the experimental probability:

Experimental probability = 19 / 119

Simplifying, we find that the experimental probability of choosing the name Ted is approximately 0.1597 or about 15.97% (rounded to the nearest hundredth).

To find the theoretical probability of choosing the name Ted, we divide the number of favorable outcomes (choosing Ted) by the total number of possible outcomes.

Theoretical probability = Number of favorable outcomes / Total number of possible outcomes

In this case, there are 9 different names in the hat, and Ted is one of them. Therefore, the theoretical probability of choosing Ted is:

Theoretical probability = 1 / 9

Simplifying, the theoretical probability of choosing the name Ted is approximately 0.1111 or about 11.11% (rounded to the nearest hundredth).

If the number of names in the hat were different, both the experimental and theoretical probabilities would change.

For the experimental probability, the number of times Ted is chosen would change based on the frequency of Ted in the new set of names. The total number of names chosen would also change.

For the theoretical probability, the total number of possible outcomes would change based on the number of names in the hat. If there were more names, the probability of choosing Ted would decrease. Conversely, if there were fewer names, the probability of choosing Ted would increase.

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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)

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 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.

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.

Learn more about conversion factor click;

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