Solve the equation for the variable given. Solve for x

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.

Given statement solution is :- It takes 4 months to pay off the card, and the total amount paid, including interest, is $4,871.78.

To find out how many months it takes to pay off the credit card and the total amount paid including interest, we can use the following steps:

Step 1: Calculate the monthly interest rate

Divide the annual percentage rate (APR) by 12 to get the monthly interest rate.

Monthly interest rate = 14.5% / 12 = 0.145 / 12 = 0.01208 (rounded to 5 decimal places)

Step 2: Set up the equation for the number of months

Let's denote the number of months it takes to pay off the card as 'n'. The monthly payment is $1,200.00, and the initial balance is $3,287.90. The monthly interest rate is 0.01208. The equation for the number of months can be written as:

(1) $3,287.90 ×[tex](1 + 0.01208)^n[/tex] - $1,200.00 ×[tex][(1 + 0.01208)^n[/tex] - 1] = 0

Step 3: Solve the equation

To find the value of 'n', we need to solve equation (1). However, solving it algebraically can be complex. Instead, we can use numerical methods like trial and error, or we can use a spreadsheet or a calculator to find the solution.

Using a spreadsheet or a calculator, we can input the values and increment 'n' until we find the point where the equation equals zero.

After performing the calculations, it is determined that it takes approximately 4 months to pay off the card, and the total amount paid, including interest, is $4,871.78.

Therefore, the answer to the original question is that it takes 4 months to pay off the card, and the total amount paid, including interest, is $4,871.78.

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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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Answer:

Option 3

(3xy³ + 5x²y⁴)  (3xy³ - 5x²y⁴)

Step-by-step explanation:

Factorize polynomials:

Use exponent law:

                   [tex]\boxed{\bf a^{m*n}=(a^m)^n} \ & \\\\\boxed{\bf a^m * b^m = (a*b)^m}[/tex]

9x²y⁶ = 3²* x² * y³*² = 3² * x² * (y³)² = (3xy³)²

25x⁴y⁸ = 5² * x²*² * y⁴*² = 5² * (x²)² * (y⁴)² = (5x²y⁴)²

Now use the identity:  a² - b² = (a +b) (a -b)

Here, a = 3xy³ & b = 5x²y⁴

9x²y⁶ - 25x⁴y⁸ = 3²x²(y³)² - 5²(x²)² (y⁴)²

                       = (3xy³)² - (5x²y⁴)²

                      = (3xy³ + 5x²y⁴)  (3xy³ - 5x²y⁴)

The Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides.

From the given Pythagoras theorem above, the mathematical representation of the theorem would be written below as follows;

C² = a²+b²

Where:

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The relationship in this problem is not proportional, as there are different ratios between the number of people and the number of cars washed.

A proportional relationship is a relationship in which a constant ratio between the output variable and the input variable is present.

The equation that defines the proportional relationship is a linear function with slope k and intercept zero given as follows:

y = kx.

The slope k is the constant of proportionality, representing the increase or decrease in the output variable y when the constant variable x is increased by one.

The ratios between the output and the inputs are given as follows:

Different ratios, hence the relationship is not proportional.

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Answer:

84

Step-by-step explanation:

a² + b² = c²

4² + b² = 10²

16 + b² = 100

b² = 100 - 16

b² = √84

Answer:

Hi

Please mark brainliest ❣️

Thanks

Step-by-step explanation:

Using Pythagorean theorem

hyp² = adj² + opp²

10² = 4² +b²

100-16 = b²

84= b²

√84 = b²

.°. b = 9.17

or b = √84

As you can see below, both expressions result in the same value of 105. This demonstrates the application of the associative law in regrouping terms and maintaining the equivalence of the expression.

The associative law in mathematics states that the grouping of numbers in an addition or multiplication operation does not affect the result. In other words, you can regroup terms within parentheses without changing the value of the expression.

Using the associative law, we can regroup the terms in the expression s + (r + 75) by removing the parentheses and rearranging the terms:

s + (r + 75) = (s + r) + 75

The expression (s + r) + 75 is equivalent to s + (r + 75) because the addition operation is associative.

Let's take an example to illustrate this:

Suppose s = 10 and r = 20.

Using the original expression:

s + (r + 75) = 10 + (20 + 75) = 10 + 95 = 105

Using the expression with regrouped terms:

(s + r) + 75 = (10 + 20) + 75 = 30 + 75 = 105

As you can see, both expressions result in the same value of 105. This demonstrates the application of the associative law in regrouping terms and maintaining the equivalence of the expression.

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a) The  two equations in terms of x and y: one for the perimeter and one for the area of the rectangle are:

y = 0.5x + 3.5

6y + 3 = x² + 14x + 48

b) The area and perimeter of the rectangle are:

Perimeter = 30 m

Area = 15 m²

The formula to find the area of a rectangle is:

Area = Length * Width

The formula to find the perimeter of a rectangle is:

Perimeter = 2(Length + Width)

We are given that:

Perimeter = 8y meters

Area = (6y + 3) square meters

From the image, we see that:

Length = x + 6

Width = x + 8

Thus:

Perimeter equation is:

8y = 2(x + 6 + x + 8)

8y = 4x + 28

y = 0.5x + 3.5

Area equation is:

6y + 3 = (x + 6)(x + 8)

6y + 3 = x² + 14x + 48

Thus:

6(0.5x + 3.5) + 3 = x² + 14x + 48

3x + 24 = x² + 14x + 48

x² + 11x + 24 = 0

Using quadratic equation calculator gives:

x = -8 or -3

Thus, we will use x = -3 and we have:

Length = -3 + 6 = 3 m

Width = -3 + 8 = 5 m

Perimeter = 2(3 + 5) = 30 m

Area = 3 * 5 = 15 m²

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The scale factor of the dilation is 4.

We are given that;

The equation of line −11x=4y+4

Now,

The center of dilation is the origin (0, 0). The line given by −11x = 4y + 4 can be rewritten as y = −11/4 x − 1. This means that it passes through the points (0, −1) and (−4, 0). The image of the line after dilation is given by −11x − 4y = 16, which can be rewritten as y = −11/4 x − 4. This means that it passes through the points (0, −4) and (−16, 0).

To find the scale factor, we can use any pair of corresponding points. For example, let’s use (0, −1) and (0, −4). The distance from the center of dilation to (0, −1) is 1 unit. The distance from the center of dilation to (0, −4) is 4 units. Therefore, the scale factor is 4/1 = 4.

Therefore, by dilation the answer will be 4

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Answer:

To determine the possible degree(s) of a function from the graph alone, you need to examine the behavior of the graph at the extremes (far left and far right) and consider the number of turning points or changes in direction. Here's a step-by-step approach:

Look at the far left side of the graph: Determine the behavior of the graph as it approaches negative infinity. Does the graph approach a specific value, such as a horizontal line (asymptote) or the x-axis? If the graph approaches a horizontal line, it suggests a polynomial function of even degree. If the graph approaches the x-axis, it indicates a polynomial function of odd degree or possibly a function with a root of multiplicity greater than one.

Look at the far right side of the graph: Determine the behavior of the graph as it approaches positive infinity. Similar to step 1, observe if the graph approaches a specific value or a horizontal line. The behavior at the far right side should be consistent with the behavior at the far left side. This can help you identify if the function is even or odd degree.

Examine the number of turning points or changes in direction: Count the number of times the graph changes direction. These points are where the slope of the graph changes from positive to negative or vice versa. The number of turning points can provide an indication of the degree of the polynomial. For example, if there are two turning points, it suggests a polynomial function of degree 3.

Remember that this method provides potential degrees, but it may not definitively determine the exact degree of the function. Additional information or analysis might be required for a more accurate determination.

Answer:

None

Step-by-step explanation:

Truly, I'm not sure what type of problem this is, but most people don't favor all the seasons. If there is more to the problem, I would be glad to help further.

Answer:

One possible way to estimate how many people out of 20 would say that they like all seasons is to use a simple random sample. A simple random sample is a subset of a population that is selected in such a way that every member of the population has an equal chance of being included. For example, one could use a random number generator to assign a number from 1 to 20 to each person in the population, and then select the first 20 numbers that appear. The sample would then consist of the people who have those numbers.

Using a simple random sample, one could ask each person in the sample whether they like all seasons or not, and then calculate the proportion of positive responses. This proportion is an estimate of the true proportion of people in the population who like all seasons. However, this estimate is not exact, and it may vary depending on the sample that is selected. To measure the uncertainty of the estimate, one could use a confidence interval. A confidence interval is a range of values that is likely to contain the true proportion with a certain level of confidence. For example, a 95% confidence interval means that if the sampling procedure was repeated many times, 95% of the intervals would contain the true proportion.

One way to construct a confidence interval for a proportion is to use the formula:

p ± z * sqrt(p * (1 - p) / n)

where p is the sample proportion, z is a critical value that depends on the level of confidence, and n is the sample size. For a 95% confidence interval, z is approximately 1.96. For example, if out of 20 people in the sample, 12 said that they like all seasons, then the sample proportion is 0.6, and the confidence interval is:

0.6 ± 1.96 * sqrt(0.6 * (1 - 0.6) / 20)

which simplifies to:

0.6 ± 0.22

or:

(0.38, 0.82)

This means that we are 95% confident that the true proportion of people who like all seasons in the population is between 0.38 and 0.82. Therefore, based on this sample and this confidence interval, we would expect between 8 and 16 people out of 20 to say that they like all seasons in the population.

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Answer:

D is the correct answer

Step-by-step explanation:

Acellus

There are a total of 3 forwards on the team.

A fraction represents a part of the whole or group of objects.In a fraction, numerator and denominator are separated by a horizontal bar known as the fractional bar

Given:

2/8 of the players are forwards.

Total number of players on the team = 12

We will determine the total number of forwards on the team by multiplying the total number of players by the fraction representing the forwards.

Fraction representing the forwards:

= 2/8

= 1/4

Total forwards on the team:

= 12 * (1/4)

= 3.

Translated question:

Of the 12 players on the team, 2/8 are forwards. What are the total forward in the team?

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<1 and <2 are same-side interior angles. Option D

<1 and <5 are same-side interior angles. Option B

<6 and <8 are corresponding angles. Option C

To determine the angles, we need to know the following;

Now, from the information given, we have that;

1. <1 and <2 are same-side interior angles because they are found on the same side of the transversal

2. We can see that < 1 and <5 are also same-side interior angles because they are two angles found on the exact same side of the transversal

3. Since corresponding angles are angles on matching corners of a transversal, < 6 and < 8 are corresponding angles

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The best way to solve an equation of the form Ax+by=c could be either substitution, completing the squares or graphical method depending on the type of equation given.

Quadratic equation

The equation in the form Ax+by=c is a quadratic problem which could be approached in different ways depending on the specific equation given and the information provided.

The methods of approach are substitution, completing the squares or graphical method which all have proven to be suitable ways to arrive at the solution.

Hence, either of the three methods are good ways to solve such equation.

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The equation that represents this situation is:

⇒ 60t = 0.0284091

where t is the time in minutes.

We have to given that,

The speed of the cheetah is 60 miles per hour

The distance timed was 50 yards

Now, We have to convert the distance to miles, because the speed is in miles per hour.

Since, 1 yard = 0.000568182 miles

Hence, 50 yards = 50 x 0.000568182 miles

50 yards = 0.0284091 miles

Now we can use the formula:

distance = speed x time

We know that,

The distance is 0.0284091 miles, and the speed is 60 miles per hour.

Hence, the time it took the cheetah to run the 50-yard distance.

So, the speed from miles per hour to miles per minute, since we want the time in minutes:

60 miles per hour = 1 mile per minute

Now we can plug in the values we know:

0.0284091 miles = (1 mile per minute) x time

Solving for time:

time = 0.0284091 / 1

time = 0.0284091 minutes

time = 0.0284091 minutes x 60 seconds per minute

time = 1.704546 seconds

Thus, the equation that represents this situation is:

⇒ 60t = 0.0284091

where t is the time in minutes.

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The expression that is equivalent to 6/q is 6q/q²

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

6/q = ?/q²

Multiply both sides of the equation by q²

So, we have

q² *  6/q = ?/q² * q²

Evaluate the products on both sides of the equation

? = 6q

Hence, the expression that is equivalent to 6/q is 6q/q²

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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

Approximately 37.5% of the data are greater than 28, 33.3% of the data are less than 23, 91.7% of the data are greater than 17.5, and 62.5% of the data are between 17.5 and 28.

To answer the questions, we can analyze the given data set.

First, let's count the number of data points that satisfy each condition:

Greater than 28:

There are 9 data points greater than 28 (29, 29, 29, 30, 29, 29, 28, 28, 28).

Less than 23:

There are 8 data points less than 23 (20, 18, 15, 17, 17, 15, 17, 19).

Greater than 17.5:

There are 22 data points greater than 17.5.

Between 17.5 and 28:

There are 15 data points between 17.5 and 28.

Now, let's calculate the approximate percentage for each condition:

Percent greater than 28:

The total number of data points is 24. Approximately, 9 out of 24 data points are greater than 28.

Percentage =[tex](9 / 24) \times 100 = 37.5[/tex]%.

Percent less than 23:

Approximately, 8 out of 24 data points are less than 23.

Percentage = [tex](8 / 24) \times 100 = 33.3[/tex]%.

Percent greater than 17.5:

Approximately, 22 out of 24 data points are greater than 17.5.

Percentage = [tex](22 / 24) \times 100 = 91.7[/tex]%.

Percent between 17.5 and 28:

Approximately, 15 out of 24 data points are between 17.5 and 28.

Percentage = [tex](15 / 24) \times 100 = 62.5[/tex]%.

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Answer:

208 units

Step-by-step explanation:

The first term is given as 16, which means a = 16.

The second term can be obtained by adding the common difference to the first term: 16 + d = 48.

The third term is obtained by adding the common difference to the second term: 48 + d = 80.

The fourth term is obtained by adding the common difference to the third term: 80 + d = 112.

We can solve these equations to find the value of 'd':

16 + d = 48

d = 48 - 16

d = 32

48 + d = 80

32 + 48 = 80 (valid)

80 + d = 112

32 + 80 = 112 (valid)

Therefore, the common difference is 32.

Now that we have the common difference, we can find the distance the body falls in the seventh second.

The formula for finding the nth term of an arithmetic progression is:

a_n = a + (n - 1) * d

where a_n is the nth term, a is the first term, n is the position of the term, and d is the common difference.

Plugging in the values, we can find the seventh term:

a_7 = 16 + (7 - 1) * 32

a_7 = 16 + 6 * 32

a_7 = 16 + 192

a_7 = 208

Therefore, the distance the body falls in the seventh second is 208 units.

The quantity of goods and services that sellers are willing and able to sell is known as the quantity supplied.

Quantity supplied is defined as the amount of goods and services that a supplier is able to produce and sell at a given market price.

The quantity supplied differs from the actual amount of supply as price changes influence how much supply producers actually put on the market.

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The distance between the line y = 5 and point C (7, 0) is 5 units.

We have,

To find the distance between a line and a point, we can use the formula for the distance between a point (x1, y1) and a line ax + by + c = 0, which is given by:

Distance = |ax1 + by1 + c| / √(a² + b²)

In this case,

The equation of the line is y = 5, which can be rewritten as 0x + 1y - 5 = 0.

Comparing this to the general form ax + by + c = 0, we have a = 0, b = 1, and c = -5.

The coordinates of point C are (7, 0), so x1 = 7 and y1 = 0.

Let's calculate the distance using the formula:

Distance = |0(7) + 1(0) - 5| / sqrt(0² + 1²)

Distance = |-5| / √(1)

Distance = 5 / 1

Distance = 5

Therefore,

The distance between the line y = 5 and point C (7, 0) is 5 units.

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Apologies, but your questions seem to have errors and incomplete information.

For the first question about Jessica and Melissa sharing 12 pieces of dried pears, it is unclear how much Jessica ate as the fraction is missing. Similarly, information about how many pears Melissa ate is also missing. To answer the question accurately, I need the missing information.

For the second question about the children playing in the park, you mentioned that one-third of the children returned home, but you didn't provide the total number of children initially in the park. Without that information, I cannot determine how many children stayed in the park.

Please provide complete and accurate information for a precise answer.

Options 2, 3, and 5 are correct representations of the line passing through the given points.

The equation y = -5/3(x - 2) represents a line passing through the points (-4, 10) and (-1, 5).

Let's verify each option:

y = -5/3x - 2: This equation does not represent the same line. The constant term is different (-2 instead of +10/3).

y = -5/3x + 10/3: This equation represents the same line. It has the same slope (-5/3) and the same y-intercept (10/3).

3y = -5x + 10: This equation represents the same line. It can be simplified by dividing both sides by 3, resulting in the same slope (-5/3) and the same y-intercept (10/3).

3x + 15y = 30: This equation does not represent the same line. The coefficients of x and y are different, resulting in a different slope.

5x + 3y = 10: This equation represents the same line. It has the same slope (-5/3) and the same y-intercept (10/3).

Therefore, options 2, 3, and 5 are correct representations of the line passing through the given points.

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The exact value of cos(-n/12) is equal to cos(n/12).

The cosine function (or cos function) in a triangle is the ratio of the adjacent side to that of the hypotenuse.

The value of cosine function is periodic with a period of 2π. Therefore, we can use the property cos(x) = cos(x + 2πk) for any integer value of k.

In this case, we have cos(-n/12). To find the exact value, we can use the fact that cos(x) = cos(-x) for any angle x. Therefore, we can rewrite cos(-n/12) as cos(n/12).

So, the exact value of cos(-n/12) is equal to cos(n/12).

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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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Answer: [tex]3x^4yz^3[/tex]

I'm hoping this is your equation: [tex](9x^{8}y^{2}z^{6})^{1/2}[/tex]

and not: [tex](9x^{8y^{2z^{z^6}}})[/tex]

Step-by-step explanation:

The square root of a number can be shown as a 1/2 power

We'll use the exponent rule:

= [tex]9^{1/2}(x^8 )^{1/2}(y^2)^{1/2}(z^6){1/2}[/tex]

Then we'll do each term individually

the square root of 9 is 3

for [tex](x^8)^{1/2}[/tex] the exponents multiply which give us [tex]x^4[/tex]

[tex](y^2)^{1/2}[/tex] gives us y

[tex](z^6)^{1/2}[/tex] gives us z^3

After doing all this we get [tex]3x^4yz^3[/tex]

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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