Pythagorean theorem answer quick please

The histogram represents the sampling distribution of the proportion of heads in 10 flips of the coin, which is an example of a binomial distribution.

In this scenario, the parameter of interest is the true population proportion of heads in a coin flip. The process of flipping the coin 10 times and recording the proportion of heads is an example of a binomial distribution, where each flip is a Bernoulli trial with a probability of success (getting heads) of 0.5.
By repeating this process many times and creating a histogram of the results, we are creating a sampling distribution of the proportion of heads in 10 flips of the coin. This allows us to see the variability of the proportion of heads we could get from different samples of the same size.
If we are using simple random sampling, meaning each possible sample of 10 coin flips has an equal chance of being chosen, then there should be no bias present in our results. However, if we are using a different sampling method, such as convenience sampling, there could be a bias present in our results.

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To estimate the sales for the coffee shop in 2008 and 2018, we first need to find the values of t for those years. Since t = 6 corresponds to 1996, we can calculate the values for 2008 and 2018 as follows:
2008: t = 6 + (2008 - 1996) = 6 + 12 = 18
2018: t = 6 + (2018 - 1996) = 6 + 22 = 28

Now, we can plug these values of t into the exponential function S(t) = 188.38(1.272)^t to estimate the sales.
For 2008:
S(18) = 188.38(1.272)^18 ≈ 5170.9
For 2018:
S(28) = 188.38(1.272)^28 ≈ 14264.5
So, the estimated sales for the coffee shop in 2008 is approximately $5,170.9 million, and for 2018, it's approximately $14,264.5 million.

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The probability that Ella chooses the picture with the dog is 1/5 or 0.2, which can also be expressed as a percentage, 20%.

Probability is a measure of the probability of an event. It measures the certainty of an event. The probability formula  is given; P(E) = number of positive results / total number of results.

The probability of choosing a picture with a dog is the ratio of the number of dog pictures  to the total number of pictures.  We learn that Ella drew 40 different images for the art exhibit, eight of which feature a dog. Therefore, the probability of choosing a picture with a dog is:

8/40

Simplifying this fraction, we get:

1/5

So the probability that Ella chooses the picture with the dog is 1/5 or 0.2, which can also be expressed as a percentage, 20%.

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Answer:

a.6×(7-5)+4=16

b.-2+24÷(12-4)=2

A price of $75 per unit will yield maximum total revenue for the wholesaler.

To find the price that will yield maximum total revenue, we need to use the concept of elasticity of demand.

Elasticity of demand measures the responsiveness of the quantity demanded to a change in price.

The formula for the price elasticity of demand is:

E = (percent change in quantity demanded) / (percent change in price)

If E is greater than 1, demand is considered elastic, meaning that a change in price will result in a more than proportional change in quantity demanded.

If E is less than 1, demand is considered inelastic, meaning that a change in price will result in a less than proportional change in quantity demanded.

If E is equal to 1, demand is unit elastic, meaning that a change in price will result in a proportional change in quantity demanded.

We know that before the price increase, the wholesaler was selling 250 units per week at a price of $40 per unit.

After the price increase, the wholesaler was selling 225 units per week at a price of $45 per unit.

Using these values, we can calculate the price elasticity of demand as follows:

E = ((225 - 250) / 250) / ((45 - 40) / 40) = -0.5

The negative sign indicates that the relationship between price and quantity demanded is inverse.

Also, we see that the absolute value of E is less than 1, indicating that demand is inelastic.

This means that a price increase results in a less than proportional decrease in quantity demanded.

Now, to find the price that will yield maximum total revenue, we can use the following formula:

P* = E / (E - 1) * C

Where P* is the price that will yield maximum total revenue, E is the price elasticity of demand, and C is the current price that the wholesaler is charging.

Plugging in the values we know, we get:

P* = (-0.5) / (-0.5 - 1) * 45 = $75.

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The marginal profit when x = 250 is -$12,500, indicating that the company is losing money at this production level.

(A) The revenue function can be obtained by multiplying the number of units sold by the selling price per unit:

R(x) = p * x = (-30x + 1400) * x = -30[tex]x^2[/tex] + 1400x

(B) The cost function consists of two components: fixed cost and variable cost. T

he fixed cost is given as $5000, and the variable cost is $250 per unit. Therefore, the cost function is:

C(x) = 5000 + 250x

(C) The profit function is obtained by subtracting the cost function from the revenue function:

P(x) = R(x) - C(x) = (-30[tex]x^2[/tex] + 1400x) - (5000 + 250x) = -30[tex]x^2[/tex] + 1150x - 5000

(D) The marginal profit is the derivative of the profit function with respect to x, evaluated at x = 250:

P'(x) = -60x + 1150

P'(250) = -60(250) + 1150 = -12500

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Answer:

The given equation is 2x

2

−10x+k=0

Here, a=2,b=−10,c=k

∴D=b

2

−4ac=(−10)

2

−4×2×k=100−8k

The given equation will have real and equal roots, if

D=0⇒100−8k=0⇒k=

8

100

​

=

2

25

​

Step-by-step explanation:

The point on the plane r + y + z = 1 with the shortest distance from the point (2, 0, -3) is (1, 1, -1), and the shortest distance is √14.


1. Given the plane equation r + y + z = 1 and the point P(2, 0, -3).
2. Introduce a point Q(r, y, z) on the plane.
3. Determine the vector PQ = .
4. Find the normal vector of the plane. Rewrite the equation as r + y + z - 1 = 0. The coefficients of r, y, and z form the normal vector N = <1, 1, 1>.
5. Calculate the projection of PQ onto N: proj_N(PQ) = (PQ.N/||N||²)N = ((r-2)+y+(z+3))/3N.
6. The shortest distance occurs when PQ is orthogonal to the plane, so proj_N(PQ) = PQ.
7. Equate the components of PQ and proj_N(PQ): r-2 = (r+1)/3, y = (y-1)/3, z+3 = (z+4)/3.
8. Solve for r, y, and z: r = 1, y = 1, z = -1.
9. Shortest distance = ||PQ|| = √((1-2)² + (1-0)² + (-1+3)²) = √14.

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The position of the object at time t = 3 is approximately 6.59 units from the origin.

The velocity of an object is defined as the rate of change of its position with respect to time. Mathematically, we can express this as:

v(t) = dx/dt

where v(t) is the velocity of the object at time t, and dx/dt is the derivative of the position function x(t) with respect to time.

In this problem, we are given the velocity function v(t) as:

v(t) = (1+t²)¹/₃

To find the position function x(t), we need to integrate the velocity function with respect to time. We can do this as follows:

x(t) = ∫v(t)dt

x(t) = ∫(1+t²)^(1/3) dt

To evaluate this integral, we can use the substitution u = 1 + t², which gives du/dt = 2t. Substituting this into the integral, we get:

x(t) = (3/2) * ∫u¹/₃ * (1/2u) du

x(t) = (3/2) * ∫u⁻¹/₆ du

x(t) = (3/2) * (u⁵/₆ / (5/6)) + C

where C is the constant of integration.

To find the value of C, we need to use the initial condition x(0) = 2. Substituting t = 0 into the position function, we get:

x(0) = (3/2) * (u⁵/₆ / (5/6)) + C

x(0) = (3/2) * (1⁵/₆ / (5/6)) + C

x(0) = 2

Therefore, C = 2 - (3/2) * (1⁵/₆ / (5/6)) = 2 - (3/2) * (1 / (5/6)) = 2 - (9/5) = 1.2

Substituting C = 1.2 into the position function, we get:

x(t) = (3/2) * (u⁵/₆ / (5/6)) + 1.2

x(t) = (3/2) * ((1+t²)⁵/₆ / (5/6)) + 1.2

Finally, to find the position of the object at time t = 3, we simply substitute t = 3 into the position function:

x(3) = (3/2) * ((1+3²)⁵/₆/ (5/6)) + 1.2

x(3) ≈ 6.59

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Answer:

82 1/2 in²

Step-by-step explanation:

First find the area of 1 planter

11/2 * 5/3 = 55/6

There is a total of 9 boxes / planters in the picture, so:

55/6 * 9 = 165/2 = 82 1/2 or as a decimal it would be 82.5

It is important to consult a healthcare provider to determine if doxycycline is appropriate for a particular individual, regardless of their age.

Doxycycline is a broad-spectrum antibiotic that is used to treat a variety of bacterial infections. However, there are certain medical conditions and factors that may contraindicate the use of doxycycline.

Some of the common medical conditions that may prevent the use of doxycycline include:

Allergy or hypersensitivity to doxycycline or other tetracycline antibiotics.

Severe liver disease or impairment.

Pregnancy or breastfeeding.

Children under the age of 8 (because doxycycline can cause permanent discoloration of teeth and affect bone growth).

Kidney disease, because doxycycline is excreted through the kidneys and may accumulate to toxic levels.

In general, the use of doxycycline is based on the patient's medical history, the severity of the infection, and other factors such as age and weight. Therefore, it is important to consult a healthcare provider to determine if doxycycline is appropriate for a particular individual, regardless of their age.

Complete question : It is important to consult a healthcare provider to determine if doxycycline is appropriate for a particular individual, regardless of their age. what dx gets doxy no matter what age?

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The minimum value of f(x,y) subject to the constraint 2x + 3y = 6 is

approximately 0.9615, and the maximum value is approximately 5.5385.

To find the minimum and maximum values of the function [tex]f(x,y) = x^2 + y^2[/tex] subject to the constraint 2x + 3y = 6, we can use the method of Lagrange multipliers.

First, we define the Lagrangian function [tex]L(x, y, \lambda) = x^2 + y^2 + λ(2x + 3y - 6).[/tex]

Next, we find the partial derivatives of L with respect to x, y, and λ, and set them equal to zero:

∂L/∂x = 2x + 2λ = 0

∂L/∂y = 2y + 3λ = 0

∂L/∂λ = 2x + 3y - 6 = 0

Solving these equations, we get x = -3/13, y = 6/13, and λ = -4/13.

To determine whether this point is a minimum or maximum, we need to

check the second partial derivatives of the Lagrangian function:

[tex]d^2L/dx^2 = 2[/tex]

[tex]d^2L/dy^2 = 2[/tex]

[tex]d^2L/dxdy = 0[/tex]

The Hessian matrix is therefore positive definite, which means that the

point (-3/13, 6/13) is a minimum value of f(x,y) subject to the constraint 2x

+ 3y = 6.

To find the maximum value, we can use the same method with a

negative sign in front of the Lagrange multiplier λ:

[tex]L(x, y, \lambda) = x^2 + y^2 - \lambda(2x + 3y - 6)[/tex]

∂L/∂x = 2x - 2λ = 0

∂L/∂y = 2y - 3λ = 0

∂L/∂λ = 2x + 3y - 6 = 0

Solving these equations, we get x = 6/13, y = -3/13, and λ = 4/13.

Again, we check the second partial derivatives of the Lagrangian function:

[tex]d^2L/dx^2 = 2\\d^2L/dy^2 = 2\\d^2L/dxdy = 0[/tex]

The Hessian matrix is positive definite, which means that the point (6/13,

-3/13) is a maximum value of f(x,y) subject to the constraint 2x + 3y = 6.

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The word AND in probability implies that we use the​ multiplication rule.

The word OR in probability implies that we use the​ addition rule.

The statement "if two events are disjoint, then they are independent" - this statement is FALSE because those events that cannot occur simultaneously, and independent events are those events that do not affect the probability of each other's occurrence.

The word "AND" in probability implies that we use the "multiplication rule." This rule states that the probability of two events occurring together is the product of their individual probabilities.

The word "OR" in probability implies that we use the "addition rule." This rule states that the probability of at least one of the events occurring is the sum of their individual probabilities, minus the probability of both events occurring simultaneously.

Two events can be either disjoint or independent, but they cannot be both at the same time. For instance, let's say we are rolling a die. The events "getting a 1" and "getting an even number" are disjoint, as they cannot occur simultaneously. However, they are not independent, as the occurrence of one event affects the probability of the other event occurring. Specifically, if we get a 1, the probability of getting an even number reduces to zero.

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The word AND in probability implies the multiplication rule, while the word OR implies the addition rule. Disjoint events are not necessarily independent.

The word AND in probability implies that we use the multiplication rule. The word OR in probability implies that we use the addition rule.

However, it is FALSE that if two events are disjoint, then they are independent. Disjoint events mean that they have no outcomes in common, while independent events mean that the occurrence of one event has no effect on the probability of the occurrence of the other event.

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The format of the data set appears to be unstacked.

Unstacked data refers to data that is organized in separate columns for each variable or category. In this case, the table likely has separate columns for the GPA of students in the morning math class and the afternoon math class.

Each row in the table would represent a student and contain separate entries for their GPA in the morning and afternoon classes. This allows for easy comparison of GPAs between the two classes as each class has its own column.

Therefore, based on the given information, the format of the data set is unstacked.

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Answer:

12.99

Step-by-step explanation:

The high temperature is given as follows:

104 ºF.

The temperatures are obtained by a system of equations, for which the variables are given as follows:

The difference between the record high and low temperatures in Chicago, Illinois is 109°F, hence:

x - y = 109.

The record low temperature was -5°F, hence:

y = -5.

Then the high temperature is obtained as follows:

x - (-5) = 109

x + 5 = 109

x = 104 ºF.

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The probability that exactly 5 out of 10 graduates become vice presidents is 0.2007 or 0.201 (rounded to three decimal places).

We can use the binomial distribution to solve this problem.

Let X be the number of graduates who become vice presidents out of the 10 selected.

Then X follows a binomial distribution with parameters n = 10 and p = 0.4. We want to find the probability that exactly 5 become vice presidents, i.e., P(X = 5).

Using the formula for the binomial probability mass function, we have:

P(X = 5) = (10 choose 5) *[tex](0.4)^5 * (0.6)^5[/tex]

P(X = 5) = (252) * (0.01024) * (0.07776)

P(X = 5) = 0.2007

Therefore,

The probability that exactly 5 out of 10 graduates become vice presidents is 0.2007 or 0.201 (rounded to three decimal places).

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Approximately 34 percent of people sleep between 6 and 7.5 hours per night.

Since the distribution is symmetric, we know that the percentage of people who sleep between 6 and 7.5 hours per night is approximately half of the total percentage of people who sleep within one standard deviation of the mean.

One standard deviation from the mean in either direction is 1.5 hours, so the percentage of people who sleep between 4.5 and 7.5 hours per night is approximately 68%. Therefore, approximately half of 68%, or 34%, of people sleep between 6 and 7.5 hours per night.

In this case, we want to find the percentage of people who sleep between 6 and 7.5 hours per night. Since the mean is 6 hours and the standard deviation is 1.5 hours, one standard deviation above the mean is 6 + 1.5 = 7.5 hours. Therefore, according to the Empirical Rule, approximately 34% of people sleep between 6 and 7.5 hours per night.

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The probability of you getting into an accident is 10%, and the probability of your neighbor getting into an accident is 20%.

The probability of both you and your neighbor getting into an accident on the way to work today is 10% x 20% = 2%.

The first scenario, we are given the probability of taking two different jobs.

Since the jobs are mutually exclusive, meaning you can only take one job, we can add the probabilities of taking each job to find the overall probability of taking either one job or the other job.
The probability of taking the first job is 50%, and the probability of taking the second job is 40%.

The probability of taking either one job or the other job is 50% + 40% = 90%.
The second scenario, we are given the probability of both you and your neighbor getting into an accident on the way to work.

Since the events are independent, meaning whether one of you has an accident does not affect the probability of the other having an accident, we can multiply the probabilities of each event happening to find the overall probability of both events happening.
These two scenarios demonstrate the importance of understanding probabilities and how they can be calculated to make informed decisions.

In the first scenario, we can use probabilities to determine the likelihood of taking either job, while in the second scenario, we can use probabilities to determine the likelihood of both you and your neighbor getting into an accident on the way to work.

By understanding probabilities, we can make more informed decisions and take appropriate actions to mitigate risks.

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Answer:

A. 3x + x = 4x

E. x + x + x + x = 4x

A and E are the correct expressions.

The s20 of the arithmetic progression is 660.

The sum of the of the 1st nth term an arithmetic progression can be determined using the formula:

Sₙ = (n/2) * (2a + (n-1)d)

where n is the number of term, a is the first term and d is the common difference

Given:

a = 14

d = 16 - 4 = 2

n = 20

Thus, sum of the of the 1st twenty term (s20) is:

s20 = (20/2) * (2*14 + (20-1)2)

s20 = 10 * (28 + 38)

s20 = 660

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By the use of trigonometric identity the distance between two cars is 431 ft.

What is trigonometric identity?

Trigonometric Identities are used whenever trigonometric functions are involved in an expression or an equation. Trigonometric Identities are true and it is proven for every value of variables occurring on both sides of an equation. These identities involve certain trigonometric functions for example sine, cosine, tangent, cotangent of one or more angles.

Previous Sarcoma PO Weight Try Again 1 Solution A police helicopter (A) and a police cruiser (B) are chasing the car (C) of a suspect. The helicopter is flying at a height of 384 feet. From the helicopter the angles of depression of the police cruiser and the suspect's car are 73 and 35 respectively.

A diagram regarding to the given problem is attached below.

From the diagram,

AB = height of the helicopter from ground = 384 ft

∠EAC and ∠EAD are angles of dispersion which are equals to 73° and 35° respectively.

Now ∠EAC and ∠ACB are alternate angles also ∠EAD and ∠ADB are alternate angles so both are equal.

So ∠ACB= 73° and ∠ADB= 35°

Now we are going to use trigonometric identities to find the value of distance between two cars.

Here we will use the trigonometric identity tangent.

tan ∠ACB = AB/BC [ since in trigonometry tan∅= perpendicular/base , where ∅ is an angle of a right angled triangle.]

tan  73° = 384/ BC

BC= 384/ tan  73°

BC= 117.4

Again,

tan ∠ADB = AB/BD

tan 35° = 384/ BD

BD= 384/ tan 35°

BD= 548.4

now BD= BC+CD

       548.4 = 117.4 + CD

So, CD = 548.4- 117.4

             = 431

Hence, the distance between two cars is 431 ft.

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The scatter plot of fees versus days taken to fill a job for Sarin Employment's sample data suggests that there may be a positive linear relationship between the two variables.

a. To construct a scatter plot in Excel, we would plot the fees on the y-axis (dependent variable) and the days taken to fill a job on the x-axis. Each data point would be represented as a dot on the graph.

Next, to calculate the correlation coefficient and test for statistical significance, we can use a statistical calculator or software. In this case, since the alpha level is given as 0.05, we will use a significance level of 0.05 for our hypothesis test.

b. The correlation coefficient, also known as the Pearson correlation coefficient, measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where -1 indicates a perfect negative linear relationship, 0 indicates no linear relationship, and 1 indicates a perfect positive linear relationship.

To test for statistical significance, we will conduct a hypothesis test with the following hypotheses:

Null Hypothesis (H0): There is no statistically significant linear relationship between fees and days taken to fill a job.

Alternative Hypothesis (H1): There is a statistically significant linear relationship between fees and days taken to fill a job.

We will use a two-tailed test since we are interested in determining whether there is any linear relationship, regardless of the direction.

If the calculated p-value is less than our chosen significance level of 0.05, we will reject the null hypothesis and conclude that there is a statistically significant linear relationship between fees and days taken to fill a job.

Therefore, by analyzing the scatter plot and conducting a hypothesis test, we can determine the apparent relationship between fees and days taken to fill a job and assess whether it is statistically significant.

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The value of p to the nearest tenth of a percent, based on the given scenario, is approximately 77.8%.

We can use the formula for the multiplier effect to solve for p

Total revenue effect = Initial spending × multiplier

Total revenue effect = $9 billion

Initial spending = $2 billion

So, multiplier = Total revenue effect / Initial spending

multiplier = $9 billion / $2 billion

multiplier = 4.5

Now we can use the formula for the multiplier to solve for p

multiplier = 1 / (1 - p)

4.5 = 1 / (1 - p)

4.5 - 4.5p = 1

-4.5p = -3.5

p = 3.5 / 4.5

p ≈ 0.7778

Therefore, p to the nearest tenth of a percent is approximately 77.8%.

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the series converges and the sum is 4/3. The answer is (b) converges, and the sum is 4/3.

The given series is (-1/-4)^(n-1), which can be rewritten as (1/4)^(n-1).

This is a geometric series with first term a=1 and common ratio r=1/4.

For a geometric series to converge, the absolute value of the common ratio must be less than 1.

|r| = |1/4| < 1

Therefore, the series converges.

To find the sum of the series, we use the formula for the sum of an infinite geometric series:

S = a/(1-r)

S = 1/(1-1/4)

S = 4/3

So the sum of the series is 4/3.
The given series is n=1 (-1/-4)^(n-1). To determine whether it converges or diverges, we can identify the type of series. This is a geometric series with a common ratio r = -1/-4 = 1/4.

For a geometric series to converge, the absolute value of the common ratio must be less than 1 (i.e., |r| < 1). In this case, |1/4| < 1, so the series converges.

To find the sum of a converging geometric series, we can use the formula:

Sum = a / (1 - r),

where 'a' is the first term of the series. In this case, a = (-1/-4)^(1-1) = 1.

Sum = 1 / (1 - 1/4) = 1 / (3/4) = 4/3.

So, the series converges and the sum is 4/3. The answer is (b) converges, and the sum is 4/3.

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The total time for the CPU to access all 100 instructions is 250 ns (for accessing the remaining 50 instructions) + 400 ns (for replacing the first 50 instructions) = 650 ns.

The total time for the CPU to access 100 program instructions can be calculated by adding the time taken to access the first 50 instructions, the time taken to replace those with the next 50, and the time taken to access those next 50 instructions.
1. Time to access the first 50 instructions: 50 instructions * 5 ns/instruction = 250 ns
2. Time to replace the first 50 instructions with the next 50: 400 ns
3. Time to access the next 50 instructions: 50 instructions * 5 ns/instruction = 250 ns
Total time = 250 ns (first 50) + 400 ns (replacement) + 250 ns (next 50) = 900 ns

The CPU needs to access a total of 100 instructions, but only the first 50 are already in memory. So, it needs to access the remaining 50 instructions from memory, which will take 50 * 5 = 250 ns. However, since the memory can only hold 50 instructions, it needs to replace the first 50 instructions with the next 50, which takes 400 ns.  Therefore, the total time for the CPU to access all 100 instructions is 250 ns (for accessing the remaining 50 instructions) + 400 ns (for replacing the first 50 instructions) = 650 ns.

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The integral of 2x(x-3) dx is ∫2x(x-3) dx = 2∫x²-3x dx = 2(x³/3 - 3x²/2) + C, where C is the constant of integration. Thus the answer is 2(x³/3 - 3x²/2) + C.

To evaluate the integral of the function 2x(x-3) dx, follow these steps:
1. Expand the function: 2x(x-3) = 2x^2 - 6x
2. Integrate term by term: ∫(2x^2 - 6x) dx = ∫2x^2 dx - ∫6x dx
3. Apply the power rule to each term:
  - For ∫2x^2 dx: (2/3)x^3 + C₁
  - For ∫6x dx: (6/2)x^2 + C₂
4. Combine the results: (2/3)x^3 + C₁ - (6/2)x^2 + C₂
5. Simplify and write the general form of the integral: (2/3)x^3 - 3x^2 + C, where C is the constant of integration.

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The U.S. Census Bureau takes a census every 10 years, which means that they attempt to count every person living in the United States and collect data on various demographic and social characteristics.

A sample is a subset of a larger population, selected in a way that it represents the characteristics of the population from which it is drawn. The purpose of sampling is to estimate or infer something about the population based on the characteristics of the sample.

On the other hand, a census is a survey or count that attempts to measure every member of a population.

In a census, data is collected on every individual or item in a population, rather than just a representative sample.

If you wanted to estimate the average income of households in a city, you could select a sample of households and estimate the average income based on the incomes of the sampled households.

This would be an example of sampling.

Alternatively, you could conduct a census of every household in the city, collecting income data from every household, and calculate the exact average income of all households in the city.

The purpose of the census is to provide a complete and accurate count of the population, which can then be used to allocate political representation and government funding, as well as to provide data for research and planning purposes.

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The area of the trapezoidal figure is 80 m²

An equation is an expression that shows how numbers and variables using mathematical operators.

The area of a figure is the amount of space it occupies in its two dimensional form

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

Hence:

Area = (1/2) * (8 m + 12 m) * 8 m = 80 m²

The area of the figure is 80 m²

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(a) The equation of the tangent line to the curve y = 4x³ – 2x³ +1 when x = 3 is y = 54x - 107.

(b)The equation of the tangent to the curve f(x) = 2x²+ 2x + 1 that has slope 8 is y = 8x - 1.

(a) To find the equation of the tangent line to the curve y = 4x³ – 2x³ +1 when x = 3, we first need to find the slope of the curve at the point (3, 25). We can do this by taking the derivative of the function y with respect to x:

y' = 12x² - 6x² = 6x²

Then, at x = 3, we have:

y' = 6(3)² = 54

So the slope of the curve at the point (3, 25) is 54. To find the equation of the tangent line, we use the point-slope form of a line:

y - y₁ = m(x - x₁)

where m is the slope and (x₁, y₁) is the point on the line. Substituting in the values we know, we get:

y - 25 = 54(x - 3)

Simplifying, we get:

y = 54x - 107

(b) To find an equation of the tangent to the curve f(x) = 2x²+ 2x + 1 that has slope 8, we need to find the point on the curve where the slope is 8. We can find this by taking the derivative of the function f(x) with respect to x:

f'(x) = 4x + 2

Setting this equal to 8, we get:

4x + 2 = 8

Solving for x, we get:

x = 3/2

So the point on the curve where the slope is 8 is (3/2, 11/2). Now we can use the point-slope form of a line as before, using the slope of 8 and the point (3/2, 11/2):

y - 11/2 = 8(x - 3/2)

Simplifying, we get:

y = 8x - 1

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