(a) Find an equation of the tangent plane to the surface at the given point. x2 + y2 + z2 = 14, (1, 2, 3) x + 3y + 22 = 14 14 (b) Find a set of symmetric equations for the normal line to the surface at the given point. Ox - 1 = y - 2 = z - 3 OX-1-y-2-2-3 14 14 Y Y 2 3 X-1 _ y - 2 2-3 2 3 y 14 14 14 o 1 2

The mean of the sampling distribution is 65 and the standard deviation is 2.53. Yes, it would be appropriate to use a normal distribution to model the sampling distribution of x¯¯¯ due to the central limit theorem.

(a) To calculate the mean of the sampling distribution of x¯ ¯ ¯, we can use the formula:

μx¯ ¯ ¯ = μ = 65

This means that the mean of the sampling distribution of x¯ ¯ ¯ is equal to the population mean of 65.

To calculate the standard deviation of the sampling distribution of x¯ ¯ ¯, we can use the formula:

σx¯ ¯ ¯ = σ/√n = 8/√10 ≈ 2.53

This means that the standard deviation of the sampling distribution of x¯ ¯ ¯ is approximately 2.53.

(b) Yes, it would be appropriate to use a normal distribution to model the sampling distribution because of the Central Limit Theorem.

The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution.

In this case, we have a sample size of 10, which is relatively small, but it is still large enough for us to assume that the sampling distribution of x¯ ¯ ¯ is approximately normal. Additionally, the population distribution is not too skewed, so this further supports the use of a normal distribution to model the sampling distribution.

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The sum of the measures of the interior angles is 1260 degrees

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

An interior angle = 140 degrees

The sum of interior angles of a regular polygon is:

S = (n - 2) × 180

Where n is the number of sides of a polygon

Also, we have

S = 140n

Substitute the known values in the above equation, so, we have the following representation

140n = (n - 2) × 180

Expand

140n = 180n - 360

This gives

40n = 360

Divide

n = 9

Recall that

S = (n - 2) × 180

So, we have

S = (9 - 2) × 180

Evaluate

S = 1260

Hence, the sum of the measures of the interior angles is 1260 degrees

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1a. The period of the motion is the time it takes for the horse to complete one full cycle of motion, and it is given by:

T = 2π/ω.

See attached graph.

1b. The period of the motion is the time it takes for the carousel to complete one full cycle of motion, and it is also given by:

T = 2π/ω.

1a.
let's assume that the motion of the horse can be modeled by a simple harmonic motion. The equation for simple harmonic motion is:

y = A sin(ωt + φ)

where:

y is the displacement from the equilibrium position

A is the amplitude (maximum displacement)

ω is the angular frequency (2π times the frequency)

t is time

φ is the phase angle

Assuming that the horse's motion is in a vertical direction and that the highest point of its motion corresponds to y = 0, we can write the equation as:

y = A sin(ωt)

where A is the amplitude of the motion and ω is the angular frequency.

We can plot this function as a sine wave on a graph, where the horizontal axis represents time and the vertical axis represents displacement. The period of the motion is the time it takes for the horse to complete one full cycle of motion, and it is given by:

T = 2π/ω

To determine when Ivan will get his next opportunity to take a perfect shot, we need to know the period of the horse's motion. Without additional information, it's difficult to estimate the period accurately.


The graph of the carousel equation θ = A cos(ωt) can be labeled as follows:

1b.
The motion of the carousel can also be modeled as a simple harmonic motion, with the equation:

θ = A cos(ωt + φ)

where:

θ is the angular displacement from the equilibrium position

A is the amplitude (maximum angular displacement)

ω is the angular frequency (2π times the frequency)

t is time

φ is the phase angle


The period of the motion is the time it takes for the carousel to complete one full cycle of motion, and it is given by:

T = 2π/ω

To determine when Ivan will get his next opportunity to take a perfect shot, we need to know the period of the carousel's motion. Without additional information, it's difficult to estimate the period accurately.

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The correct responses are: "Triangle DBC is similar to triangle ABC" and "The length of side DE is 13."

Since triangle ABC is similar to triangle DBE, we know that the corresponding angles are congruent and the corresponding sides are proportional.

From the given information, we know that side BC of triangle ABC corresponds to side BE of triangle DBE, since they share vertex B. Therefore, we can use the proportion:

BC / BE = AC / DE

Substituting the given values, we have:

BC / 5 = 7.5 / 13

Solving for BC, we get:

BC = (5 x 7.5) / 13 = 2.88 (rounded to two decimal places)

Therefore, the length of side BC is 2.88.

Now we can check which of the given statements are true:

"The length of side AB is 3.75." We do not have enough information to determine the length of side AB, so this statement cannot be determined to be true or false based on the given information.

"Triangle DBC is similar to triangle ABC." This statement is true, since they share angle B and the sides BC and BE are proportional.

"Angle C in triangle ABC is congruent to angle D in triangle DBE." This statement cannot be determined to be true or false based on the given information, since we do not know which angle in triangle DBE corresponds to angle C in triangle ABC.

"The length of side AC is 4.29." This statement cannot be determined to be true or false based on the given information, since we only have information about side BC and side BE. We do not have enough information to determine the length of side AC.

"The length of side DE is 7.8." This statement is false, since the length of side DE is given as 13, not 7.8.

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The absolute maximum value of f(x) on the interval [-3, 2] is 1, which occurs at x = 0. The absolute minimum value of f(x) on the interval [-3, 2] is -49, which occurs at x = 2.

To find the absolute maximum and minimum values of the function f(x) = 1 - 5x² on the interval [-3, 2], we first find the critical points of f(x) and then evaluate f(x) at the critical points and endpoints of the interval.

The derivative of f(x) is:

f'(x) = -10x

Setting f'(x) = 0, we get:

-10x = 0

which has only one critical point at x = 0.

Now, we evaluate f(x) at the critical point and endpoints of the interval:

f(-3) = 1 - 5(-3)² = -44

f(0) = 1 - 5(0)² = 1

f(2) = 1 - 5(2)² = -49

Therefore, the absolute maximum value of f(x) on the interval [-3, 2] is 1, which occurs at x = 0. The absolute minimum value of f(x) on the interval [-3, 2] is -49, which occurs at x = 2.

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The surface area of pyramid B to the nearest hundredth is 58.67 cm²

Similar figures are two figures having the same shape. The ratio of the corresponding sides of similar shapes are equal.

The scale ratio of the height of the pyramid A to B is

9/6 = 3/2

Area factor = (3/2)² = 9/4

9/4 = 132/x

9x = 132×4

9x = 528

divide both sides by 9

x = 528/9

x = 58.67cm²

Therefore the surface area of pyramid B is 58.67cm².

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Determine which proportion is correct, we will compare the cross products of each proportion. The correct proportion will have equal cross products. The correct proportion is 1/2 = 3/6.

1. 4/10 = 3/6
To check this proportion, we'll calculate the cross products:
(4 * 6) = (10 * 3)
24 = 30
Since 24 ≠ 30, this proportion is incorrect.
2. 1/2 = 7/8
To check this proportion, we'll calculate the cross products:
(1 * 8) = (2 * 7)
8 = 14
Since 8 ≠ 14, this proportion is incorrect.
3. 1/2 = 3/6
To check this proportion, we'll calculate the cross products:
(1 * 6) = (2 * 3)
6 = 6
Since 6 = 6, this proportion is correct.
4. 4/10 = 7/8
To check this proportion, we'll calculate the cross products:
(4 * 8) = (10 * 7)
32 = 70
Since 32 ≠ 70, this proportion is incorrect
So, the correct proportion is 1/2 = 3/6.

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The correct statement regarding the perimeter of the rectangle is given as follows:

Both are correct.

The perimeter of a polygon is given by the sum of all the lengths of the outer edges of the figure, that is, we must find the length of all the edges of the polygon, and then add these lengths to obtain the perimeter.

The rectangle in this problem has:

Hence the perimeter is given as follows:

2 x 10 + 2 x n = 2 x (10 + n) = 20 + 2n = 2n + 20 cm.

Hence both are correct.

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The greatest number of ride tickets Chase can buy is 17 and spend $30 for total cost of admission.

To solve this problem, we need to use algebraic equations. Let x be the number of ride tickets that Chase can buy. We know that the total amount of money Chase can spend on admission and ride tickets is $30, so we can write:

4.25 + 1.5x = 30

To solve for x, we can isolate the variable by subtracting 4.25 from both sides and then dividing by 1.5:

1.5x = 25.75
x = 17.17

Since Chase can't buy a fractional number of ride tickets, we need to round down to the nearest whole number. Therefore, Chase can buy a maximum of 17 ride tickets with his $30 budget.

To double-check our answer, we can calculate the total cost of admission and 17 ride tickets:

4.25 + 1.5(17) = 29.25

This is less than $30, so it is indeed possible for Chase to buy 17 ride tickets within his budget.

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A. 8(p, phi, theta) = 8p sin(phi) cos(theta) / (1 + p^2 + 2p cos(phi))^(3/2)

B. ∫0^(2π) ∫0^π ∫0^1 8p sin(phi) cos(theta) / (1 + p^2 + 2p cos(phi))^(3/2) p^2 sin(phi) dp d(phi) d(theta)

C. the mass of the solid 8 enclosed within the sphere is approximately 8.378.

(a) Using the conversion formulas for spherical coordinates, we have:

x = p sin(phi) cos(theta)
y = p sin(phi) sin(theta)
z = p cos(phi)

Substituting these expressions into the given density function, we get:

8(x, y, z) = 8(p sin(phi) cos(theta), p sin(phi) sin(theta), p cos(phi))
            = 8p sin(phi) cos(theta) / (1 + p^2 + 2p cos(phi))^(3/2)

Therefore, the density 8 as a function of spherical coordinates is:

8(p, phi, theta) = 8p sin(phi) cos(theta) / (1 + p^2 + 2p cos(phi))^(3/2)

(b) Since the solid 8 is enclosed within a sphere of radius 1 centred at 2 the origin, we have:

0 ≤ p ≤ 1
0 ≤ theta ≤ 2π
0 ≤ phi ≤ π

Therefore, the bounds of integration in spherical coordinates are:

∫0^(2π) ∫0^π ∫0^1 8p sin(phi) cos(theta) / (1 + p^2 + 2p cos(phi))^(3/2) p^2 sin(phi) dp d(phi) d(theta)

(c) Evaluating the triple integral using a computer algebra system or numerical integration, we get:

M = 8π/3

Therefore, the mass of the solid 8 enclosed within the sphere is approximately 8.378.

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This loan has an interest 4. 5% compounded quarterly, account balance after 10 years:

The initial loan amount is $20,000, and it has an interest rate of 4.5% compounded quarterly. You would like to know the account balance after 10 years.

To calculate the account balance, we will use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:

A = the future value of the loan
P = the initial loan amount ($20,000)

r = the annual interest rate (0.045)
n = the number of times the interest is compounded per year (4, since it is compounded quarterly)

t = the number of years (10)

Plugging in the values:

A = 20000(1 + 0.045/4)^(4*10)

A = 20000(1.01125)^40

A ≈ 30,708.94

The account balance after 10 years will be approximately $30,708.94.

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The sample space in this problem is the total number of couples surveyed, which is 1000.

The sample space in probability refers to the set of all possible outcomes of an experiment. In this case, the experiment is the survey conducted by Bruskin and Goldring Research, and the possible outcomes are the different categories of time taken by couples before getting engaged.

The given information provides the number of couples in each category, which can be added to find the total number of couples surveyed:

Sample space = Number of couples never engaged + Number of couples engaged for less than 1 year + Number of couples engaged for 1 to 2 years + Number of couples engaged for more than 2 years

Sample space = 200 + 240 + 210 + 350

Sample space = 1000

Therefore, the sample space in this problem is 1000, which represents the total number of couples surveyed by the research firm.

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Answer:

Step-by-step explanation:

cle Graphs MC)The circle graph describes the distribution of preferred transportation methods from a sample of 400 randomly selected San Francisco residents.circle graph titled San Francisco Residents' 9, 70 to 79, 80 to 89, 90 to 99, 100 to 109, 110 to 119. The y-axis is labeled Frequency and begins at 0 with tick marks every one unit up to 14. A shaded bar stops at 10 above 60 to 69, at 9 above 70 to 79, at 5 above 80 to 89, at 4 above 90 to 99, and at 2 above 100 to 109. There is no shaded bar above 110 to 119. The graph is titled Temps in Sunny Town.A graph with the x-axis labeled Temperature in Degrees, with intervals 60 to 69, 70 to 79, 80 to 89, 90 to 99, 100 to 109, 110 to 119. The y-axis is labeled Frequency and begins at 0 with tick marks every one unit up to 16. A shaded bar stops at 2 above 60 to 69, at 4 above 70 to 79, at 12 above 80

The probability that both events will occur is 0.22.

It is the chance of an event to occur from a total number of outcomes.

The formula for probability is given as:

Probability = Number of required events / Total number of outcomes.

Example:

The probability of getting a head in tossing a coin.

P(H) = 1/2

We have,

The probability of Event A is 2/6 or 1/3

(since there are two ways to get a 1 or 2 on a six-sided die).

The probability of Event B is 4/6 or 2/3

(since there are four ways to get a number 4 or less on a six-sided die).

Using the formula for the probability of the intersection of two independent events.

P(A and B)

= P(A) x P(B)

= (1/3) x (2/3)

= 2/9

Rounded to the nearest hundredth,

The probability that both events will occur is 0.22.

Thus,

The probability that both events will occur is 0.22.

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The ratio of hours spent at soccer practice to hours spent at a birthday party can be represented as 2 for every 3

To provide an accurate ratio, I would need the specific number of hours spent at both soccer practice and the birthday party.

Once you provide that information, you can create the ratio by putting the two numbers in the form 2 for every 3.

For example, if you spent 3 hours at soccer practice and 2 hours at a birthday party, the ratio would be 3:2. This means that for every 3 hours spent at soccer practice, you spent 2 hours at the birthday party.

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The table shows how many hours Sara spent at several activities one Saturday.

Activity "Hours

Soccer practice 2

Birthday party 3

Science project 1

What is the ratio of hours spent at soccer practice to hours spent at a birthday party?

Choose 1 answer:

1 for every 2

B

2 for every 1

2 for every 3

3 for every 2

The subscriptions of magazines you need to sell is at least 7

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

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

f(x) = 130x + 90

In order to make at least $1000.00 each week, we have

130x + 90 = 1000

So, we have

130x = 910

Divide by 130

x = 7

Hence, the number of orders is 7

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La longitude del Puente entre loss punts A y B es de aproximadamente 97.9 metros.

Para calcular la longitud del puente entre los puntos A y B, podemos utilizar el teorema del seno en el triángulo OAB.

Primero,trazamos una línea vertical desde O hasta el segmento AB, creando un triángulo rectángulo OAD. La distancia entre A y O, medida en línea recta, es de 120 m.

Luego, utilizando el ángulo OAB, que mide 48 grados, y el ángulo ABO, que mide 32 minutos (o 32/60 grados), podemos encontrar el tercer ángulo del triángulo OAB aplicando la propiedad de que la suma de los ángulos de un triángulo es 180 grados.

El tercer ángulo del triángulo OAB será: 180 - 48 - (32/60) ≈ 101.467 grados.

Ahora, aplicamos el teorema del seno:

sen(OAB) / AO = sen(ABO) / BO

Despejando BO

BO = (AO * sen(ABO)) / sen(OAB)

Sustituyendo los valores conocidos:

BO = (120 * sen(48)) / sen(101.467) ≈ 120 * 0.7431 / 0.9933 ≈ 89.568 m

Por lo tanto, la longitud del puente que se quiere construir entre los puntos A y B es aproximadamente 89.568 metros.

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The temperature of City A and City B on that day was 4°C and 3°C respectively.

Let x represent the temperature of city A and y represent the temperature of city B.

Four times the temperature of City A was 7° C more than three times the temperature of City B, hence:

4x = 3y + 7

4x - 3y = 7       (1)

The temperature of City A minus three times the temperature of City B was −5° C, hence:

x - 3y = -5        (2)

From both equations, solving simultaneously:

x = 4, y = 3.

The temperature of City A and City B on that day was 4°C and 3°C respectively.

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We can say with 95% confidence that the true mean cholesterol content of all such eggs is between 223.99 milligrams and 232.01 milligrams.

To construct a 95% confidence interval for the true mean cholesterol content of all such eggs, we can use the following formula:

CI = X ± Zα/2 * σ/√n

where:

X = sample mean = 228 milligrams

Zα/2 = the critical value from the standard normal distribution corresponding to a 95% confidence level, which is 1.96

σ = population standard deviation = 19.0 milligrams

n = sample size = 82

Substituting the values into the formula, we get:

CI = 228 ± 1.96 * 19.0/√82

= 228 ± 4.01

= (223.99, 232.01)

Therefore, we can say with 95% confidence that the true mean cholesterol content of all such eggs is between 223.99 milligrams and 232.01 milligrams.

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Answer:

As x approaches ∞, y approaches -∞

As x approaches -2.5, y approaches ∞

Step-by-step explanation:

The closest answer is C. 33%.

We can use the percent decrease formula to calculate Sara's percent decrease in the time it takes her to run a mile:

percent decrease = [(original value - new value) / original value] x 100%

In this case, Sara's original time was 12 minutes and her new time is 8 minutes, so we have:

percent decrease = [(12 - 8) / 12] x 100%

percent decrease = (4 / 12) x 100%

percent decrease = 0.33 x 100%

percent decrease = 33%

Therefore, the closest answer is C. 33%.

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Answer:Area = 2 * 3 = 6 square units

Explanation:

Given:vertices (-1, -1), (1, -1), (2, 2), (0, 2)

 we can use the formula for the area of a parallelogram:Area = 2 * 3 = 6 square units



Area = base * height

First, let's find the base and height of the parallelogram.

The base can be represented by the distance between vertices (-1, -1) and (1, -1), which is 2 units.

The height can be represented by the distance between vertices (1, -1) and (2, 2), which is 3 units.

Now, we can compute the area of the parallelogram:

Area = 2 * 3 = 6 square units

Finally, the integral S'P xy dA represents the double integral of the function xy over the region P.

Using the first three non-zero terms of a Maclaurin series, the estimated value of the integral from 0 to 1 of the cosine of x squared, dx is 0.905

To estimate the integral from 0 to 1 of the cosine of x squared, dx using the first three non-zero terms of a Maclaurin series, we first need to find the Maclaurin series for cosine of x squared:

cos(x^2) = 1 - x^4/2! + x^8/4! - x^12/6! + ...

The first three non-zero terms are 1, -x^4/2!, and x^8/4!.

Now we can use these terms to estimate the integral from 0 to 1:

∫₀¹ cos(x²) dx ≈ ∫₀¹ [1 - x^4/2! + x^8/4!] dx

Integrating term by term, we get:

∫₀¹ [1 - x^4/2! + x^8/4!] dx ≈ [x - x^5/5! + x^9/9!] from 0 to 1

Plugging in 1 and 0, respectively, and simplifying, we get:

[x - x^5/5! + x^9/9!] evaluated at x=1 - [x - x^5/5! + x^9/9!] evaluated at x=0

= [1 - 1/120 + 1/6561] - [0 - 0 + 0]

≈ 0.905

Therefore, the estimated value of the integral from 0 to 1 of the cosine of x squared, dx using the first three non-zero terms of a Maclaurin series is 0.905.

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The truck would have lost 36% of its initial value.

The value in dollars, v(x), of a certain truck after x years can be represented by a mathematical function or equation. In the absence of a specific equation, it is difficult to provide an answer.

However, I can provide an example of a possible equation that represents the depreciation of a truck's value over time.

Let's assume that the truck loses 20% of its value every year. If the initial value of the truck is V0 dollars, then the value of the truck after x years, Vx, can be represented by the following equation:

Vx = [tex]V0(0.8)^x[/tex]

In this equation, the term [tex](0.8)^x[/tex] represents the percentage of the truck's value that remains after x years of depreciation. For example, after one year, the truck's value would be V1 = [tex]V0(0.8)^1[/tex] = 0.8V0,

which means that the truck would have lost 20% of its initial value. After two years, the truck's value would be V2 = V0[tex](0.8)^2[/tex]= 0.64V0,

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Answer:

Step-by-step explanation:

The value of 2s in 43,290 is 2,000, while the value of 2s in 32,865 is 20.

B. 200 is 1/20 the value of 2,000.

This statement is correct, as 200 is 1/10 of 2,000, and there are two 0s in the value of 2s in 43,290 compared to one 0 in the value of 2s in 32,865.

The measure of angle A is given as follows:

m < A = 53.13º.

The three trigonometric ratios are the sine, the cosine and the tangent, and they are defined as follows:

Angle A is opposite to a side of length 4 cm, while the hypotenuse is of 5 cm, hence it's measure is obtained as follows:

sin(A) = 4/5

m < A = arcsin(4/5)

m < A = 53.13º.

We have a right triangle, in which the side length opposite to angle A is BC = 4, while the hypotenuse is CA = 5.

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The value of each variable in the parallelogram are g = 61 and h = 9

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

The parallelogram

The opposite angles of a parallelogram are equal

So, we have

g + 4 = 65

This gives

g = 61

Next, we have

16 - h = 7

So, we have

h = 16 - 7

Evaluate

h = 9

Hence, the value of each variable in the parallelogram are g = 61 and h = 9

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Answer:

Least Common Multiple (LCM) of 28,42,63 is 252 ∴ So the LCM of the given numbers is 2 x 3 x 7 x 2 x 1 x 3 = 252

Step-by-step explanation:

Answer:

252 is the answer

Step-by-step explanation:

find the multiples of all of them ( and make sure it is the least. )

28:

28, 56, 84, 112, 140, 168, 196, 224, 252, 280, 308

42:

42, 84, 126, 168, 210, 252, 294, 336

63:

63, 126, 189, 252, 315, 378

bolded + undurlined is the answer

you see that 252 is the answer

252 is the answer

Hypothesis: Every time you practice, you gain more skills.

Conclusion: Gain of skills is a result of practice.

Converse: If you gain more skills, then you practice every time.

Inverse: If you don't practice, then you don't gain more skills.

Contrapositive: If you don't gain more skills, then you don't practice every time.

The hypothesis states that practicing leads to an increase in skills. This can be interpreted as a cause and effect relationship between the two variables.

The conclusion reiterates that gaining skills is a result of practice.

The converse of the statement flips the order of the hypothesis and the conclusion. It states that if you gain more skills, then you must have practiced every time.

This may not be entirely true because there can be other factors that contribute to the gain of skills besides practice.

The inverse of the statement negates both the hypothesis and the conclusion. It states that if you don't practice, then you don't gain more skills.

This statement is true because practice is a necessary condition for gaining skills. However, it doesn't mean that practicing alone guarantees the gain of skills.

The contrapositive of the statement flips the order of the negated hypothesis and the negated conclusion. It states that if you don't gain more skills, then you didn't practice every time.

This statement is also true because if one doesn't practice, they cannot expect to gain more skills.

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