Let n be an odd integer. Prove that n3−n is divisible by 24 . The following two problems are only appropriate if you took at least two semesters of calculus. Though you may have worked these before, the idea is to work them again paying close attention to the final presentation. Make sure you define all variables. Use complete sentences, with proper punctuation.

The rectangular horse pasture's largest area, in square feet, can be enclosed in \(80000m^{2}\)

Let the sides of the rectangle-shaped fence be a and b.

We have to use wire for the three sides of the fence.

\(a+b*2=800\)

Area A is given by

\(A=a*b\)

Also, we can write A as a function of b

\(A=A(b)\\=(800-2*b)b=800b-2b^{2}\)

Taking the derivative of A

\(A'=800-4b\\A'=0\\b=200m\)

Then the length of the other side

\(a=800-2*200\\=400m\)

The largest area

\(A=400*200\\=80000m^{2}\)

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Answer:

C. 3^9

Step-by-step explanation:

You would simply just add the exponents.

Answer:

3^9

Step-by-step explanation:

5+4=9

The length of the longer diagonal of the given rhombus with side length 51units and shorter diagonal 48 units is equal to 90units.

As given in the question,

Side length of the given rhombus 's' = 51 units

Length of the shorter diagonal of given rhombus 'a' = 48 units

Let us consider length of the longer diagonal of the given rhombus be 'b'

As per the property of the rhombus:

Diagonals of the rhombus bisects each other at right angle

Triangle formed by side length bisecting diagonals is right angled triangle.

Using Pythagoras theorem we have,

(Side length)² = (a/2)² + (b/2)²

Substitute the values

⇒ (51)² = (48/2)² + (b/2)²

⇒2601 = 576 + (b/2)²

⇒(b/2)² = 2601 -576

⇒(b/2)² = 2025

⇒b/2 = √2025

⇒ b/2 =45

⇒ b = 90units

Therefore, the length of the longer diagonal of the given rhombus is equal to 90units.

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There are 35 experimental units are in the experiment.

It is defined as the operation in which we do the addition of numbers, subtraction, multiplication, and division. It has basic four operators that are +, -, ×, and ÷.

It is given there are 2 types of paints, and 3 types of metal, Thirty sheets of metal were used in the experiment.

We have to apply the addition operation in order to find the number of experimental units.

The number of experimental units in the experiment is found as,

=2+3+30

=5+30

=35

Thus, there are 35 experimental units are in the experiment.

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yes proper subset  is different from a regular subset. The empty set is a proper subset of every set except for the empty set

A proper subset of a set A is a subset of A that cannot be equal to A. In other words, if B is a proper subset of A, then all elements of B are in A but A contains at least one element that is not in B.

Subset: If A and B are sets and every element of A is also an element of B, then:

A is a subset of B, denoted by A ⊆ B.

or equivalently, B is a superset of A, denoted by B ⊇ A.

Any set is considered to be a subset of itself. No set is a proper subset of itself. The empty set is a subset of every set. The empty set is a proper subset of every set except for the empty set.

For example, A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

A = {1, 2, 3} and B = {1, 2, 3}

Here, A is a subset of B, or we can say that B is the superset of A.

Proper Subset: If A is a subset of B, but A is not equal to B (that is, there exists at least one element of B which is not an element of A), then

A is also a proper (or strict) subset of B; this is written as A ⊊ B.

For example A = {1, 2, 3} and B = {1, 2, 3, 4}.

Clearly, A is not equal to B and element {4} belongs to set B but is absent in set A, so we have one element in set B which is not an element of set A. Thus, A can be called a proper subset of B.

Hence, a proper subset of a set A is a subset of A that is not equal to A. In other words, if B is a proper subset of A, then all elements of B are in A but A contains at least one element that is not in B, thus the proper subset may or may not be the same.

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Answer:

To find the value of X for which 70.54% of the area under the distribution curve lies to the right of it, we need to find the z-score that corresponds to this percentile and then use it to calculate the value of X.

Let z be the z-score that corresponds to the 70.54th percentile of the standard normal distribution. We can find this z-score using a standard normal table or a calculator:

z = 0.5484

This means that 70.54% of the area under the standard normal curve lies to the left of z = 0.5484, and the remaining 29.46% of the area lies to the right of it.

We can now use the formula for standardizing a normal random variable to calculate the corresponding value of X:

z = (X - μ) / σ

where μ is the mean and σ is the standard deviation.

Rearranging this formula to solve for X, we get:

X = μ + z * σ

Substituting the values given in the problem, we get:

X = 10 + 0.5484 * 4

X = 12.1936

Therefore, the value of X for which 70.54% of the area under the distribution curve lies to the right of it is approximately 12.1936.

|x−3|+2=−3|x-3|+2=-3

Move all terms not containing |x−3||x-3| to the right side of the equation.

|x−3|=−5

Remove the absolute value term. This creates a ±± on the right side of the equation because |x|=±x|x|=±x.

x−3=±(−5)

Set up the positive portion of the ± solution.

x−3=−5

Move all terms not containing x to the right side of the equation.

x= −2

Set up the negative portion of the ± solution.

x−3 =5

Move all terms not containing x to the right side of the equation.

x=8

The solution to the equation includes both the positive and negative portions of the solution.

x=−2, 8

Answer:

1) x < 12

2) x > 0.5/7

3) x < 5.83

4) x <= 7.2

5) x <= -6

Step-by-step explanation:

1)

x + 10 < 22

x < 22 - 10

x < 12

2)

x * 7 >= 0.5

x > 0.5/7

3)

x/3 <= 17.5

x <= 17.5/3

x < 5.83

4)

8.2 >= x + 1

x <= 8.2 - 1

x <= 7.2

5)

9 - x >= 15

9 >= 15 + x

x <= 9 -15

x <= -6

Answer:8 gumballs 3 jaw breakers

Step-by-step explanation:

Using an inequality to represent the expression and it's constraint, the required expression would be 0.50g + 0.30j ≤ 3

Given the Parameters :

(Cost of gumball × number of gumballs + Cost of Jawbreaker × number of gumbreakers) ≤ maximum cost

0.50g + 0.30j ≤ 3

Hence, the required inequality expression is 0.50g + 0.30j ≤ 3

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The  equivalent expression to 8k - (5 + 2k) is 6k-5.

An expression is combination of variables, numbers and operators.

The given expression is 8k-(5+2k)

Eight times k minus five plus two times of k.

We need to find the equivalent expression to the given expression.

K is the variable in the expression.

Equivalent expressions are expressions that work the same even though they look different.

Apply distributive property

8k-5-2k

Add the like terms

6k-5

Hence, 6k - 5 is the equivalent expression of 8k-(5+2k).

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Answer:

2/5

Step-by-step explanation:

40 divided from 20100 divided by 20 is 25

Answer:

2/5

Step-by-step explanation:

40/100

factor

(2*2*2*5)/(2*2*5*5)

cross off like factors

2/5

3m+6=24

3m=18

m=6

So mike traveled 6 miles by taxi.

We know that the taxi driver chargers $3 per mile, so we say M is the mile variable, and for every M there is $3. So 3m+6 (initial fee) is the first part of the formula.

We set 3m+6=24 (24 being the total amount the taxi driver is charging)

Answer:33

Step-by-step explanation:

The correct conclusion is that we do not have enough evidence to suggest that the average test score obtained by early finishers is significantly different from the overall average.

The significance level (alpha) is a probability value used to specify the threshold at which the null hypothesis can be denied. It is often referred to as the alpha level, alpha value, or significance level. The significance level specifies how strong the evidence must be for the null hypothesis to be denied. The significance level, alpha, is typically set at 0.05 (5%) or 0.01 (1%) in many scientific investigations, including hypothesis testing.

Any p value less than the α significance level is considered statistically significant and the null hypothesis is denied. Therefore, we need to look at the p-value to decide whether the mean test score of the first finishers is significantly different from the overall means. The null and alternative hypotheses for the given question can be stated as follows:

Null hypothesis: H0: The mean test score obtained by early finishers is not significantly different from the overall mean. Alternative hypothesis:

Ha: The average test score obtained by the first finishers is significantly different from the overall average.

The test that is reinforced in this situation is a one-sample t-test, which compares the mean of the first-finishers tests with the population means. If the p-value is found to be less than the 0.05 significance level, we will deny the null hypothesis and conclude that there is sufficient evidence to believe that the average score of the early finishers test is significantly different from the overall mean.

In this case, the p-value is greater than the alpha level of 0.05, so we cannot deny the null hypothesis. Thus, we can infer that the average test score obtained by early finishers is not significantly different from the overall average. Therefore, the correct conclusion is that we do not have enough evidence to suggest that the average test score obtained by early finishers is significantly different from the overall average.

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In order to determine if the point (20,13) is on the line, it is necessary to write the equation of the line.

The general form of a linear equation is:

y = mx + b

where b is the y-intercept and m is the slope. Y-intercept is the value of y when x = 0. You can observe in the graph that b = 3.

The slope m is conputed by using the following formula:

where (x1,y1) and (x2,y2) are two points of the line. Use the points (0,3) and (6,6), you can select any other two points. Replace these values into the formula for m:

Then, the equation of the line is:

Now, replace the value of x = 20 in the previous equation, if y = 13, then the point (20,13) in on the line:

Hence, the point (20,13) is on the line

Step-by-step explanation:

bit.^{}

ly/3gVQKw3 here in this link

Answer: the answer is 31 I just submitted it

Step-by-step explanation:

Answer:

18 units

Step-by-step explanation:

since the y- coordinates are equal then the 2 points lie on a horizontal line and the distance (d) between them is the absolute value of the x- coordinates, that is

d = | - 4 - 14 | = | - 18 | = 18

or

d = |  14 - (- 4) | = | 14 + 4 | =   18  = 18

Answer:

A

Step-by-step explanation:

We can calculate the equation of the lines based on two points. First line m:

y = ax +b

where a is the slope and b is the y-intercept. To find the slope we use the x and y of the points as:

a = \(a = \frac{y_1-y_2}{x_1-x_2} = \frac{4- -32}{2- -4} = \frac{36}{6} = 6.\)

So:

y = 6x + b

we can find the y intercept by substituting the x and y for one of the points:

solving for b:

b=-32-(6)(-4)= -8.

So the equation is:

y = 6x - 8.

We can change it to:

y + 8 = 6x

8 = 6x - y.

So it’s (A).

Hope that helps!

Answer:

B I think

Step-by-step explanation:

theorem 1 : angles opposite to equal sides in a triangle are equal

theorem 2 : A diameter in a circle makes 90°on the circumference

theorem 3 : A tangent makes 90° with the radius

(1) two sides (radii) are equal...hence

a = 50°

a + 50° + ? = 180° ( ang-sum prp)

? = 180 - 50 - 50 = 80°

b = 360° - 80° ( rflx ang)

b = 280°

(2) a = 90° ( theorem 2)

b = 180 - 90 - 60° ( ang sum prp)

b = 30°

(3) ? = a ( theorem 1)

a + ? + 40 = 180 ( ang sum prp)

2a = 180 - 40

a = 70°

since...the larger triangle is a Right triangle ( theorem 2)

the remaining Angles add up to 90°

a + b = 90°

b = 90 - 70

b = 20°

(4) ? + 35 = 90 ( theorem 2)

? = 90 - 35

? = 55°

now ..

a = 55° ( theorem 1)

a + 55 + b = 180 ( ang sum prp)

55 + 55 + b = 180

b = 180 - 110

b = 70°

(5) a = b ( theorem 1)

a + b + 130 = 180 ( ang sum prp)

a + a + 130 = 180

2a = 180 - 130

a = 25°; b = 25°

a + c = 90° ( theorem 2)

c = 90 - 25

c = 65°

(6) x + 2x = 90° ( theroem 2)

3x = 90

x = 30°

(7) a = 90° ( theorem 3)

b + 60 = 90 ( theorem 3)

b = 90 - 60

b = 30°

b + c = 90 ( theorem 2)

30 + c = 90

c = 90 - 30

c = 60°

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To obtain the residuals from the fit in 8.4a and plot them against y and x, as well as prepare a normal plot, we need the specific details of the fit and the data used.

WE know that residuals represent the differences between the observed values and the predicted values from a statistical model or regression analysis.

Since the residuals against y and x can help identify patterns or trends in the data that may indicate issues with the model's fit.

A normal plot, known as a Q-Q plot, compares the distribution of the residuals to a theoretical normal distribution. If the residuals closely follow a straight line in the normal plot, the residuals are normally distributed, which is an assumption of many statistical models.

Interpreting these plots involves examining the patterns and deviations from expected behavior. If the residuals exhibit a consistent pattern, it might indicate that the model does not capture all the relevant information in the data.

Thus if the residuals appear randomly scattered around zero with no discernible pattern, it suggests that the model adequately explains the data. Deviations in the normal plot may indicate departures from the assumption of normality in the residuals, which could impact the reliability of statistical inferences.

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Answer:  No it does not .

Step-by-step explanation:

First use the coordinates of point A and B to write an equation.

A (-3,-2)

B(2,1)

We will need to use these points to find the slope first and by doing that we will find the difference in the y values and divide it by the difference in the x values.

-2-1= -3

-3-2 = -5

-3/-5= 3/5  

The slope is 3/5 and now we need to find the y intercept using the same points. As you can see the equation will have to be in slope intercept form which uses the formula y=mx+b  where m is the slope and b is the y-intercept.

We will plot the first points in and solve for b.

-2=3/5(-3) + b

-2= -9/5 + b  

+9/5  +9/5

b = -1/5  

We will now write the equation  y=3/5x - 1/5   .

Now using the equation try point c to see if it fits the graph.  

Point C is (5,3)  so again plot in the x and y coordinates and solve,

3= 3/5(5) - 1/5

3 = 3 - 1/5

3\(\neq\) 14/5

In this case 3 does not equal 14 over 5 so Point c does not fit the graph

Answer:

-0.6

Step-by-step explanation:

The values of the missing sides are;

a.  x = 35. 6 degrees

b. x = 15

c. x = 22. 7 ft

d. x = 31. 7 degrees

To determine the values, we have;

a. Using the tangent identity;

tan x = 5/7

Divide the values

tan x = 0. 7143

x = 35. 6 degrees

b. Using the Pythagorean theorem

x² = 9² + 12²

find the square

x² = 225

x = 15

c. Using the sine identity

sin 29= 11/x

cross multiply the values

x = 11/0. 4848

x = 22. 7 ft

d. sin x = 3.1/5.9

sin x = 0. 5254

x = 31. 7 degrees

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Answer:

2/3 Cups

Step-by-step explanation:

2/3 cups would fit but there would still be 1/6 cup left over.

The correct match of each ordered pair with each function is:

(-9, 74) for y = -8x + 2

(2,-6) for y = -4x + 2

(9, 70) for y = 7x + 7

(-4, 23) for y = -7x - 5

To match each function with the correct ordered pair, we need to substitute the x-values from the ordered pairs into each function and see which one gives the corresponding y-value.

Substitute the x value of (-9, 74) into y = -8x + 2:

y = -8(-9) + 2

y = 74

Substitute the x value of (2,-6) into y = -4x + 2:

y = -4(2) + 2

y = -6

Substitute the x value of (9, 70) into y = 7x + 7:

y = 7(9) + 7

y = 70

Substitute the x value of (-4, 23) into y = -7x - 5:

y = -7(-4) - 5

y = 23

Therefore, the correct matching is:

(-9, 74) for y = -8x + 2

(2,-6) for y = -4x + 2

(9, 70) for y = 7x + 7

(-4, 23) for y = -7x - 5

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Answer:

The constant of proportionality of the graph is 0.75

Step-by-step explanation:

The coordinates of the points on the lines are;

(400, 300), (250, 187.5), and (120, 90)

Which can be written in a tabular form as follows;

Number of US dollars   \({}\)                       Number of euros

120                                 \({}\)                        90

250                                \({}\)                        187.5

400                                \({}\)                        300

Given that the graph is a straight line graph, we can represent the graph with the straight line equation, y = m·x + c

Where;

c = The y-intercept (where the graph meets the y-axis) = 0

m = The slope = The constant of proportionality

\(Slope, \, m =\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

\(m =\dfrac{300-90}{400-120} = \dfrac{210}{280} = 0.75\)

We have, y = m·x = 0.75·x.

Therefore, the constant of proportionality of the graph = 0.75.

In sampling distribution, it is not necessarily true that the expected value of j will be equal to the expected value of k, and the variability of j will be equal to the variability of k.

The properties of the sampling distribution of an estimator depend on the sample size, the population distribution, and the estimator itself. While it is possible that the expected value and variability of j and k may be equal under certain conditions, this cannot be assumed without further information.

More information about the specific estimator and the population distribution is needed to determine the properties of the sampling distributions for different sample sizes.

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The robot moves in the positive direction along a straight line so that after t minutes its distance is s=6t^4 feet from the origin. (a) Find the average velocity of the robot over intervals 2, 4. We have the following data: Initial time, t₁ = 2 min.

Final time, t₂ = 4 min.The distance from the origin is given by s = 6t^4Therefore, s₁ = s(2) = 6(2^4) = 6(16) = 96 feet s₂ = s(4) = 6(4^4) = 6(256) = 1536 feet

We can find the average velocity of the robot over the interval 2, 4 as follows: Average velocity = (s₂ - s₁) / (t₂ - t₁)Average velocity = (1536 - 96) / (4 - 2)Average velocity = 1440 / 2Average velocity = 720 feet per minute(b) Find the instantaneous velocity at t=2.To find the instantaneous velocity at t = 2 min, we need to take the derivative of the distance function with respect to time. We have the distance function as:s = 6t^4 Taking derivative of s with respect to t gives the velocity function:v = ds / dt Therefore,v = 24t³At t = 2, the instantaneous velocity is:v(2) = 24(2)³v(2) = 24(8)v(2) = 192 feet per minute Therefore, the instantaneous velocity at t = 2 min is 192 feet per minute.

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Answer:

x^4+x³+7x²-6x+1

Step-by-step explanation:

(x²+2x+8)(x²-x+1)

distribute

x²(x²-x+1)

x^4-x³+x²

2x(x²-x+1)

2x³-2x²+2x

8(x²-x+1)

8x²-8x+1

x^4-x³+x²+2x³-2x²+2x+8x²-8x+1

combine like terms

x^4+x³+7x²-6x+1

The absolute value function f(x) = |x – 2| takes a non-negative value for all real numbers x, and it returns 0 only when x = 2.

The graph of f(x) is symmetric about the vertical line x = 2 and has a vertical asymptote at x = 2.

We have,

The absolute value function f(x) = |x – 2| is a piecewise function that returns the positive distance between the input value x and the number 2.

Geometrically,

It represents the distance of a point on the number line from point 2.

On the coordinate plane,

The graph of the absolute value function is V-shaped, with its vertex at the point (2, 0).

The two arms of the V extend indefinitely in opposite directions, passing through the points (-∞, 2) and (2, +∞) on the positive and negative sides of the x-axis, respectively.

Thus,

The absolute value function f(x) = |x – 2| takes a non-negative value for all real numbers x, and it returns 0 only when x = 2.

The graph of f(x) is symmetric about the vertical line x = 2 and has a vertical asymptote at x = 2.

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