consider the curve defined by the equation y+cosy=x+1

The grapefruit's volume is approximately 1097.4 cubic centimeters.

The volume of sphere is the capacity it has. It is the space occupied by the sphere. The volume of sphere is measured in cubic units, such as m3, cm3, in3, etc. The shape of the sphere is round and three-dimensional. It has three axes as x-axis, y-axis and z-axis which defines its shape.

To find the volume of a sphere, we can use the formula:

[tex]V = (4/3) * π * r^3[/tex]

where V represents the volume and r represents the radius of the sphere.

In this case, Chase measured the radius of the grapefruit as 6.4 cm. Plugging this value into the formula, we have:

V = (4/3) * π * [tex](6.4 cm)^3[/tex]

V = (4/3) * π * [tex](262.144 cm^3)[/tex]

V ≈ [tex]1097.445 cm^3[/tex]

Rounding this value to the nearest tenth, the grapefruit's volume is approximately 1097.4 cubic centimeters.

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Answer:

The steps to solve the equation x^2 - 18x + 8 = 0 by completing the square are:

1. Move the constant term to the right side: x^2 - 18x = -8

2. Take half of the coefficient of x, square it, and add it to both sides of the equation:

x^2 - 18x + (-18/2)^2 = -8 + (-18/2)^2

x^2 - 18x + 81 = -8 + 81

3. Simplify the left side and the right side of the equation:

(x - 9)^2 = 73

4. Take the square root of both sides of the equation, remembering to include both the positive and negative square roots:

x - 9 = ± √73

5. Add 9 to both sides of the equation:

x = 9 ± √73

Therefore, the solutions to the equation x^2 - 18x + 8 = 0 are x = 9 + √73 and x = 9 - √73.

Step-by-step explanation:

The correct representation of Kirchhoff's junction rule would be:

ΣIᵢ = 0

This equation states that the sum of all currents (Iᵢ) flowing into a junction or node is equal to zero.

Kirchhoff's junction rule, also known as Kirchhoff's current law (KCL), states that the algebraic sum of currents flowing into any junction or node in an electrical circuit is equal to zero.

Among the equations you provided, none of them accurately represents Kirchhoff's junction rule. The correct representation of Kirchhoff's junction rule would be:

ΣIᵢ = 0

This equation states that the sum of all currents (Iᵢ) flowing into a junction or node is equal to zero.

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The value of tan θ for the given angle is option (b): tan θ = negative four times square root of thirty-three divided by thirty-three.

To determine the value of tan θ, we need to evaluate the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

By examining the options, we can see that option (b) matches this criteria: tan θ = negative four times square root of thirty-three divided by thirty-three.

The negative sign indicates that the angle is in the third quadrant or the second quadrant, where the tangent function is negative.

The square root of thirty-three represents the length of the side opposite the angle, and the denominator, thirty-three, represents the length of the side adjacent to the angle.

Therefore, the correct answer is option (b): tan θ = negative four times square root of thirty-three divided by thirty-three.

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The answers are A. the standard matrix A for the linear transformation T is: A = [[√2/2, -√2/2], [√2/2, √2/2]] and B. the image of the vector v under the linear transformation T is T(v) = (0, 5√2).

(a) To find the standard matrix A for the linear transformation T, which represents a counterclockwise rotation of 45° in R2, we can consider the effect of the transformation on the standard basis vectors.

T maps the standard basis vector i = (1, 0) to a new vector that is rotated counterclockwise by 45°. This new vector is (√2/2, √2/2) since it has equal components along the x and y axes.

Similarly, T maps the standard basis vector j = (0, 1) to a vector that is also rotated counterclockwise by 45°. This vector is (-√2/2, √2/2) as it has equal components along the negative x and positive y axes.

Therefore, the standard matrix A for the linear transformation T is:

A = [[√2/2, -√2/2], [√2/2, √2/2]].

(b) To find the image of the vector v = (5, 5) under the linear transformation T, we multiply the standard matrix A by the vector v:

T(v) = A * v = [[√2/2, -√2/2], [√2/2, √2/2]] * [5, 5].

Performing the matrix multiplication yields:

T(v) = [(√2/2)*5 + (-√2/2)*5, (√2/2)*5 + (√2/2)*5]

= [(5√2/2 - 5√2/2), (5√2/2 + 5√2/2)]

= [0, 5√2].

Therefore, the image of the vector v under the linear transformation T is T(v) = (0, 5√2).

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Answer: 6 + 4/x

Step-by-step explanation:

Since a quotient is division, the algebraic expression is basically saying that you divide 4 with a number (or variable) and add 6 more to it.

6 + 4/x

Therefore, the way to rephrase "6 more than the quotient of 4 and a number' would be 6 + 4/x. I understand sometimes it's very easy to forget the basic things, for example, in my class :). Hope this helps!

-From A 5th Grade Honors Student who loves Algebra!

The estimated margin of error can be found using the confidence interval provided by the store manager. The confidence interval of (44.9, 82.3) represents a range within which the true population mean price for textbooks is estimated to lie.

To find the estimated margin of error, we take half of the width of the confidence interval. The width of the confidence interval is obtained by subtracting the lower bound from the upper bound: 82.3 - 44.9 = 37.4. Since the margin of error is half the width, we divide this value by 2: 37.4 / 2 = 18.7.

Therefore, the estimated margin of error is 18.7 dollars. This means that, based on the provided confidence interval, the store manager estimates that the mean price for the population of textbooks is within 18.7 dollars of the sample mean.

However, the sample mean itself is not directly provided in the given information.

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The equation of the plane that is equidistant from points A and B is:

-6x - 4y - 2z = 0

To find the equation of the plane equidistant from points A = (3, 2, 1) and B = (-3, -2, -1), we can follow these steps:

1. Find the midpoint M between points A and B:

  M = ((3 + (-3)) / 2, (2 + (-2)) / 2, (1 + (-1)) / 2)

    = (0, 0, 0)

2. Find the vector v from point A to point B:

  v = B - A = (-3 - 3, -2 - 2, -1 - 1)

      = (-6, -4, -2)

3. Since the plane is equidistant from A and B, any vector that lies on the plane must be orthogonal (perpendicular) to the vector v. Therefore, the normal vector of the plane is the same as the vector v:

  n = v = (-6, -4, -2)

4. Now we have the normal vector n = (-6, -4, -2) and a point on the plane M = (0, 0, 0). We can use the point-normal form of the equation of a plane to obtain the equation:

  Ax + By + Cz = D

  Plugging in the values, we have:

  -6x - 4y - 2z = D

  To find the value of D, we substitute the coordinates of the midpoint M into the equation:

  -6(0) - 4(0) - 2(0) = D

  0 = D

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From the given triangle ABC, the measure of side BC is 9 yards.

From the given triangle ABC, AB=12 yards and AC=15 yards.

By using Pythagoras theorem, we get

AC²=AB²+BC²

15²=12²+BC²

225=144+BC²

BC²=225-144

BC²=81

BC=√81

BC=9 yards

Therefore, from the given triangle ABC, the measure of side BC is 9 yards.

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The type of sampling used in this scenario is a multistage cluster sampling. In this type the population is divided into clusters or groups.

In multistage cluster sampling, the population is divided into clusters or groups, and a sample is taken from each cluster. In this case, the clusters are the states, and a simple random sample of four states is selected.

Within each selected state, another level of sampling occurs where two colleges or universities are randomly selected from each state. This forms the second stage of sampling.

Finally, from each of the eight selected colleges or universities, a simple random sample of 20 undergraduates is taken. This forms the third stage of sampling.

By combining these stages, a final sample of 160 undergraduates is obtained.

Multistage cluster sampling is commonly used when the population is large and geographically dispersed. It offers a practical approach to obtain a representative sample by selecting clusters at different stages, which helps to reduce time, cost, and logistical challenges.

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To solve this problem, we can use the binomial distribution formula. Let's define success as a student who sleeps at least 7 hours per night. Then, the probability of success for each student is 0.45, and the probability of failure is 0.55. The number of trials is 25, since we have a sample of 25 students.

Now, we want to find the probability that fewer than 40% of the students in a sample would respond that they spend at least 7 hours per night sleeping. This means we want to find P(X < 0.4*25), where X is the number of students in the sample who sleep at least 7 hours per night.

Using a binomial calculator or table, we find that P(X < 10) = 0.0638. Therefore, the answer closest to the probability is 0.0638.

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Rounding to the nearest meter, the distance required to stop a car at 80 km/hr is approximately 87 meters.

According to the given information, the distance needed to stop a car varies directly with the square of its speed. Let's denote the distance as "d" and the speed as "s".

We can write the proportional relationship as:

d = k[tex]s^2[/tex]

where "k" is the constant of variation.

In this case, we are given that it requires 120 m to stop a car at 70 km/hr. Let's convert the speed to meters per second for consistency. There are 1000 meters in a kilometer and 3600 seconds in an hour, so:

70 km/hr = (70 ×1000) / 3600 m/s ≈ 19.44 m/s

Now, we can substitute the values into the equation to find the constant "k":

120 = k × (19.44[tex])^2[/tex]

Solving for "k":

k = 120 / (19.44[tex])^2[/tex] ≈ 0.156

Now that we have the constant of variation, we can determine the distance required to stop a car at 80 km/hr. Again, let's convert the speed to meters per second:

80 km/hr = (80 × 1000) / 3600 m/s ≈ 22.22 m/s

Substituting the values into the equation:

d = 0.156 × (22.22[tex])^2[/tex]

Calculating:

d ≈ 87.34 m

Rounding to the nearest meter, the distance required to stop a car at 80 km/hr is approximately 87 meters.

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Yes, Dijkstra's algorithm can find the shortest paths in a directed acyclic graph (DAG). Dijkstra's algorithm is a popular algorithm used to solve the single-source shortest path problem in graphs with non-negative edge weights.

In a DAG, there are no cycles, meaning there are no paths that loop back to the same node. This absence of cycles ensures that there are no negative weight cycles that would cause the algorithm to fail. Since Dijkstra's algorithm relies on non-negative edge weights, it works perfectly fine in a DAG.

When applied to a DAG, Dijkstra's algorithm will efficiently compute the shortest paths from a given source vertex to all other vertices in the graph. It iteratively explores the graph, updating the distances to each vertex until the shortest paths to all vertices have been determined.

Therefore, if you have a directed acyclic graph and you want to find the shortest paths from a source vertex to all other vertices, you can confidently use Dijkstra's algorithm to achieve that.

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Hello !

[tex]\frac{3}{4} *\frac{3}{8} \\\\= \frac{3*3}{4*8} \\\\= \frac{9}{36} \\\\\boxed{= \frac{1}{4} }[/tex]

[tex]\frac{3}{5} - \frac{1}{4} \\\\= \frac{3*4}{5*4} - \frac{1*5}{4*5}\\\\= \frac{12}{20} - \frac{5}{20} \\\\\boxed{= \frac{7}{20} }[/tex]

Rounding the result to the nearest whole number, we find that approximately 24,108 bacteria are expected to be present after 7 hours.

To solve this problem, we can use the exponential growth formula for population growth:

N(t) = N₀ * e^(kt),

where:

N(t) is the population at time t,

N₀ is the initial population,

e is the base of the natural logarithm (approximately 2.71828),

k is the growth rate constant,

t is the time in hours.

We are given the initial population N₀ = 70 and the population after 4 hours N(4) = 360.

Using this information, we can solve for the growth rate constant k:

N(4) = 70 * e^(4k) = 360

Dividing both sides of the equation by 70:

e^(4k) = 360/70

Taking the natural logarithm (ln) of both sides:

4k = ln(360/70)

Simplifying:

k = ln(360/70) / 4

Now that we have the value of k, we can find the population N(7) after 7 hours:

N(7) = 70 * e^(7k)

Substituting the value of k:

N(7) = 70 * e^(7 * ln(360/70) / 4)

Using a calculator, we can evaluate this expression:

N(7) ≈ 70 * e^(7 * ln(360/70) / 4) ≈ 70 * 345.828 ≈ 24107.96

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To find the nullity of matrix a, we need to determine the number of linearly independent vectors in the null space of matrix a.

Given that the set {u1, u2, u3} is a basis for [tex]R^3\\[/tex], it means that these vectors span the entire space and are linearly independent.

Matrix a is formed by arranging these basis vectors as columns:

a = [u1 u2 u3]

Since the vectors in the basis {u1, u2, u3} are linearly independent, the null space of matrix a will contain only the zero vector. In other words, the only vector that satisfies the equation a * x = 0 is the zero vector, where x is a column vector.

Therefore, the nullity of matrix a is zero, as there are no linearly independent vectors in the null space.

In summary, nullity(a) = 0.

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These are the computed values of beta and p-value for the given alternative value p = 0.21 and sample sizes.

To compute the beta (β) for the given alternative value p = 0.21 and sample sizes n = 81, 900, 14,400, 40,000, and 90,000, we need to determine the probability of Type II error. Beta is the probability of failing to reject the null hypothesis (H₀) when the alternative hypothesis (Hₐ) is true.

To calculate beta, we need to find the corresponding z-value for the given significance level α = 0.01, which corresponds to a z-value of 2.33 (approximate value).

Using the formula for the standard deviation of the sample proportion:

σ = sqrt((p₀ * (1 - p₀)) / n)

where p₀ is the null hypothesis value, p = 0.2, and n is the sample size.

Then we can calculate the z-score for the alternative value p = 0.21 using the formula:

z = (p - p₀) / σ

Finally, we can compute beta using the standard normal distribution table or a statistical calculator.

Let's calculate beta for the given sample sizes:

For n = 81:

σ = sqrt((0.2 * (1 - 0.2)) / 81) ≈ 0.0447

z = (0.21 - 0.2) / 0.0447 ≈ 0.2494

beta = P(Z > 0.2494) ≈ 0.4013

For n = 900:

σ = sqrt((0.2 * (1 - 0.2)) / 900) ≈ 0.0143

z = (0.21 - 0.2) / 0.0143 ≈ 0.7692

beta = P(Z > 0.7692) ≈ 0.2206

For n = 14,400:

σ = sqrt((0.2 * (1 - 0.2)) / 14,400) ≈ 0.0071

z = (0.21 - 0.2) / 0.0071 ≈ 1.5493

beta = P(Z > 1.5493) ≈ 0.0606

For n = 40,000:

σ = sqrt((0.2 * (1 - 0.2)) / 40,000) ≈ 0.0050

z = (0.21 - 0.2) / 0.0050 ≈ 2.2000

beta = P(Z > 2.2000) ≈ 0.0139

For n = 90,000:

σ = sqrt((0.2 * (1 - 0.2)) / 90,000) ≈ 0.0033

z = (0.21 - 0.2) / 0.0033 ≈ 3.0303

beta = P(Z > 3.0303) ≈ 0.0012

Next, let's calculate the p-value for the alternative value P = 0.21 and sample sizes n = 81, 900, 14,400, and 40,000.

The z-score for the given P can be calculated using the formula:

z = (P - p₀) / σ

Using the standard normal distribution table or a statistical calculator, we can find the area under the curve beyond the calculated z-score to obtain the p-value.

For n = 81:

σ = sqrt((0.2 * (1 - 0.2)) / 81) ≈ 0.0447

z = (0.21 - 0.2) / 0.0447 ≈ 0.2494

p-value = P(Z > 0.2494) ≈ 0.4013

For n = 900:

σ = sqrt((0.2 * (1 - 0.2)) / 900) ≈ 0.0143

z = (0.21 - 0.2) / 0.0143 ≈ 0.7692

p-value = P(Z > 0.7692) ≈ 0.2206

For n = 14,400:

σ = sqrt((0.2 * (1 - 0.2)) / 14,400) ≈ 0.0071

z = (0.21 - 0.2) / 0.0071 ≈ 1.5493

p-value = P(Z > 1.5493) ≈ 0.0606

For n = 40,000:

σ = sqrt((0.2 * (1 - 0.2)) / 40,000) ≈ 0.0050

z = (0.21 - 0.2) / 0.0050 ≈ 2.2000

p-value = P(Z > 2.2000) ≈ 0.0139

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To find the Taylor polynomial T3(x) for the function f(x) = ex sinx at a = 0, we can use the Taylor series expansion. The Taylor polynomial of degree 3 is given by:

[tex]T3(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3[/tex]

First, let's find the derivatives of f(x):

f(x) = ex sinx

f'(x) = ex cosx + ex sinx

f''(x) = 2ex cosx

f'''(x) = 2ex cosx - 2ex sinx

Evaluate the derivatives at x = 0:

[tex]f(0) = e^0 sin(0) = 0\\f'(0) = e^0 cos(0) + e^0 sin(0) = 1\\f''(0) = 2e^0 cos(0) = 2\\f'''(0) = 2e^0 cos(0) - 2e^0 sin(0) = 2[/tex]

Now, substitute these values into the Taylor polynomial formula:

[tex]T3(x) = 0 + 1x + (2/2!)x^2 + (2/3!)x^3[/tex]

Simplifying:

[tex]T3(x) = x + x^2 + (1/3)x^3[/tex]

Now, let's evaluate the limit:

[tex]lim(x- > 0) (e^x sinx / x^3)[/tex]

We can use the Taylor polynomial T3(x) to approximate the function [tex]e^x sinx[/tex]  as x approaches 0.

The term [tex]e^x sinx[/tex] can be approximated as x when x is close to 0. Thus, as x approaches 0, the limit becomes:

[tex]lim(x- > 0) (x / x^3) = lim(x- > 0) (1 / x^2)[/tex]

However, the limit of [tex]1 / x^2[/tex]as x approaches 0 is infinity.

Therefore, the limit is infinity.

Please note that this is an approximation using the Taylor polynomial, and the actual behavior of the function may differ.

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Answer:

  6%

Step-by-step explanation:

You want the simple interest rate that results in an interest charge of $1800 on a $5000 loan for 6 years.

Your list of financial formulas will tell you that simple interest is computed as ...

  I = Prt . . . . interest on principal P at rate r for t years

Using the given information, we can fill in the values like this:

  1800 = 5000 × r × 6

Dividing by the coefficient of r gives ...

  1800/30000 = r = 6/100 = 6%

The annual interest rate for her loan was 6%.

The statement is True. In a continuous probability distribution, the probability of a single point, such as an endpoint, has a zero probability, so including or excluding them does not affect the overall probability of the interval.

Continuous probability refers to the probability distribution associated with continuous random variables. In contrast to discrete probability, where the random variable can only take on a finite or countably infinite set of values, continuous random variables can take on any value within a certain range or interval.

The probability distribution for a continuous random variable is described by a probability density function (PDF), often denoted as f(x). The PDF represents the relative likelihood of different values of the random variable occurring. Unlike the probability mass function (PMF) used for discrete random variables, the PDF does not directly give the probability of a specific value but rather the probability density at a given point.

Your question is asking if, for a continuous probability distribution, the probability of x being between a and b remains the same whether you include the endpoints (a and b) or not. The statement is True. In a continuous probability distribution, the probability of a single point, such as an endpoint, has a zero probability, so including or excluding them does not affect the overall probability of the interval.

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The correct Bonferroni adjustment to make two comparisons with an overall experimental error rate of 0.05 is 0.025.

The Bonferroni adjustment is a method used to control the family-wise error rate (FWER) when multiple hypothesis tests are conducted. It works by dividing the desired significance level by the number of tests being conducted. In this case, we have two comparisons, so the adjusted significance level for each test would be 0.025.

This means that for each comparison, we would reject the null hypothesis if the p-value is less than 0.025, instead of the usual 0.05. This adjustment ensures that the overall FWER does not exceed the desired level of 0.05.

Therefore, option b is the correct answer.

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The value of  x of chord = 5

By definition of circle,

The chord of a circle is defined as the line segment connecting any two locations on the circle's perimeter; nevertheless, the diameter is the longest chord of a circle that goes through the centre of the circle.

The chord is one of the several line segments that may be made in a circle, and its endpoints are on the circumference.

⇒ 6 (6 + x) = 7 (7 + 11)

Solve for x;

⇒ 36 + 6x = 7 × 18

⇒ 36 + 6x = 126

⇒ 6x = 126 - 36

⇒ 6x = 90

⇒ x = 15

Thus, The value of x = 5

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To find E(X1X10X22), the expected value of the product of the first, tenth, and twenty-second picks from an urn containing 30 chips numbered from 1 to 30, we can use the concept of linearity of expectation.

The expected value of the product is the product of the expected values of the individual picks.

The expected value of a single pick can be computed by taking the sum of all possible values multiplied by their respective probabilities. In this case, the expected value of a single pick is (1+2+3+...+30)/30 = 15.5.

Using the linearity of expectation, the expected value of the product X1X10X22 is E(X1) * E(X10) * E(X22) = 15.5 * 15.5 * 15.5 = 3759.875. Therefore, the expected value of the product of the first, tenth, and twenty-second picks is approximately 3759.875.

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Answer:

4j^2-12j+8

Step-by-step explanation:

Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Correct answer is option (c).

To determine which set of numbers could represent the three sides of a right triangle, we can use the Pythagorean theorem:

a. {13, 48, 50}:

Using the Pythagorean theorem: 13² + 48² = 169 + 2304 = 2473 ≠ 50²

Therefore, this set of numbers does not represent the sides of a right triangle.

b. {49, 55, 73}:

Using the Pythagorean theorem: 49² + 55² = 2401 + 3025 = 5426 ≠ 73²

Therefore, this set of numbers does not represent the sides of a right triangle.

c. {16, 63, 65}:

Using the Pythagorean theorem: 16² + 63² = 256 + 3969 = 4225 = 65^²

Therefore, this set of numbers represents the sides of a right triangle.

d. {20, 72, 75}:

Using the Pythagorean theorem: 20² + 72² = 400 + 5184 = 5584 ≠ 75²

Therefore, this set of numbers does not represent the sides of a right triangle.

Based on the Pythagorean theorem, the set of numbers {16, 63, 65} represents the sides of a right triangle.

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△CMN is equilateral because CM = CN (midpoint property) and ∠CMN = ∠CNM = ∠MNC (corresponding angles).

To prove that △CMN is an equilateral triangle, we need to show that all three sides are equal in length and that all three angles are equal.

Let's start by analyzing the given information and the properties of the figure.

△ABC is an equilateral triangle, meaning all three sides (AB, BC, and CA) are equal in length.

△CDE is an equilateral triangle, implying that all three sides (CD, DE, and EC) are equal in length.

Point M is the midpoint of AD, so AM = MD.

Point N is the midpoint of BE, so BN = NE.

Now, let's proceed with the proof:

Show that CM = CN.

Since M is the midpoint of AD and N is the midpoint of BE, we can write:

AM = MD (definition of M being the midpoint)

BN = NE (definition of N being the midpoint)

By adding these two equations, we get:

AM + BN = MD + NE

Now, let's examine the left-hand side of the equation. The sum AM + BN represents the length of AB, as AM and BN are the midpoints of AD and BE respectively. Since AB is a side of the equilateral triangle △ABC, it is equal in length to BC and CA. Therefore, we can rewrite the equation as:

AB = MD + NE

Next, let's consider the right-hand side of the equation. MD + NE represents the length of DE, which is a side of the equilateral triangle △CDE. As mentioned earlier, all sides of △CDE are equal in length. Therefore, we can rewrite the equation as:

AB = DE

Since AB = BC = CA (because △ABC is equilateral), and DE = CD = EC (because △CDE is equilateral), we can conclude that:

BC = DE

This implies that the line segments BC and DE have the same length. Moreover, since BC is parallel to DE, the line segments BC and DE are congruent (have equal length) according to the properties of parallel lines. Therefore, we have:

BC = DE = CM + MN + NE

Now, let's examine the right-hand side of the equation. CM + MN + NE represents the length of CN. We've just established that BC = DE, so we can substitute these equal lengths in the equation:

BC = CM + MN + NE

Simplifying the equation, we have:

BC = CN

Therefore, we've shown that CM = CN, meaning that two sides of △CMN are equal.

Show that ∠CMN = ∠CNM = ∠MNC.

To prove that all three angles of △CMN are equal, we need to show that ∠CMN = ∠CNM = ∠MNC.

First, let's consider △CME. Since △CDE is equilateral, the angle ∠CME is 60 degrees. As MN is parallel to CD, we can conclude that ∠CMN is congruent to ∠CME (corresponding angles). Therefore, ∠CMN = ∠CME = 60 degrees.

Next, let's consider △CNB. Since △ABC is equilateral, the angle ∠ACB is 60 degrees. As MN is parallel to AB, we can conclude that ∠CNM is congruent to ∠CNB (corresponding angles). Therefore, ∠CNM = ∠CNB = 60 degrees.

Since ∠CMN = ∠CME = 60 degrees and ∠CNM

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Step-by-step explanation:

Composite area:

   Consists of a rectangle,    area  12 x 10 = 120 cm^2

        and two half-circles ( making a whole circle)    Area = pi r^2

                          = pi (5^2) = 78.5 cm^2

Total area = 198.5 cm^2

Answer:

Omar's share is $420in $540

Step-by-step explanation:

Given :

Amount of money that Omar and Jamil shares = $540

Let share of Omar and Jamil in the money is 7x and 2x

Sum of share = 7x + 2x =9x

Omar's share = (7x /9x ) ×$540

                     = ( 7 × 540 )/9

                     = 7 × 60 = $420

Step-by-step explanation:

To find the general solution of the system x' = 10 −5 8 −12 x, we first need to find the eigenvalues and eigenvectors of the coefficient matrix.

The characteristic equation is det(A - λI) = 0, where A is the coefficient matrix, λ is the eigenvalue, and I is the identity matrix. So, we have:

det(10-λ -5 8 -12-λ) = 0
(10-λ)(-12-λ) - (-5)(8) = 0
λ[tex]^2[/tex] - 2λ - 64 = 0
(λ - 8)(λ + 8) = 0
λ1 = 8, λ2 = -8

Next, we need to find the eigenvectors corresponding to each eigenvalue. For λ1 = 8, we have:

(10-8)x1 - 5y1 + 8z1 = 0
-5x1 + (8-8)y1 + 8z1 = 0
8x1 + 8y1 + (-12-8)z1 = 0

Simplifying the system, we get:

2x1 - y1 + 4z1 = 0
-5x1 = 0
8x1 + 8y1 - 20z1 = 0

Solving for x1, y1, and z1, we get:

x1 = 0
y1 = 0
z1 = t

So, the eigenvector corresponding to λ1 = 8 is [0, 0, t].

For λ2 = -8, we have:

(10+8)x2 - 5y2 + 8z2 = 0
-5x2 + (8+8)y2 + 8z2 = 0
8x2 + 8y2 + (-12+8)z2 = 0

Simplifying the system, we get:

18x2 - 5y2 + 8z2 = 0
-5x2 + 16y2 + 8z2 = 0
8x2 + 8y2 - 4z2 = 0

Solving for x2, y2, and z2, we get:

x2 = 2t
y2 = 5t
z2 = -2t

So, the eigenvector corresponding to λ2 = -8 is [2t, 5t, -2t].

Now that we have the eigenvalues and eigenvectors, we can write the general solution as:

[tex]x(t) = c1[0, 0, t]e^{(8t)} + c2[2t, 5t, -2t]e^{(-8t)}[/tex]
where c1 and c2 are constants determined by initial conditions.

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Answer:

  V = x(8 -2x)(10 -2x) = 4x³ -36x² +80x

Step-by-step explanation:

You want an equation for the volume (V) in terms of x, the side length of the square corner removed from an 8" by 10" piece of cardboard. After the square is removed, the cardboard is folded to make an open-top box.

The depth of the box is the x dimension of the flap left after the corner is removed.

The side flap in the 8" direction will have a remaining dimension of (8 -2x) inches.

The side flap in the 10" direction will have a remaining dimension of (10-2x) inches.

Then the bottom area is (8 -2x)(10 -2x).

The volume is the product of the depth and the area of the bottom of the box:

  V = x(8 -2x)(10 -2x)

This can be expanded to the cubic ...

  V = 4x(4 -x)(5 -x) = 4x(x² -9x +20)

  V = 4x³ -36x² +80x

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Additional comment

The volume is maximized when x = 3-(√21)/3 ≈  1.472 inches.

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