hannah wants to find the perimeter of a placemat she is making. the diagram shows the placemat. which expressions can hannah use to find the perimeter of the placemat? select all the correct answers.

The probability of at least 3 computer shutdowns during the year is 0.6472 or 64.72%. The probability of at least 3 months with exactly 1 shutdown is 0.5886 or 58.86%.

The number of shutdowns during the year has a Poisson Distribution with parameter λ = 0.25 * 12 = 3. We want to find the probability of at least 3 shutdowns during the year, which is:

P(X >= 3) = 1 - P(X < 3)

Using the cumulative distribution function (CDF) for the Poisson Distribution, we get:

P(X < 3) = F(2) = e^(-λ) * (λ^0/0! + λ^1/1!) + e^(-λ) * (λ^2/2!) = e^(-3) * (1 + 3 + 9/2) = 0.3528

Therefore, P(X >= 3) = 1 - P(X < 3) = 1 - 0.3528 = 0.6472

The number of months with exactly 1 shutdown during the year has a Poisson Distribution with parameter λ = 0.25. The probability of a month having exactly 1 shutdown is:

P(X = 1) = e^(-λ) * (λ^1/1!) = 0.25 * e^(-0.25)

The number of months with exactly 1 shutdown in a year has a Binomial Distribution with parameters n = 12 and p = P(X = 1). We want to find the probability of at least 3 months with exactly 1 shutdown, which is:

P(X >= 3) = 1 - P(X < 3)

Using the cumulative distribution function (CDF) for the Binomial Distribution, we get:

P(X < 3) = F(2) = Σ (12 choose k) * p^k * (1-p)^(12-k), for k = 0 to 2

P(X < 3) = (12 choose 0) * 0.25^0 * 0.75^12 + (12 choose 1) * 0.25^1 * 0.75^11 + (12 choose 2) * 0.25^2 * 0.75^10

P(X < 3) = 0.4114

Therefore, P(X >= 3) = 1 - P(X < 3) = 1 - 0.4114 = 0.5886

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(a) The cube can be described as a set of points in 3-dimensional space where each coordinate (x, y, z) satisfies -1 ≤ x ≤ 1, -1 ≤ y ≤ 1, and -1 ≤ z ≤ 1.

(b) Let O be the origin of the coordinate system. Then,The vector −→OE can be written as OE = 1i + 0j + 0k.The vector −−→OD can be written as OD = 0i + 1j + 0k.The vector −−→OF can be written as OF = 0i + 0j + 1k.The vector −−→OG can be written as OG = 1i + 1j + 1k.

(c) The center point of the cube can be found by taking the average of the coordinates of opposite corners. Let A = (-1, -1, -1) be one corner of the cube and B = (1, 1, 1) be the opposite corner. Then the coordinates of the center point C are:C = ((-1 + 1)/2)i + ((-1 + 1)/2)j + ((-1 + 1)/2)k = 0i + 0j + 0k = (0, 0, 0)

(d) Let D, E, and F be the vertices of the cube that are adjacent to A, B, and C respectively. Then the point P that is the vector sum of −→OA, −−→OB and −−→OC is:P = A + OD + OF = (-1, -1, -1) + (0, 1, 0) + (0, 0, 1) = (-1, 0, 0)

(e) The plane of all linear combinations of −→OA and −→OA + −−→OB is the plane that passes through points A, O, and A + B. The normal vector of this plane is the cross product of vectors OA and (OA + OB), which is:OA x (OA + OB) = (-1i - 1j - 1k) x (0i + 0j + 2k) = 2i - 2jTherefore, the equation of the plane is:2x - 2y + z = 0This plane passes through the origin O, which is the point (0, 0, 0) in xyz space.

Ka and Kb are both acid-base equilibrium constants, but they apply to different types of reactions. Ka (acid dissociation constant) is used to describe the dissociation of an acid in water, while Kb (base dissociation constant) is used to describe the dissociation of a base in water.

Specifically, Ka is a measure of the strength of an acid in terms of how easily it donates a proton (H+) to a solvent like water. A higher Ka value indicates a stronger acid, while a lower Ka value indicates a weaker acid. Kb, on the other hand, is a measure of the strength of a base in terms of how easily it accepts a proton (H+) from a solvent like water. A higher Kb value indicates a stronger base, while a lower Kb value indicates a weaker base. There is a mathematical relationship between Ka and Kb for conjugate acid-base pairs, which are molecules or ions that differ by the presence or absence of a single proton. The relationship is expressed by the equation: Ka x Kb = Kw, where Kw is the ion product constant for water, which is 1.0 x 10⁻¹⁴ at 25°C. This relationship shows that the stronger the acid, the weaker its conjugate base, and vice versa.

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The best snack for Tyrell to help keep his brain sharp during capture the flag would be walnuts.

Walnuts are a great source of omega-3 fatty acids, which are essential for brain function and cognitive health. They also contain antioxidants that can help protect the brain from oxidative damage and improve memory and learning ability.

On the other hand, candy bars are often high in sugar and unhealthy fats, which can cause a quick energy spike followed by a crash, leaving Tyrell feeling tired and sluggish.

Oatmeal and whole-grain crackers are healthier options, as they contain complex carbohydrates that provide sustained energy and fiber that helps regulate blood sugar levels. However, they do not offer the same brain-boosting benefits as walnuts do.

In summary, walnuts are the best snack for Tyrell to help keep his brain sharp during capture the flag, as they provide important nutrients for brain function and cognitive health.

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Answer:

Step-by-step explanation:

18%

Answer:

18%

Step-by-step explanation:

We can work this out in a simple way by first converting the fraction 9/50 to a decimal. To do that, we simply divide the numerator by the denominator:9/50 = 0.18 Once we have the answer to that division, we can multiply the answer by 100 to make it a percentage:0.18 x 100 = 18%

Therefore , the solution of the given problem of quadrilateral comes out to be a trapezoid with four sides is the final possible number of sides for the polygon.

A rectangular is a four-sided, four-cornered object in geometry. Latin terms either "quad" but rather "array of ingenious thoughts" were used to create the expression (meaning "side"). The three parts of a rectangle are four regions, four corners, and four corners. The two main types of concave as well as convex shapes are convex and concave. Subdivisions of isosceles triangles, geometric figures, angles, rhombuses, but instead squares are also considered to be convex quadrilaterals.

Here,

Any polygon whose two sides are parallel line segments must be a trapezoid.

The other two sides of a trapezoid must intersect since the shape only has one pair of parallel sides. A trapezoid has four sides as a result.

Drawing every possible diagonal from one vertex results in n-2 triangles for a polygon with n sides.

The interior angle sum of the complete polygon is (n-2) * 180 degrees because each triangle has an interior angle sum of 180 degrees.

The interior angle sum of a trapezoid is (4-2) * 180 degrees, which equals 360 degrees.

which each have interior angles that add up to 180 degrees.

Therefore, a trapezoid with four sides is the final possible number of sides for the polygon.

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Answer:

4.5 feet

Step-by-step explanation:

To find the width of Mrs. Jones' garden, we can use the formula for the area of a rectangle:

Area = length x width

We are given the area of the garden as 63 square feet and the length as 14 feet. Plugging these values into the formula, we get:

63 = 14 x width

To solve for the width, we can divide both sides by 14:

width = 63/14

Simplifying this expression, we get:

width = 4.5 feet

Therefore, the width of Mrs. Jones' garden is 4.5 feet.

Theorem 8.1: If f is continuous on the closed interval [a, b], then f is bounded on [a, b]. 

 If f is discontinuous at any point in the interval [a, b], then the conclusion of Theorem 8.1 is false. To prove this, let us consider an example of a discontinuous function f defined on the closed interval [a, b]. Let f(x) = {1, for x ∈ [a, b] and 0, for x ∉ [a, b]}.

Clearly, f is discontinuous at any point in the interval [a, b] and is not bounded. Thus, the conclusion of Theorem 8.1 is false.

If f is continuous at all but a finite number of points in the interval [a, b] and is bounded, then the conclusion of Theorem 8.1 is true. To prove this, let us consider a function f defined on the closed interval [a, b].

Let f(x) = x2.

Clearly, f is continuous at all but a finite number of points in the interval [a, b] and is bounded. Thus, the conclusion of Theorem 8.1 is true.

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answer is a 55° is and how to

Option (b) [tex]r = 14 + 14\ cos \theta[/tex] is symmetric with respect to the polar axis. So we have to know meaning of symmetry.

In terms of geometry, symmetry is said to as a balanced and proportionate similarity between two (2) halves of an item.

Let, Start with option (a)symmetric with respect to the line  [tex]\theta =\frac{\pi }{2}[/tex].

So , in this polar equation we have to place [tex]\theta\ by\ \pi -\theta[/tex],

[tex]r = 14 + 14\ cos (\pi -\theta)[/tex]

Now solve by the formula, cos(α-β) = cosα*cosβ + sinα*sinβ

[tex]r= 14 + 14[cos\pi *cos\theta + sin\pi *sin\theta][/tex]

Here, [tex]sin\pi +sin\theta\ become\ 0\ and\ cos \pi =-1[/tex]

[tex]r=14+14 cos(-cos\theta)[/tex]

[tex]r = 14 - 14\ cos \theta[/tex]

Since it is not equivalent to my original , so it not symmetric with respect to the line [tex]\theta =\frac{\pi }{2}[/tex].

Now, Start with option (b) i.e. symmetric with respect to polar axis.

[tex]r = 14 + 14\ cos \theta[/tex]

So,  in this polar equation we have to place [tex]\theta\ by\ -\theta[/tex]

[tex]r=14+14\ cos(-\theta)[/tex]

[tex]r = 14 + 14\ cos \theta[/tex]

So, it is symmetric with the polar axis.

Hence our option is (b) i.e. symmetric with respect to polar axis.

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Answer:

Step-by-step explanation:

1. (x+6)(x+6) expand the brackets to x²+6x+6x+36 then if you need to simplify add the 6x +6x= 12x

2. (x+8)(x-5) expand to x²-5x+8x-45 then if needed simplify by adding -5x + 8x=3x

3. (x-5)(x-2) expand the brackets to x²-2x-5x+10 if you need to simplify take the -5x + -2x=-7

The volumes of the solids generated by rotating the given region around the given axis of rotation are 200π, 120π, 200π, and 80π, respectively.

Pappus' Theorem is a mathematical theorem that relates the volume of a solid generated by rotating a plane region about an external axis to the area of the region and the distance traveled by the region's centroid during the rotation. The theorem can be applied to find the volume of solids generated by rotating the given region around the given axis of rotation.

a. If region R having an area of 20 and centroid (3,5) is rotated around x-axis, the volume of the solid generated is given by:
V = 2π(5)(20) = 200π

b. If region R having an area of 20 and centroid (3,5) is rotated around y-axis, the volume of the solid generated is given by:
V = 2π(3)(20) = 120π

c. If region R having an area of 20 and centroid (3,5) is rotated around x=2, the volume of the solid generated is given by:
V = 2π(5)(20) = 200π

d. If region R having an area of 20 and centroid (3,5) is rotated around y=1, the volume of the solid generated is given by:
V = 2π(2)(20) = 80π

Therefore, the volumes of the solids generated by rotating the given region around the given axis of rotation are 200π, 120π, 200π, and 80π, respectively.

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Proposition 1.13(a) states that "Let m be an integer, and m ≥ 1 be an integer, and suppose that a1 ≡ a2 (mod m) and b1 ≡ b2 (mod m). Then, a1 ± b1 ≡ a2 ± b2 (mod m) and a1 · b1 ≡ a2 · b2 (mod m)."

Proof of Proposition 1.13(a):

We have that a1 ≡ a2 (mod m) implies m|(a1 - a2) and b1 ≡ b2 (mod m) implies m|(b1 - b2). Therefore, a1 - a2 = km and b1 - b2 = jm, for some integers k and j, by the definition of the modulo operation.

Adding these two equations, we get that a1 - a2 + b1 - b2 = (k + j)m, which implies that a1 + b1 ≡ a2 + b2 (mod m).

Multiplying a1 - a2 = km and b1 - b2 = jm, we get a1 b1 - a2 b1 + a2 b1 - a2 b2 = kmjm. Subtracting these two equations, we obtain a1 b1 - a2 b2 = (a1 - a2)b2 + (b1 - b2)a2, which implies a1 b1 ≡ a2 b2 (mod m).

Therefore, Proposition 1.13(a) is proven.

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The question asks us to prove that if X and Y are Bernoulli random variables and f : {0, 1} → {0, 1} is a function, then f(X) and Y are independent. To prove this statement, we will make use of the fact that if two random variables X and Y are independent, then the joint probability of X and Y is equal to the product of the two marginal probabilities.

The probability of X taking a value of 0 or 1 can be given as P(X = 0) = p and P(X = 1) = q. Similarly, the probability of Y taking a value of 0 or 1 can be given as P(Y = 0) = r and P(Y = 1) = s. Then, the joint probability of X and Y can be given as P(X = 0, Y = 0) = pr, P(X = 0, Y = 1) = ps, P(X = 1, Y = 0) = qr, and P(X = 1, Y = 1) = qs.

Now, if X and Y are independent, then P(X = 0, Y = 0) = pr = P(X = 0) * P(Y = 0) = p * r and similarly P(X = 0, Y = 1) = ps = P(X = 0) * P(Y = 1) = p * s, P(X = 1, Y = 0) = qr = P(X = 1) * P(Y = 0) = q * r and P(X = 1, Y = 1) = qs = P(X = 1) * P(Y = 1) = q * s. Now, if f : {0, 1} → {0, 1} is a function and f(X) and Y are independent, then the joint probability of f(X) and Y can be given as P(f(X) = 0, Y = 0) = P(X = 0, Y = 0) = pr, P(f(X) = 0, Y = 1) = P(X = 0, Y = 1) = ps, P(f(X) = 1, Y = 0) = P(X = 1, Y = 0) = qr and P(f(X) = 1, Y = 1) = P(X = 1, Y = 1) = qs.

Therefore, we can conclude that if X and Y are Bernoulli random variables and f : {0, 1} → {0, 1} is a function, then f(X) and Y are independent if and only if P(f(X) = 0, Y = 0) = P(X = 0, Y = 0) = pr, P(f(X) = 0, Y = 1) = P(X = 0, Y = 1) = ps, P(f(X) = 1, Y = 0) = P(X = 1, Y = 0) = qr and P(f(X) = 1, Y = 1) = P(X = 1, Y = 1) = qs. This concludes the proof.

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The length of the rectangular plot of land is 168 ft.

To find the length of the rectangular plot of land, we need to use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

In the given diagram, we can see that the plot of land forms a right triangle, where the length of one leg is 874 ft and the length of the diagonal is 890 ft. Therefore, the length of the rectangular plot can be found as follows:

[tex]h^2 = a^2 +b^2\\b^2 = h^2 - a^2\\b = \sqrt{h^2-a^2} \\b= \sqrt{(890)^2- (874)^2}\\ b= \sqrt{792100- 763876}\\ b= \sqrt{28224}\\ b= 168[/tex]

So, the length of the rectangular plot of land is 168 ft.

The lengths of the legs of a right triangle are related to the lengths of the sides of a rectangle in the following way:

If we draw a rectangle with sides of length "a" and "b", then the diagonal of the rectangle (which is the hypotenuse of the right triangle formed by the sides of the rectangle) will have a length equal to the square root of ([tex]a^2 + b^2[/tex]).

Conversely, if we have a right triangle with legs of length "a" and "b", then we can form a rectangle by making the legs of the triangle the sides of the rectangle. The length of one side of the rectangle will be "a" and the length of the other side will be "b".

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For the integral test, the integral have to be positive function then itself is called as convergence of the series.

Let's take an example to understand this concept better. Consider the infinite series:

∑n=1 to ∞ (1/n²)

To apply the integral test, we need to compare this series with a corresponding integral function. The integral function for this series is:

∫1 to ∞ (1/x²)dx

Now, although the function 1/x² is positive, continuous, and decreasing, the integral of the function does not converge. The integral function can be evaluated as:

∫1 to ∞ (1/x²)dx = [(-1/x)] from 1 to ∞

= (1/1) - (1/∞)

= 1

Since the integral function diverges, we can conclude that the series also diverges. Therefore, we use the integral test to determine the convergence of this series is positive.

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the number of vectors (x1, x2 . . ., xn) such that each xi is either 0 or 1 and xn i=1 xi ≥ are:

(0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1).

The number of vectors (x1, x2, ..., xn) such that each xi is either 0 or 1 and xn i=1 xi ≥ is 2^n,

where n is the number of elements in the vector.

For example, if n=3,

then there would be 8 possible vectors: (0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1).

A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector and with an arrow indicating the direction. The direction of the vector is from its tail to its head.

Two vectors are the same if they have the same magnitude and direction. This means that if we take a vector and translate it to a new position (without rotating it), then the vector we obtain at the end of this process is the same vector we had in the beginning.

Two examples of vectors are those that represent force and velocity. Both force and velocity are in a particular direction. The magnitude of the vector would indicate the strength of the force, or the speed associated with the velocity.

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Answer:

c

Step-by-step explanation:

the axis meets as that point.

Answer: C

Step-by-step explanation: the answer is C trust me I’m learning this in school :)

Answer:

it is a decrease of 12%

Step-by-step explanation:

The correct solution for this question is  (b) $25.20.

How to solve and what is sale tax?

To calculate the total cost of the DVD including tax, we need to add the sales tax to the original price.

If the DVD costs $24, then the sales tax is 5% of $24, which is:

0.05 x $24 = $1.20

Therefore, the total cost of the DVD including tax is:

$24 + $1.20 = $25.20

So the answer is (b) $25.20.

Sales tax is a tax imposed by the government on the sale of goods and services. The tax is usually calculated as a percentage of the sale price and is added to the price that the buyer pays.

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The correct answer is "One solution" at (−2, 1)

A mathematical equation known as a linear equation denotes a straight line on a graph. It is an algebraic equation where the variable's largest exponent is 1. A linear equation's standard form is:

ax + by = c

where x and y are variables and a, b, and c are constants.

To determine the number of solutions to the system of linear equations shown on the graph, we need to find the point where the two lines intersect.

From the graph, we can see that the two lines intersect at the point (−2, 1). Therefore, the system of linear equations has one unique solution at (−2, 1).

Therefore, the correct answer is: One solution at (−2, 1).

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Answer:

x = 1, y = 2

Step-by-step explanation:

Solve the following system:

{2 y + 3 x = 7 | (equation 1)

4 y - 3 x = 5 | (equation 2)

Add equation 1 to equation 2:

{3 x + 2 y = 7 | (equation 1)

0 x + 6 y = 12 | (equation 2)

Divide equation 2 by 6:

{3 x + 2 y = 7 | (equation 1)

0 x + y = 2 | (equation 2)

Subtract 2 × (equation 2) from equation 1:

{3 x + 0 y = 3 | (equation 1)

0 x + y = 2 | (equation 2)

Divide equation 1 by 3:

{x + 0 y = 1 | (equation 1)

0 x + y = 2 | (equation 2)

Collect results:

Answer: {x = 1, y = 2

An investment strategy with an expected return of 14 percent and a standard deviation of 8 percent has the potential for higher returns, but also comes with a higher level of risk. It is important to carefully consider the trade-off between risk and return before making an investment decision.

An investment strategy with an expected return of 14 percent and a standard deviation of 8 percent means that the investment has a higher potential for return, but also a higher risk.

The standard deviation is a measure of the variation or dispersion of a set of data values. In this case, the standard deviation of 8 percent indicates the degree of variation in the investment returns. A higher standard deviation means that the investment returns are more spread out and there is a higher chance of extreme values, both positive and negative.

In terms of investment strategy, it is important to consider the trade-off between risk and return. While a higher expected return is desirable, it often comes with a higher level of risk. Therefore, it is important to carefully consider the standard deviation and the potential risks associated with an investment before making a decision.

In conclusion, an investment strategy with an expected return of 14 percent and a standard deviation of 8 percent has the potential for higher returns, but also comes with a higher level of risk. It is important to carefully consider the trade-off between risk and return before making an investment decision.

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1. The coordinates of the point P on the line P(x,y,z) is (−1, 4, -7)  and a vector v parallel to the line is 2i − j.

2. The set of parametric equations of the line is given as, x= −1 + 2t, y= 4 - t, z= -7

The parametric equation of the line is given by the following equation,

x= x0 + vt, y= y0 + vt, and z= z0 + vt

Where (x0, y0, z0) is a point on the line and v is a vector parallel to the line.

We are given the following details about the line,

x=5t, y=6-t, z=1+4t

P(x,y,z)=v=2.

And the line passes through the point (−1, 4, -7) and is parallel to v = 2i − j.

Coordinates of the point P: We are given that x = 5t, y = 6 - t and z = 1 + 4t.

The point P(x,y,z) is on the line.

We can substitute the given values in the equation to find the coordinates of the point P.

Hence, P(x,y,z) = (5t, 6 - t, 1 + 4t).

Vector v: We are given that vector v is parallel to the line and v = 2.i + (-1).j = (2,-1,0)

Let A be the point (−1, 4, -7) and v = (2,-1,0) be the direction of the line.

Then, the vector equation of the line is given by: r = A + tv = (−1, 4, -7) + t(2,-1,0) = (−1 + 2t, 4 − t, −7)

Parametric equation of the line: x= −1 + 2t, y= 4 - t, z= -7

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To solve the recurrence relation T(n) = T(n/2) + 2^n using the master theorem, we need to compare the function 2^n with n^(log_b a) = n^(log_2 1) = n^0 = 1.

Since 2^n grows much faster than n^0, we can apply case 3 of the master theorem, which states that if f(n) = 2^n, and if there exists a constant ε > 0 such that f(n) = Ω(n^(log_b a + ε)), then T(n) = Θ(f(n)).

In our case, since 2^n = Ω(n^(0 + ε)) for any ε > 0, we can conclude that T(n) = Θ(2^n).

Therefore, the solution to the recurrence relation T(n) = T(n/2) + 2^n is T(n) = Θ(2^n).

Therefore, the measures of the angles in quadrilateral QRST are Q = 121 degrees, S = 123 degrees, R = 58 degrees, and T = 58 degrees. Therefore, the measures of the angles in quadrilateral ABCDE are A = 90 degrees, B = 90 degrees, C = 64 degrees, D = 42 degrees, and E = 74 degrees.

In geometry, an angle is a figure formed by two rays, called the sides of the angle, that share a common endpoint, called the vertex of the angle. The sides of the angle can be thought of as two line segments that extend in opposite directions from the vertex. Angles are usually measured in degrees or radians. One complete revolution around a circle is equal to 360 degrees or 2π radians. Angles can be classified as acute (less than 90 degrees), right (exactly 90 degrees), obtuse (greater than 90 degrees but less than 180 degrees), straight (exactly 180 degrees), or reflex (greater than 180 degrees but less than 360 degrees). Angles are used in many areas of mathematics, science, and engineering to model relationships between lines, shapes, and other geometric objects. They are also used in trigonometry, which is the study of the relationships between angles and the sides and ratios of triangles. Angles are an important concept in many fields, including physics, astronomy, and engineering.

Here,

16. In any quadrilateral, the sum of the angles is equal to 360 degrees. So, we have:

Q + S + R + T = 360

Substituting the given angle measures, we get:

(2x+5) + (2x+7) + x + x = 360

Simplifying the equation, we get:

6x + 12 = 360

Subtracting 12 from both sides, we get:

6x = 348

Dividing both sides by 6, we get:

x = 58

Now we can substitute x back into the expressions for the angles to find their measures:

Q = 2x+5 = 2(58)+5

= 121 degrees

S = 2x+7 = 2(58)+7

= 123 degrees

R = x = 58 degrees

T = x = 58 degrees

18. In any quadrilateral, the sum of the angles is equal to 360 degrees. So, we have:

A + B + C + D + E = 360

Substituting the given angle measures, we get:

90 + 90 + (2x-20) + x + (2x-10) = 360

Simplifying the equation, we get:

5x + 150 = 360

Subtracting 150 from both sides, we get:

5x = 210

Dividing both sides by 5, we get:

x = 42

Now we can substitute x back into the expressions for the angles to find their measures:

A = 90 degrees

B = 90 degrees

C = 2x-20 = 2(42)-20

= 64 degrees

D = x = 42 degrees

E = 2x-10 = 2(42)-10

= 74 degrees

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Answer: x + 8 = 30

Step-by-step explanation:

More than indicates addition (in this context) and is indicates equals.

Hope this helps!

Answer:

Step-by-step explanation:

x>8 or 30 i think

Answer:

Proofs attached to answer

Step-by-step explanation:

Proofs attached to answer

Let g be the function defined by g(x)=∫x−1(−12+cos(t3+2t))ⅆt for 0.471. The answer is option (a).

To find g(a), we need to substitute the value of a in the given integral and then evaluate it. So,

g(a) = ∫a⁻¹ (-12 + cos(t³+2t)) dt

Using substitution u = t³ + 2t, du/dt = 3t² + 2, dt = du/(3t²+2)

When t = a, u = a³ + 2a, and when t = -1, u = -1

So, we have

g(a) = ∫(a³+2a)⁻¹ (-12 + cos(u)) / (3t²+2) du

Using the properties of definite integrals, we can write this as

g(a) = [-1/3 ∫(a³+2a)⁻¹ cos(u) du] - 12∫(a³+2a)⁻¹ [1/(3t²+2)] du

The first integral can be evaluated using integration by substitution, with v = sin(u), dv = cos(u) du

g(a) = [-1/3 sin((a³+2a))] - [4√3/9 arctan((√3/3)(a²+2)/√2)] - [8/9 ln((√3a²+2a+2)/(√2))] - [4/9 ln(2+√6)]

Using the given value of a, we can substitute and evaluate g(a).

g(0.5) = [-1/3 sin((0.5³+2(0.5)))] - [4√3/9 arctan((√3/3)(0.5²+2)/√2)] - [8/9 ln((√3(0.5)²+2(0.5)+2)/(√2))] - [4/9 ln(2+√6)]

≈ 0.471

Therefore, the value of g(a) at a=0.5 is approximately 0.471. The answer is option (a).

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The value of f(5) is 37. This can be answered by the use the fundamental theorem of calculus.

This states that the integral of the derivative of a function equals the difference between the value of the function at the upper limit and the lower limit of the integral.

In this case,

the integral from 3 to 5 of f'(x)dx is equal to 24.

Therefore, f(5) = f(3) + 24. As f(3) is 13, f(5) is 37.

To know more about theorem of calculus, refer here:

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