between smokers and nonsmokers, random samples from each group were selected. the sample proportion of people with

Answer:

  C. RP = 4.5

Step-by-step explanation:

You want to know what condition is sufficient to show ∆ABC ~ ∆QPR, given  three sides and 2 angles in ∆ABC, and 2 sides in ∆QPR.

Similarity can be shown if all three sides are proportional, or if two angles are congruent.

The offered answer choices only list one angle, so none of those will work. The answer choice that makes the third side of ∆QPR be in the same proportion as the corresponding side of ∆ABC is the condition of interest.

  C. RP = 4.5

__

Additional comment

The side ratios in the two triangles are ...

  AB : BC : CA = 10 : 9 : 6

  QP : PR : RQ = 5 : PR : 3

For these ratios to be the same, PR must be half of BC, just as the other segments in ∆QPR are half their counterparts in ∆ABC.

The fraction Bret needs to turn the logo is 1 / 2.

Bret needs to put a logo on a business card.  He prints the logo the wrong way up by mistake. Bret needs to turn the logo the correct way up.

Therefore, the fraction he needs to turn the logo can be represented as follows:

For a complete turning, it will be 360 degrees. according to the diagram he has already turned the logo 180 degrees.

Therefore, for Bret to turn the logo correctly, he needs to turn it 180 degrees.

Hence, the fraction he needs to turn the logo is as follows:

180 / 360 = 18  /36 = 1 / 2

Hence, Bret needs to turn the logo half way(1 / 2)up.

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Answer: 1704 square inches

Step-by-step explanation:

The surface area of a square pyramid is given by the formula:

Surface Area = base area + 4 × (1/2 × slant height × base length)

The base area of the pyramid is:

base area = length × width = 24 in × 24 in = 576 in²

The slant height of the pyramid can be found using the Pythagorean theorem:

slant height = sqrt(20² + 12²) = 236/5 in

Therefore, the surface area of the whole pyramid is:

Surface Area = 576 in² + 4 × (1/2 × 236/5 in × 24 in) = 1152 in² + 2256 in² = 3408 in²

Half of this area is black, so the area that is yellow is:

Yellow Area = 1/2 × 3408 in² = 1704 in²

Therefore, the surface area of the toy that is yellow is 1704 square inches.

As per have proved that the metric space (x,d) has the property that every closed and bounded subset of x is compact.

First, let us define a sequence of positive real numbers εₙ as follows: εₙ = 1/2ⁿ. Since εₙ → 0 as n → ∞, we have that for each n ∈ ℕ, there exists an Nₙ ∈ ℕ such that if n, a > Nₙ, then d(xₙ, xₐ) < εₙ.

To see that A is bounded, note that for any two elements xₙ, x_{Nₙ} in A, we have d(xₙ, x_{Nₙ}) ≤ d(xₙ, x_{Nₙ-1}) + d(x_{Nₙ-1}, x_{Nₙ}) ≤ εₙ + ε_{Nₙ} ≤ 2εₙ. Therefore, the diameter of A is bounded by 2εₙ for any n ∈ ℕ. Since εₙ → 0 as n → ∞, we have that A is bounded.

To see that A is closed, let us consider a point x in the closure of A. If x is not in A, then there exists a positive real number δ such that the open ball B(x, δ) does not intersect A.

Let N be such that 1/2ⁿ < δ/2, and let y be an element in A. If y is equal to x_{Nₙ} for some n ∈ ℕ, then d(x, y) ≥ δ/2 > ε_{Nₙ}. Otherwise, y is equal to xₐ for some a ∈ ℕ.

If a > N, then d(x, y) > εₙ > δ/2. If a ≤ N, then d(y, xₙ) ≤ d(y, xₐ) + d(xₐ, xₙ) ≤ εₐ + εₙ < δ/2 + δ/2 = δ, which contradicts the assumption that B(x, δ) does not intersect A. Therefore, we must have x ∈ A, and so A is closed.

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Answer:

d

Step-by-step explanation:

the theoretical probability that the coin lands on the same side every time is 0.0625.

What is Probability?

The area of mathematics known as probability is concerned with how random events turn out. The definition of probability is chance or potential for a result. It clarifies the likelihood of a specific occurrence. We regularly use words like - 'It will probably rain today, 'he will probably pass the test', 'there is very less possibility of receiving a storm tonight', and 'most certainly the price of onion will go high again. In essence, probability is the forecasting of an outcome that is either based on the analysis of past data or the variety and quantity of alternative outcomes.

The theoretical probability of getting the same side every time in five coin tosses is:

Since the coin has two sides, there are 2^5 = 32 possible outcomes in total. Out of these outcomes, there are only two ways to get the same side every time (either all heads or all tails). Therefore, the probability of getting the same side every time is:

P(E) = favorable outcomes / total outcomes

     =  2/32

     = 1/16 = 0.0625 or 6.25%

So, the theoretical probability of getting the same side every time in five coin tosses is 0.0625 or 6.25%.

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Step-by-step explanation:

1. 2/10

2.3/15

3.4/20

4. 5/25

5.6/30

6.7/35

The volume of the square pyramid is 9.12 m.

A square pyramid is a pyramid, in geοmetry, that has a square base and fοur lateral faces. A Pyramid is a pοlyhedrοn that has a base and 3 οr greater triangular faces that meet at a pοint abοve the base (the apex). A square pyramid is a three-dimensiοnal shape that has a tοtal οf five faces, hence called a pentahedrοn.  A mοst famοus example οf such a pyramid in real life is the Great Pyramid οf Giza.

[tex]$ \rm V = \frac{1}{3} a^2H[/tex]

Where v is the volume

a is the base side and

H is the height

Here,

a = 12 cm

H = 19 cm

Then,

[tex]$ \rm V = \frac{1}{3} 12 \times 12 \times 19[/tex]

[tex]$ \rm V = 4 \times 12 \times 19[/tex]

V = 912 cm or 9.12  m

Thus, The volume of the square pyramid is 9.12 m.

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Answer:

A. Using the lens equation, 1/p + 1/q = 1/f, and substituting f = 5 cm and q = 7 cm, we can solve for p:

1/p + 1/7 = 1/5

Multiplying both sides by 35p, we get:

35 + 5p = 7p

Simplifying and rearranging, we get:

2p = 35

Therefore, the distance from the lens to the object, p, is:

p = 35/2 cm

B. Solving the lens equation, 1/p + 1/q = 1/f, for q, we get:

1/q = 1/f - 1/p

Substituting f = 5 cm, we get:

1/q = 1/5 - 1/p

Multiplying both sides by 5qp, we get:

5p = qp - 5q

Simplifying and rearranging, we get:

q = 5p / (p - 5)

Therefore, the expression that gives q as a function of p is:

q = 5p / (p - 5)

C. Here is a sketch of the graph of q(p):

The graph is a hyperbola with vertical asymptote at p = 5 and horizontal asymptote at q = 5. The image distance q is positive for object distances p greater than 5, which corresponds to a real image. The image distance q is negative for object distances p less than 5, which corresponds to a virtual image.

D. Taking the limit of q as p approaches infinity, we get:

lim q(p) = 5

This represents the horizontal asymptote of the graph. As the object distance becomes very large, the image distance approaches the focal length of the lens, which is 5 cm.

Taking the limit of q as p approaches 5 from the positive side, we get:

lim q(p) = -infinity

This represents the vertical asymptote of the graph. As the object distance approaches the focal length of the lens, the image distance becomes infinitely large, indicating that the lens is no longer able to form a real image.

In order for the lens to form a real image, the object distance p must be greater than the focal length f. When the object distance is less than the focal length, the lens forms a virtual image.

The solutions to the quadratic function 9x² - 15x - 6 = 0 are given as follows:

x = -1/3 and x = 2.

The quadratic equation for this problem is defined as follows:

9x² - 15x - 6 = 0.

The coefficients of the function are given as follows:

a = 9, b = -15 and c = -6.

The discriminant of the function is obtained as follows:

D = b² - 4ac

D = (-15)² - 4 x 9 x (-6)

D = 441.

Then the solutions to the quadratic function are obtained as follows:

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Answer:

D) Kite

Step-by-step explanation:

A kite is a 4 sides figure that a pair of adjacent sides have the same length.

Helping in the name of Jesus.

Answer:

D) kite.

Step-by-step explanation:

Refer to the picture, when you combine the quadrilateral, it's more precise to make a kite.

The correct explanation of the confidence interval for the regression slope is: "The average Tip is expected to increase by between $0.080 and $0.174 for every dollar increase in Sale amount." The correct option is A).

The question refers to a scatterplot and regression analysis of data on tips and sale amounts.

In particular, the model's multiple R, R square, and adjusted R square indicate the proportion of variability in the dependent variable (tips) that can be explained by the independent variable (sale amounts). The standard error reflects the precision of the model's predictions.

The confidence interval for the slope provides a range of values within which we can be confident that the true population slope lies. In this case, we can say with 95% confidence that the average tip is expected to increase by between $0.080 and $0.174 for every dollar increase in sale amount. The correct answer is A).

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_____The given question is incomplete, the compelete question is given below:

SLRI SONNY'S SCATTERPLOT TIP ($) 0 100 200 300 400 900 1000 1100 1200 500 600 700 800 SALE AMOUNT (S) © 2020 Radha Bese Florida State University Department of States SUMMARY OUTPUT SONNY'S Regression Statistics Multiple R 0.58064672 R Square 0.337150613 Adjusted R Square 0.325722175 Standard Error 28.73607083 Observations 60 ANOVA MS df 1 F Significance F 29.50102381 1.15405E-06 Regression Residual Total SS 24360.81754 47894.18246 72255 24360.81754 825.7617666 Intercept Sale Amount Coefficients Standard Error Stat 56.95659334 12.27319518 4.640730674 0.126993685 0.023381027 5.431484494 P-value 2.02919E-05 1.15405E-06 Lower 95% 32.38912416 0.080191475 Upper 95% 81.52406252 0.173795894 Which of the following is a correct explanation of the confidence interval for the regression slope? Choose the BEST option.

A The average Tip is expected to increase by between $0.080 and $0.174 for every dollar increase in Sale amount.

B The average Sale amount is expected to increase by between $32.389 and $81.525 for every dollar increase in Tip.

C The average Tip is expected to increase by between $32.389 and $81.525 for every dollar increase in Sale amount.

D The average Sale amount is expected to decrease by between $0.080 and $0.174 for every dollar increase in Tip.

E The Tip is expected to increase by between $32.389 and $81.525 for every dollar increase in Sale amount.

F The Tip is expected to decrease by between $0.080 and $0.174 for every dollar increase in Sale amount

Answer:

95.07[tex]m^{2}[/tex]

Step-by-step explanation:

You need to work out the area of a cicle, since the goat can only move 5.5m round

Area of a cicle = π r²

π = 22/7

r=5.5m

[tex]\frac{22}{7}[/tex] x [tex]5.5^{2}[/tex] = 95.07142857 = 95.07[tex]m^{2}[/tex] (to 2d.p)

Answer:

94.79 square meters.

Step-by-step explanation:

Assuming the goat is tied to the peg in the ground, the area that the goat can graze on can be represented by a circle with radius equal to the length of the rope.

Given that the length of the rope is 5.5 meters, the radius of the circle can be calculated as:

r = length of rope = 5.5 meters

The area of a circle is given by the formula:

A = πr²

where π (pi) is approximately equal to 22/7 or 3.14.

So, substituting the value of r, we get:

A = π × (5.5)²

= 3.14 × (5.5)²

= 3.14 × 30.25

= 94.79 square meters (approx.)

Therefore, the area of grass that the goat can eat is approximately 94.79 square meters.

Answer:

Step-by-step explanation:

|x-a|=b

x-a=±b

x=a±b

|x-2|=6

x-2=±6

either x-2=6

x=2+6=8

or

x-2=-6

x=2-6

x=-4

d

The answer on Probability to draw probability tree and calculate the final solution the answers are (a) the tree is given below , (b) i) P(R1 ∩ W2) means the probability of drawing a red marble on the first draw and a white marble on the second draw , ii) The solution is P(R1) * P(W2|R1) = (4/9) * (5/8) = 5/18. (C) 17/36. (d) the tree is given below.

In probability theory,  experiment is  process or  situation that produces  set of possible outcomes or events, each with  associated probability of occurrence.

In  experiment, the set of all possible outcomes is called  sample space, denoted by  symbol Ω. The outcomes in  sample space can be either discrete, meaning they are countable and separate, or continuous, meaning they are uncountable and form range or interval.

a) Probability tree:

       P(W1) = 5/9

         /              \

      P(W2|W1)         P(R2|W1) = 4/9

      /      \          /      \

  P(W3|W1W2)  P(R3|W1W2) P(W3|R1W2)  P(R3|R1W2)

  /   \       /   \       /    \        /    \

P(W4|W1W2W3) P(R4|W1W2R3) P(W4|R1W2W3) P(R4|R1W2R3)

b)

i) P(R1 ∩ W2) means the probability of drawing a red marble on the first draw and a white marble on the second draw.

ii) The solution is P(R1) * P(W2|R1) = (4/9) * (5/8) = 5/18.

c)

i) There are two ways to get at least one red ball: drawing a red ball on the first draw or drawing a white ball on the first draw and a red ball on the second draw.

ii) Angela is not correct. To calculate the probability of drawing at least one red ball, we need to add the probabilities of the two ways to get at least one red ball:

P(at least one red ball) = P(R1) + P(W1) * P(R2|W1) = (4/9) + (5/9) * (4/8) = 17/36.

d) If the first ball is replaced before the second draw, then the probability tree would have the same probabilities for each draw. The new probability tree would be:

         P(W1) = 5/9

         /              \

      P(W2)             P(R2) = 4/9

      /      \          /      \

  P(W3)      P(R3)     P(W3)     P(R3)

  /   \       /   \    /    \     /    \

P(W4) P(R4)  P(W4) P(R4) P(W4)  P(R4)

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It mean X will be a better choice for all inputs except small inputs when we say that an algorithm X is asymptotically more efficient than Y. So the option B is correct.

Asymptotic analysis takes algorithm growth in terms of input size into account. If an algorithm X runs faster than Y for all input sizes n greater than or equal to n₀, then X is said to be asymptotically better than Y.

When we claim that a method X is asymptotically more efficient than Y, we indicate that X will be a better option for all inputs except small inputs. The better choice would depend on the specific problem and algorithm.

Generally, if X is asymptotically more efficient than Y, then X should be the preferred choice for larger inputs, while Y may be better suited for small inputs. So the option B is correct.

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The complete question is:

What does it mean when we say that an algorithm X is asymptotically more efficient than Y?

a) X will be a better choice for all inputs

b) X will be a better choice for all inputs except small inputs

c) X will be a better choice for all inputs except large inputs

d) Y will be a better choice for small inputs

Answer:

To find the Taylor series of a function f(x) about a point a, we can use the following formula:

f(x) = f(a) + f'(a)(x-a)/1! + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...

where f'(a), f''(a), f'''(a), ... denote the first, second, third, ... derivatives of f evaluated at a.

In this case, we have:

f(x) = 1 + 3e^(x+1)

To find the Taylor series about x=0, we need to evaluate the function and its derivatives at x=0.

f(0) = 1 + 3e^(0+1) = 1 + 3e

f'(x) = 3e^(x+1)

f'(0) = 3e^(0+1) = 3e

f''(x) = 3e^(x+1)

f''(0) = 3e^(0+1) = 3e

f'''(x) = 3e^(x+1)

f'''(0) = 3e^(0+1) = 3e

and so on.

Substituting these values into the formula for the Taylor series, we get:

f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + ...

= (1 + 3e) + 3ex + 3ex^2/2! + 3ex^3/3! + ...

= 1 + (3 + 3e)x + (3/2)e x^2 + (1/2)e x^3 + ...

Therefore, the Taylor series for 1+3e^x+1 about x=0 is:

1 + (3 + 3e)x + (3/2)e x^2 + (1/2)e x^3 + ...

Answer:

All three answers are correct, but they represent different algebraic expressions that are equivalent to x + 8.

Step-by-step explanation:

i) 1/4(x + 8) is equivalent to x/4 + 2. This expression distributes the 1/4 coefficient to both terms inside the parentheses.

ii) x + 2 is equivalent to 1x + 2. This expression simplifies x + 8 by subtracting 6 from the constant term.

iii) 1/4x + 2 is equivalent to 2 + 1/4x. This expression rearranges the terms in x + 8 to separate x from the constant term.

Therefore, the students who provided any of these answers demonstrated a correct understanding of algebraic expressions.

Answer:

Step-by-step explanation:

u got this

The equation of the line in slope-intercept form is y = -2x + 8 and the equation of the line in standard form is 2x + y = 8.

What is slope-intercept form and standard form of a straight line ?

Slope-intercept form and standard form are two common ways to represent the equation of a straight line in two dimensions.

Slope-intercept form of a line is y = mx + b, where m is the slope of the line, and b is the y-intercept (where the line intersects the y-axis). This form makes it easy to determine the slope and y-intercept of the line from the equation, and to graph the line using those values.

Standard form of a line is Ax + By = C, where A, B, and C are constants, and A and B are not both zero. This form makes it easy to compare the coefficients of x and y, and to determine the x-intercept and y-intercept of the line.

Find the equation of a line passing through a given point and parallel to the given line in slope intercept and standard form :

Given point is (2,4) and line is  6x+3y=3, we first need to determine the slope of the given line. To do this, we can rearrange the equation into slope-intercept form (y = mx + b), where m is the slope:

3y = -6x + 3

y = -2x + 1

The slope of this line is -2.

Since the line we want to find is parallel to this line, it will also have a slope of -2.

We can use the point-slope form of the equation of a line to find the equation of the line passing through (2,4) with slope -2:

y - 4 = -2(x - 2)

Expanding and simplifying, we get:

y - 4 = -2x + 4

y = -2x + 8

This is the equation of the line in slope-intercept form.

To write it in standard form, we can rearrange the equation into the form Ax + By = C:

2x + y = 8

This is the equation of the line in standard form.

Graph of the line -

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Answer:

g h hh h

Step-by-step explanation:

By answering the presented question, we may conclude that Since n expression must be a whole number, the closest integer to 3.32193 is 3. Therefore, there are 3 terms in the series.

An expression in mathematics is a collection of representations, numbers, and conglomerates that mimic a statistical correlation or regularity. A real number, a mutable, or a mix of the two can be used as an expression. Mathematical operators include addition, subtraction, fast spread, division, and exponentiation. Expressions are often used in arithmetic, mathematic, and form. They are used in the representation of mathematical formulas, the solution of equations, and the simplification of mathematical relationships.

We can use the formula for the sum of a finite geometric series to solve this problem:

[tex]S_n = a1(1 - r^n)/(1 - r)\\S_n = 8,190, r = 1/2, and a1 = 4,096. \\8,190 = 4,096(1 - (1/2)^n)/(1 - 1/2)\\16,380 = 4,096(1 - (1/2)^n)\\4 = 1 - (1/2)^n\\5/4 = (1/2)^n\\n = log(5/4) / log(1/2)\\n= 3.32193\\[/tex]

Since n must be a whole number, the closest integer to 3.32193 is 3. Therefore, there are 3 terms in the series.

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Answer:

Genevieve should use a three digit code instead of a two digit code for her locker combination because a three digit code has more possible combinations than a two digit code, which makes it more secure.

With a two digit code, there are only 100 possible combinations (10 possible digits for the first number and 10 possible digits for the second number). On the other hand, with a three digit code, there are 1,000 possible combinations (10 possible digits for each of the three numbers). This means that it would take a potential thief 10 times longer to guess the correct combination for a three digit code compared to a two digit code, assuming they are using a brute force method of trying all possible combinations.

Therefore, using a three digit code increases the security of the locker and decreases the likelihood of someone gaining unauthorized access.

Answer:

more security.

Step-by-step explanation:

If you have a 2 digit code people will find it. but with 3, it makes it harder to find so people can't open her locker.

So, in this case, the length of each of the equal sides is: 17 / [tex]\sqrt{2}[/tex] ≈ 12.02

An isosceles triangle is a triangle in which two sides have the same length. This means that two of the three angles in the triangle are also equal in measure. The side that is not congruent to the other two sides is called the base of the triangle, and the angles opposite the congruent sides are called the base angles. In an isosceles triangle, the base angles are always equal in measure.

Isosceles triangles are a common shape in geometry and can be found in many real-world applications. They have special properties that make them useful in various fields of mathematics, such as trigonometry and geometry. For example, the base angles of an isosceles triangle are always equal, so if you know the measure of one of the base angles, you can determine the measure of the other base angle using the fact that the sum of the angles in a triangle is 180 degrees.

given by the question.

In an isosceles triangle with two equal angles of 45 degrees and a third angle of 90 degrees, the two equal sides are opposite the 45 degree angles, and the third side is opposite the 90 degree angle. Let's call the length of the missing side "x".

Using the Pythagorean theorem, we know that in a right triangle with legs of equal length (which is the case for the two 45-degree angles), the length of the hypotenuse is. [tex]\sqrt{2}[/tex] times the length of the legs.

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Hey!
A: 50% Of people = 150 people prefer oranges.

B: 10% Of people = 15 people prefer pineapple.

C: 15% Of people = 20 people prefer blueberries.

D: 5% Of people = 5 people prefer apples.

E: 20% Of people = 22 people prefer strawberries

In the graph, Student B and C are both 10 years old and student C has a shoe size of 5. The coordinates of D are (12,6)

In graph theory, a graph is a framework that consists of a collection of objects, some of which are paired together to form "related" objects. The objects are represented by mathematical abstractions known as vertices (also known as nodes or points), and each set of connected vertices is known as an edge (also called link or line). A graph is typically shown diagrammatically as a collection of dots or circles representing the centres and lines or curves representing the edges.

Both directed and undirected lines are possible. For instance, if the edges between two individuals are handshakes, then the graph is undirected because any individual A can only shake hands with an individual B if B also holds hands with A. The graph is directed, however, if an edge from person A to person B indicates that A owes money to B because borrowing money is not always returned.

In the given graph,

Student B and C are both 10 years old and student C has a shoe size of 5.

The coordinates of D are (12,6)

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Answer:

Math Quotient Verification

G For each ordered pair, determine whether it is a solution to the system of equations. 9x+2y=-5 2x-3y=-8 (x, y) (1, -7) (0, -4) (5,6) (-1,2) Is it a solution? Yes No X 5

To check if an ordered pair is a solution to a system of equations, we substitute the values of x and y into both equations and see if both equations are satisfied.

Let's check each ordered pair one by one:

(1, -7):

9x + 2y = -5 becomes 9(1) + 2(-7) = -5, which is false.

2x - 3y = -8 becomes 2(1) - 3(-7) = -8, which is true.

Therefore, (1, -7) is not a solution to the system of equations.

(0, -4):

9x + 2y = -5 becomes 9(0) + 2(-4) = -8, which is false.

2x - 3y = -8 becomes 2(0) - 3(-4) = 12, which is false.

Therefore, (0, -4) is not a solution to the system of equations.

(5, 6):

9x + 2y = -5 becomes 9(5) + 2(6) = 41, which is false.

2x - 3y = -8 becomes 2(5) - 3(6) = -8, which is true.

Therefore, (5, 6) is not a solution to the system of equations.

(-1, 2):

9x + 2y = -5 becomes 9(-1) + 2(2) = -11, which is false.

2x - 3y = -8 becomes 2(-1) - 3(2) = -8, which is true.

Therefore, (-1, 2) is not a solution to the system of equations.

Therefore, the answer is "No" for all the ordered pairs given in the problem.

(please mark my answer as brainliest)

(a) There are 100,000 different ways for Fiona to select her lunches for the week.

(b) There are 200,000 different ways for Fiona to select her lunches and drinks for the week.

(a) Since Fiona selects one lunch special each day, there are 10 options for Monday, 10 options for Tuesday, and so on, for a total of 10 options for each of the 5 weekdays. Therefore, the total number of ways for Fiona to select her lunches for the week is:

10 × 10 × 10 × 10 × 10 = 10^5 = 100,000

(b) In addition to the 10 lunch specials, Fiona now has 2 options for her drink, either water or tea. So for each of the 100,000 possible lunch selections from part (a), there are 2 possible drink options. Therefore, the total number of ways for Fiona to select her lunches and drinks for the week is:

100,000 × 2 = 200,000

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Answer: To solve this problem, we can use the formula:

Total = Stocks + Bonds - Both

where "Total" represents the total number of people who invested in either stocks or bonds.

Plugging in the given values, we get:

Total = 700 + 400 - 300

Total = 800

Therefore, 800 people invested in stocks or bonds.

Step-by-step explanation:

The integral of the function expressed as sum of partial frictions is -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C.

First, factor out 2x from the denominator to obtain:

∫[(x + 3)/(2x³ - 8x)] dx = ∫[(x + 3)/(2x)(x² - 4)] dx

Next, we use partial fractions to express the integrand as a sum of simpler fractions. To do this, we need to factor the denominator of the integrand:

2x(x² - 4) = 2x(x + 2)(x - 2)

Therefore, we can write:

(x + 3)/(2x)(x² - 4) = A/(2x) + B/(x + 2) + C/(x - 2)

Multiplying both sides by the denominator, we get:

x + 3 = A(x + 2)(x - 2) + B(2x)(x - 2) + C(2x)(x + 2)

Now, we need to find the values of A, B, and C. We can do this by equating coefficients of like terms:

x = A(x² - 4) + B(2x² - 4x) + C(2x² + 4x)

x = (A + 2B + 2C)x² + (-4A - 4B + 4C)x - 4A

Equating coefficients of x², x, and the constant term, respectively, we get:

A + 2B + 2C = 0

-4A - 4B + 4C = 1

-4A = 3

Solving for A, B, and C, we find:

A = -3/4

B = 7/16

C = -1/16

Therefore, the partial fraction decomposition is:

(x + 3)/(2x)(x² - 4) = -3/(4(2x)) + 7/(16(x + 2)) - 1/(16(x - 2))

The integral becomes:

∫[(x + 3)/(2x³ - 8x)] dx = ∫[-3/(8x) + 7/(8(x + 2)) - 1/(8(x - 2))] dx

Integrating each term separately gives:

∫[-3/(8x) + 7/(8(x + 2)) - 1/(8(x - 2))] dx

= -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C

where;

Therefore, the final answer is:

∫[(x + 3)/(2x³ - 8x)] dx = -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C

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Answer: We can first find the dot product of vec r with hat j and hat k:

vec r * hat j = (3 hat i - 2 hat j + 6 hat k) * (- hat j) = -2

vec r * hat k = (3 hat i - 2 hat j + 6 hat k) * hat k = 6

Substituting these values into the expression given, we get:

(vec r * hat j).(vec r * hat k) - 12 = (-2) * (6) - 12 = -24

Therefore, the value of (vec r * hat j).(vec r * hat k) - 12 is -24.

Step-by-step explanation: