The average rate of change of demand when the number of items demanded increases from 2 thousand units to 4 thousand units is 401.13 and if it increase from 1 thousand units to 5 thousand units is 201.13
What is Algebraic expression ?
Algebraic expression can be defined as the combination of variables and constants.
Given,
The expression, p(x) = 0.2 x^2 + 1.13x+5.2
where x is given in units of a thousand .
The average rate of change of demand when the number of items demanded increases from 2 thousand units to 4 thousand units.
p(x) = 0.2x^2 + 1.13x+5.2
p(2000 ) = 0.2 (2000*2000) + 1.13 (2000) + 5.2
= 800000 + 2260 + 5.2
= 802265.2
p(4000 ) = 0.2 (4000*4000 ) + 1.13 (4000) + 5.2
= 1600000 + 4520 + 5.2
= 1604525.2
p(2000) - p (4000) / 2000-4000 = 802265.2 - 1604525.2 / -2000
= -8,02,260 / -2000
= 401.13
p (1000) = 0.2 (1000*1000) + 1.13 (1000) + 5.2
= 200000 + 1130 + 5.2
= 201135.2
p (5000 ) = 0.2 (5000*5000) + 1.13(5000) + 5.2
= 1000000 + 5650 + 5.2
= 1005655.2
p (1000) - p (5000) / 1000 - 5000 = 201135.2 - 1005655.2 / -4000
= 804520 / 4000
= 201.13
Hence, The average rate of change of demand when the number of items demanded increases from 2 thousand units to 4 thousand units is 401.13 and if it increase from 1 thousand units to 5 thousand units is 201.13.
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A healthy dog will drink between 50 and 60 milliliters of
water per kilogram of body weight each day.
Approximately how many fluid ounces of water will a
35-pound dog drink in a day? Express your answer as a
range of values, rounded to the nearest whole fluid
ounce.
Pls answer quickly!
The range of values for how many fluid ounces of water will a 35 pound dog will drink in a day is given as follows:
27 oz < x < 32 oz.
How to obtain the range of values?The range of values is obtained applying the proportions in the context of this problem.
The conversion of milliliters to fluid ounces is given as follows:
1 ml = 0.0338 oz.
Hence the amounts of oz per kilogram are given as follows:
50 x 0.0338 = 1.69 oz.60 x 0.0338 = 2.03 oz.The conversion of pounds to kg is given as follows:
1 kg = 2.20462 pounds.
Hence the weight in kg of the dog is given as follows:
35/2.20462 = 15.87 kg.
The bounds of the interval are given as follows:
15.87 x 1.69 = 27 oz.15.87 x 2.03 = 32 oz.More can be learned about proportions at https://brainly.com/question/24372153
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4) Find the 12th term in the sequence -2, 6, -18, 54, ...
A. f(n) = 2,048
OB. f(n)=512
C. f(n) = 354,294
D. f(n) = 781,250
The 12th term in the sequence is (c) f(n) = 354294
How to determine the 12th term in the sequence?From the question, we have the following parameters that can be used in our computation:
The definition of the function is given as
-2, 6, -18, 54, ...
The above definitions imply that we simply multiply -3 to the previous term to get the current term
Using the above as a guide,
so, we have the following representation
f(n) = -2(-3)^n-1
Where
-2 is the first term and -3 is the common ratio
For the 12th term, we have
f(n) = -2(-3)^11
Evaluate
f(n) = 354294
Hence, the 12th term is (c) f(n) = 354294
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in how many ways can i distribute $6$ identical gummy bears and $6$ identical chocolates to $4$ students, if each student must receive exactly $3$ pieces of candy?
If we divide the candy according to numbers rather than type, I can give 4 pupils 6 identical chocolates and 6 identical gummy bears, ensuring that each student will get exactly 3 pieces of candy.
We have to find in how many ways I can distribute 6 identical gummy bears and 6 identical chocolates to 4 students, if each student must receive exactly 3 pieces of candy.
Gummy Bear = 6
Chocolates = 6
Total numbers of candy = Gummy bear + Chocolates
Total numbers of candy = 6 + 6
Total numbers of candy = 12
We have to distribute the tablet in between 4 students. So each student must receive exactly 3 pieces of candy if we distribute the candy by the numbers not their type.
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Solution set for the inequality for 4 - 5G greater than 39
Answer:
G < -7
Step-by-step explanation:
We can create an equation first with what the problem provides:
4 - 5G > 39
Now, since the question asks for the solution, we can solve for G:
-5G > 39-4
-5G > 35
-G > 7
Since G is still a negative, we need to switch the inequality around:
(-1)(-G) > (7)(-1) (To cancel out the negative)
G < -7
Find the amount in the account for the given principal, interest rate, time, and compounding period. P = $3,650, r= 5.5%, t= 15 years; compounded monthly
Answer:
$14390.75
Step-by-step explanation:
p=3650
i = 5.5% is this annual rate or monthly rate? assuming annual rate
t=15 years compounding monthly = 15*12 = 300 periods
(1+5.5%/12)^300=3.94267155
multiply the principal 3650 *3.94267155 = $14390.75
what is the lowest value of the rangee of the function shown on the graph? -♾️, -2, 0, 3
The correct option is -2. The range of function is found to be minimum is -2.
Explain the term minima of the function?The highest and smallest values of a function, respectively, can either be found inside a specific range or across the entire domain. They are sometimes referred to as the function's extrema. The plural forms of a function's maximum and minimum are called maxima and minima, respectively. Let's first examine the function's local maximum and minimal values before delving further into maxima and minima.A function's global minimum is its lowest value across the entire function's range, whereas a localized minimum is its lowest value within a specific local area.
From the given graph of the function:
To get the lowest point of the range of the function, check the value where graph is turning.
This, shows the minima of the function which is -2 at x = 3.
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The complete question is attached-
A daycare charges a base fee of 333 dollars plus 0. 500. 500, point, 50 dollars per minute for late (after closing time) pick-ups. Albin had to pay 10. 5010. 5010, point, 50 dollars for a late pick-up. Albin uses the equation, 10. 50=0. 50a+310. 50=0. 50a+310, point, 50, equals, 0, point, 50, a, plus, 3 to represent the situation
It means Albin was 15 minutes late to a pickup.
a represents the Number of Minutes Late, (a=15 in this case)
Now, According to the question:
A daycare charges a base fee of 333 dollars plus 0. 500. 500, point, 50 dollars per minute for late (after closing time) pick-ups.
Albin had to pay 10. 5010. 5010, point, 50 dollars for a late pick-up.
Albin uses the equation,
10.50 = 0.50a+3
0.50a = 10.50-3
0.50a = 7.50
a = 7.50/0.5
a= 15
Hence, It means Albin was 15 minutes late to a pickup.
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Garrick is building a swimming pool. He wants the length to be at least 20 feet, but the perimeter cannot exceed 100 feet. What are the 2 resulting equations out of this question?
The 2 resulting inequality equations out of this question are;
L ≥ 20 ft
(L + W) ≤ 50
How to express inequality equations?The formula for the perimeter of a rectangle is expressed as;
Perimeter = 2(Length + Width)
We are told that the length has to be at least 20 feet, which means it must not be lesser than 20 feet. Thus;
L ≥ 20 ft
Now, we are told that the perimeter cannot exceed 100 feet. Thus;
2(L + W) ≤ 100
Solving that by dividing both sides by 2 gives;
(L + W) ≤ 50
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Look at the picture please (IM GIVING OUT 50 POINTS TO WHOEVER ANSWER THIS CORRECTLY AND I WILL MARK BARINLIEST)
3. Use your graph to answer the following questions
3a. If King sets the price at $2.50 per doll how many disappointing customers will each store have during the week?
3b. If King sets the price at $3.80 per doll how many unsold dolls will remain at each store at the end of the week?
3c. According to this graph, if the company could give the dolls away, how many would each door need per week?
3d. According to this graph, what price would make the doll supply so tight that the average number available to each store would be zero?
3e. Estimate the price where supply and demand will be in equilibrium
4. Complete the following using equations.
A. Use two of the points, giving to find the equation for supply (S) as a function of price (P)
B. Use two of the points given to find the equation for demand (D) as a function of price (P)
C solve the system of supply and demand equations to find the price in exact equilibrium. How does this price compare with your answer in question 4e above?
a. The number of disappointed customers is 157. b. Unsold dolls = 135. c. The dolls needed per week is 660. d. The price of $0.63 will make the supply of the doll tight. e. The price at which the demand and supply will be at equilibrium is $3.20.
What are functions?A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.
a. The supply and the demand associated from the graphs are:
Supply: 95(2.50) - 60 = 178
Demand: -130(2.50) + 660 = 335
Subtracting the Demand and supply:
335 - 178 = 157
The number of disappointed customers is 157.
b. For unsold dolls set the equation for supply and demand:
Supply: 95(3.80) - 60 = 301
Demand: -130(3.80) + 660 = 166
Subtract the value of demand and supply:
301 - 166 = 135
Unsold dolls = 135
c. To give the dolls away, the cost of the doll is $0.
The demand equation thus becomes:
y = -130(0) + 660
y = 660
The dolls needed per week is 660.
d. Let the supply be 0.
y = 95x - 60
0 = 95x -60
x = $0.63
The price of $0.63 will make the supply of the doll tight.
e. The supply an demand will be in equilibrium when Demand = Supply, that is:
95x - 60 = -130x + 660
95x + 130x = 720
225x = 720
x = $3.20
The price at which the demand and supply will be at equilibrium is $3.20.
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Rationalise the denominator 7/2-root 5
On rationalizing the expression 7/(2-√5) 7/(2-√5), the value is obtained as -7 (2 + √5).
What is rationalization?
In order to get rid of the irrational number in the denominator, the procedure of rationalisation is performed in elementary algebra. There are numerous rationalisation strategies that are used on the denominator.
The given expression is - 7/(2-√5).
Multiply the denominator and numerator by the (2 + √5).
= 7 × (2 + √5)/(2-√5) × (2 + √5)
= (14 + 7√5)/(2)² - (√5)²
Simplify it further -
= (14 + 7√5)/4 - 5
= (14 + 7√5)/-1
= -(14 + 7√5)
= -7 (2 + √5)
Therefore, the simplified expression is -7 (2 + √5).
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Part 6
What is the mean absolute deviation for Doctor As data set on contacts? What is the mean absolute deviation for Doctor Bs data set on contacts
Write a sentence comparing these two data sets using their mean absolute deviations.
There is more variability on glasses of doctor B's dataset than the glasses of doctor A's dataset
Now, According to the question:
The doctors' mean absolute deviation on glasses
The dataset of doctor A is given as:
643 634 670 658 636 624 641 655 649 629
The dataset of doctor B is given as:
651 625 622 624 631 621 653 647 646 646
Using a statistical calculator, we have:
Doctor A
Doctor A: Mean Absolute Deviation (MAD) = 11.28
Doctor B: Mean Absolute Deviation (MAD) = 12
Hence, the mean absolute deviation is 11.28 for Doctor A’s and 12 for Doctor B’s data set on glasses
The variation of the two data
In a dataset, the larger the mean absolute deviations value, the larger the variation.
By comparison, 12 is greater than 11.28
Hence, there is more variability on glasses of doctor B's dataset than the glasses of doctor A's dataset
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The given question is incomplete, complete question is:
What is the mean absolute deviation for Doctor A’s data set on glasses? What is the mean absolute deviation for Doctor B’s data set on glasses? Write a sentence comparing the variation of the two data sets using their mean absolute deviations.
Doctor A Doctor B
Corrective Lenses Glasses Contacts Corrective Lenses Glasses Contacts
745 643 102 763 651 112
726 634 92 736 625 111
769 670 99 735 622 113
765 658 107 759 624 135
756 636 120 748 631 117
742 624 118 756 621 135
757 641 116 765 653 112
748 655 93 761 647 114
770 649 121 768 646 122
738 629 109 761 646 115
Use the figure to write a two-column proof.
Given: PSUR is a rectangle.
PQ=TU
Prove: QS=RT
*THE QUESTION IN THE PHOTO IS JUST NUMBER 10
Answer:Given Prove
PSUR is a rectangle QS = RT
PQ = TU
Proof:
Since PSUR is a rectangle, we know that opposite sides are parallel.
Therefore, PQ and TU are parallel.
Also, PSUR is a rectangle, so the angles P and R are right angles.
Therefore, angles Q and S are also right angles.
Since PQ and TU are parallel, and angles Q and S are right angles, we know that QS and RT are corresponding angles.
Therefore, QS=RT
QED.
Step-by-step explanation:
What appears to be the domain of the part of the exponential function graphed?
The domain of the function is x ≤ 1.
x ≤ 1 because your domain cannot have h(x) because that is your function and would be your range but in this case it is your domain and your domain, also known as the x coordinate, would have to be the number on the x axis that your line starts at, which would be your domain.
Your line starts on line 1 on the x axis, which would be your domain.
How do you determine a function's domain?Consider the function y = f(x), which has the independent variable x and the dependent variable y. A value for x is said to be in the domain of a function f if it successfully allows the production of a single value y using another value for x.
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The volume of 10 drops of a liquid is 0.1 fluid ounces.
What is the volume of 100 drops?
Answer:
1 fluid ounce
Step-by-step explanation:
if 10 drops is equal to 0.1 fl and we multiply 10 by 10 to get 100, we have to multiply 0.1 by 10 as well, which is 1 fluid ounce.
Can someone help me with this please?
Answer:
Scalene Triangle
Step-by-step explanation:
Right triangles have 90 degree angles. Isosceles triangles have 2 equal sides. While, equilateral triangles have 3 equal sides.
How many terms are in the following Algebraic Expression? -
(7 x 6) + 8
Answer: 3
Step-by-step explanation: First you have to do 7 times 6 which is 42. Then you add 8 and get 50!
Aplicación de expresions algebraica
En la vida m cotidiana
Answer:
Aprende a leer expresiones algebraicas y comprende conceptos como variable, coeficiente u operador, entre otros.
Step-by-step explanation:
Ejemplo: la mitad de un número “x” si x= 8 x/2= 8/2= +4
) Wanda, a caterer, is investing some money in equipment and employees
to help grow her business. Recently she spent $89 on equipment and hired a
server who makes $16 per hour. Wanda is hoping to make up these expense
at the next job that is scheduled, which pays a base of $90 in addition to $15
per hour that the server works. In theory, this event could pay enough to
cancel out Wanda's expenditures. How much would the job pay? How long
would the job have to be?
In theory, this event could pay enough to cancel out Wanda's expenditures the job pay 16 trips.
How much would the job pay?Establish a variable.
Utilizing the variable, write an equation.
Make the equation work.
Let C be the total cost of each transaction.
T, the quantity of slides down the water slide,
Since the price is the same regardless of how much she rides, C = 89 for the first transaction.
For the second agreement, C = 16plus T.
Set the two costs equal to one another, then use T to determine when they are equal.
89 = 16 + T + T = 18 trips.
Given that the first contract can only have a cost of $90, the price must be $15.
90 = 15 + T + T = 16 trips.
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Jake earns 7.50 per hour working at a local car wash. The function f (x) = 7.50X
The inverse of the function that indicates the amount Jake earns per hour at a local car wash is [tex]\mathbf{y =\dfrac{7.50}{x}}[/tex]
What is a function?In mathematics, a function is represented as a rule that produces a distinct result for each input x. In mathematics, a function is indicated by a map or transformation. Typically, these functions are identified by letters like f, g, and h. etc
A function is an operation or a relation that links each individual element of a non-empty set X, to at least one element of another non-empty set Y. In mathematics, a relation f between two sets, X and Y, is referred to as the function's domain and co-domain and can be expressed for all values of x and y as:
f = {(x,y)| for all x ∈ X, y ∈ Y}
The objective of this question is to write the inverse of the given function.
The function f(x) = 7.50x correlates Jake's pay to the number of hours he puts in the work.
The inverse of the function can be written as:
y = 7.50x
Interchange the variables
x = 7.5y
Divide both sides by 7.5
x/7.5 = 7.5y/7.5
Make y the subject of the formula:
[tex]\mathbf{y =\dfrac{7.50}{x}}[/tex]
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Convert 9y^2+72y=72=6x-x^2 to standard form.
A) (x-3)^2/81 + (y-4)^2/ 9=1
B) (x+3)^2/81+(y+4)^2/9=1
C) (x-3)^2/81 + (y+4)^2/9=1
D) doesnt matter the answer is C
The standard form of the equation 9y^2+72y=72=6x-x^2 will be x-3)^2/81 + (y+4)^2/9=1.
Imagine that you are collaborating on a math assignment with students Andrea and Susan. To represent a word problem, you are looking for an equation. You each work on the problem independently, then compare your solutions. The formula 2x - y = -1 was created by you. The formula y = 2x + 1 was created by Andrea. Susan's final solution was the equation y - 3 = 2(x - 1).The given equation is :
[tex]x^{2} + 9y^{2} -6x +72 y = 72[/tex]
[tex]x (x-6) +9y (y+6) = 72[/tex] +1 +36
x(x-6) / 109 + 9y(y+6) / 109 = 1
Thus, by simplifying above equation
(x-3)^2/81 + (y+4)^2/9=1
In this case, Option C is correct.
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Answer:
C) (x-3)^2/81 + (y+4)^2/9=1
Step-by-step explanation:
Sally has a swimming pool that needs to be drained. Her pool holds
9,774.5 gallons of water and will need to drain completely in 9.1 hours.
What is the change in the water level per hour for Sally's swimming
pool? Round your answer to the nearest hundredth.
The required change in the water level per hour for Sally's swimming pool is 1074 gallons.
What is rate change per hour?Rate of change is equal to (Change in completed assignments) / (Change in hours) Change in rate = (6-3) / (2-1) Change in rate = (3) / (1) The change rate is 3/1, or 3 assignments every hour. The rate of change is 3.0, or three assignments are completed every hour when time is measured in hours.
According to question:We have,
Her pool holds 9,774.5 gallons of water and will need to drain completely in 9.1 hours.
Number of gallons per hour are
gallons per hour = 9,774.5/9.1
gallons per hour = 1074 approx
Thus, required number of gallons per hours 1074.
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What would 2x - y = (-5)
Y = -x -5 be?
(Solving using substitution)
The trajectory of a golf ball hit from a tee on the ground at an angle of 50 degrees with an initial speed of 60 meters per second can be modeled by the parabola f(x) = 1. 19x − 0. 0033x2, where the x-axis is the ground. Find the height of the highest point of the trajectory and the horizontal distance the golf ball travels before hitting the ground.
PLEASE HELP!
The height of the highest point of the trajectory and the horizontal distance the golf ball travels before hitting the ground will be 360 .6 meters
Apparently, x is the distance from the place where the ball is impacted on the ground.
The ball's horizontal speed, given by s, can be calculated using the equation cos(50) = s/60, where s = 38.5 mps.
Size is represented by f(x). The highest point of the parabola would be at -b/2a = -1.19/-.0066 = 180.3 metres, meaning the ball has moved 180.3 metres.
Input 107.28 metres into the formula to find the height.
In order to calculate the distance travelled, the ball must be on the ground, hence f(x) = 0.
-.0033x^2 + 1.19x = 0
x*(-.0033x + 1.19) = 0
x= 1.19/.0033 = 360 .6 meters
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If f(x)=3x, g(x)=x+4 , and h(x)=x2−1 , find g[h(0)]
the value of function g[h(0)] at x = 0 is 3.
What is Function?A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output. Each function has a range, codomain, and domain.
Given that,the functions f(x) = 3x, g(x) = x + 4 , and h(x) = x² − 1
Since,
G(h(x)) = x² - 1 +4
G(h(x)) = x² + 3
Thus at x = 0
G(h(0)) = 0² + 3
G(h(0)) = 3
therefore, the value of function g[h(0)] at x = 0 is 3.
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A pizza company charges $60 for 12 pizzas.
Using the above situation write a complete answer for the following questions.
li) Write the ratio
iii) Write the unit rate or m
iv)write the equation(y=Mx)
v) create a table from the above question
Answer:
Step-by-step explanation:
ii) ratio = 60:12 = 5:1
iii) unit rate is 5
iv) 60 = 12M
I don't how to resolve this
[tex]\cfrac{(2x^{-1} y^2)^{\frac{1}{4}}}{(12x^2 y^{-1})^{\frac{1}{2}}}\implies \cfrac{2^{\frac{1}{4}} x^{-\frac{1}{4}} y^{\frac{2}{4}}}{12^{\frac{1}{2}} x^{\frac{2}{2}} y^{-\frac{1}{2}}}\implies \cfrac{2^{\frac{1}{4}} x^{-\frac{1}{4}} y^{\frac{1}{2}}}{12^{\frac{1}{2}} x y^{-\frac{1}{2}}}\implies \cfrac{2^{\frac{1}{4}} y^{\frac{1}{2}}y^{\frac{1}{2}}}{12^{\frac{1}{2}} xx^{\frac{1}{4}} }[/tex]
[tex]\cfrac{2^{\frac{1}{4}} y^{\frac{1}{2}+\frac{1}{2}}}{12^{\frac{1}{2}} x^{1+\frac{1}{4}} }\implies \cfrac{2^{\frac{1}{4}} y}{12^{\frac{1}{2}} x^{\frac{5}{4}} }\implies \cfrac{2^{\frac{1}{4}} y}{12^{\frac{2}{4}} x^{\frac{5}{4}} }\implies \cfrac{2^{\frac{1}{4}}}{(12^2)^{\frac{1}{4}}}\cdot \cfrac{y}{x^{\frac{5}{4}}}[/tex]
[tex]\left( \cfrac{2}{12^2} \right)^{\frac{1}{4}}\cdot \cfrac{y}{x^{\frac{5}{4}}}\implies \left( \cfrac{1}{72} \right)^{\frac{1}{4}}\cdot \cfrac{y}{x^{\frac{5}{4}}}\implies {\Large \begin{array}{llll} \cfrac{y}{(72)^{\frac{1}{4}}x^{\frac{5}{4}}} \end{array}}[/tex]
OUTCASTS OF POKER FLAT QUESTIONS: ENGLISH
PUT THIS IN THE WRONG CATEGORY.
1. Explain three details from the story that establishes its regionalism. You must write at least three complete sentences--no sentence fragments or phrases.
2. What culture about the West does the author, Harte, convey?
3. If you were living in the East in the late 1800s, what would you find interesting about this story?
4. Although the characters in the story have little in common, they band together. Why do you think this is?
5. The circumstances these characters find themselves in seems to bring out their true characters. Which one surprised you the most? Why?
1. One detail that establishes the story's regionalism is the setting, which takes place in a small mining town in the West called Poker Flat.
2. The author, Harte, conveys a culture about the West that emphasizes the harsh and unforgiving nature of the land and its people. The characters in the story struggle against the harshness of the environment and the moral ambiguity of the mining town.
3. If you were living in the East in the late 1800s, you would find interesting the story's depiction of the West as a land of opportunity and adventure, but also one of danger and hardship. The story would also provide a glimpse into the everyday lives of people living in the West, which would likely be quite different from what they were familiar with in the East.
4. The characters band together because they are all outcasts, each in their own way, and they find comfort in each other's company. They are all looking for a sense of community and belonging that they have been denied in Poker Flat.
5. The character that surprised me the most was Mother Shipton. I was surprised by her sudden change of character when she revealed her past and her true identity as a former prostitute, which was quite different from the religious and moral woman she had pretended to be. I think this was surprising because it shows how people can hide their past and their true selves in order to fit in and be accepted by society.
What is the regionalism about?Another detail is the use of Western colloquialism and dialect in the dialogue of the characters.
Lastly, the story also includes a description of the harsh winter conditions and rugged terrain of the Sierra Nevada mountains, which is specific to that region.
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Graphing piecewise functions
The graph of the piece-wise function f(x) is given by the image presented at the end of the answer.
What is a piece-wise function?A piece-wise function is a function that has different definitions, depending on the input of the function.
The extreme points of the function in this problem are given as follows:
f(-1) = -1 + 3 = 2.f(2) = 2 + 3 = 5.f(2) = -2 + 9 = 7.f(6) = -6 + 9 = 3.Hence the graph of the function is composed by:
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Find an expression to represent the area of the triangle below
The formula for the area of a triangle is (1/4)√(105a?)
How to find the area of a triangle?By applying the triangle inequality theorem, we can verify that the given formula represents the length of the sides of a triangle. This theorem states that the sum of his two sides of a triangle must be greater than his third side.
Given the expressions 2a?, 6a + 5, and 10a?, applicable
2a? + 6a + 5 > 10a? (sum of first two pages greater than third)
2a?+10a?> 6a+5 (1st and his 3rd sides sum greater than 2nd)
6a + 5 + 10a? >2a? (sum of last 2 pages greater than first page)
Therefore, the given formula can express the length of the sides of a triangle.
According to Heron's formula, the area of a triangle with side lengths a, b, and c is
Area = √(s(s-a)(s-b)(s-c))
where s is the semicircle of the triangle and is defined as
s = (a + b + c) / 2
Replacing the given formula with a, b, c, we get:
s = (2a? + 6a + 5 + 10a?) / 2
= 6a? + 3a + 5/2
Find the value of s to calculate the area of the triangle.
Area = √((6a? + 3a + 5/2)((6a? + 3a + 5/2)-(2a? + 6a + 5))((6a? + 3a + 5/2)-(10a?)))
= √((6a? + 3a + 5/2)(2a? - 3a + 5/2)(-4a? + 3a + 5/2))
= √((15/4)(a?)(-5/4)(-a?)(-7/4)(a?))
= √(105a?/16)
= (1/4)√(105a?)
So the formula for the area of a triangle is (1/4)√(105a?).
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The following games are played with 8 cards: the four kings and the four aces. The dealer fans the cards with their back to you and asks you to take one, but not to look at it. He tells you that you will win a prize if your chosen card is an ace. There are two versions of what happens next.
In version 1, he closes up the remaining 7 cards, shuffles them and reveals to you the top card face up on the table which turns out randomly to be the King of Spades. He then deals the next card face down on the table. He then offers you the chance to trade the card you hold for the face-down card that was just dealt. So you have two choices: stay or switch. What are your chances of winning under each choice?
In version 1, if you choose to stay with the card you have initially chosen, your chances of winning are 1 in 8, since you have a 1 in 8 chance of having selected an ace initially.
If you choose to switch, your chances of winning are 7 in 8, since the card you are switching to is randomly selected from the remaining 7 cards, 4 of which are aces. However, if the card you have initially chosen is an ace, you should stay with it since you have already won the prize. If it is not an ace, it may be beneficial to switch as you have a higher chance of getting an ace with the face-down card.
In either case, the dealer has a 7-in-8 chance of winning since he has access to all the cards and can choose the card to be revealed at random. This means that he has a higher chance of winning than the player does. Overall, the game is quite balanced in terms of the chances of winning for either the player or the dealer.
The dealer has an advantage due to knowing the cards and being able to select the card to be revealed, but the player still has a chance of winning depending on their decision to switch or stay with their initial selection.
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