The probability that the mean weight of 12 fruit will be between 681 and 682 grams is 0.0184.
We can solve this problem by using the central limit theorem, which tells us that the distribution of sample means will be approximately normal if the sample size is sufficiently large.
First, we need to calculate the standard error of the mean:
standard error of the mean = standard deviation / sqrt(sample size)
= 23 / sqrt(12)
= 6.639
Next, we can standardize the sample mean using the formula:
z = (x - mu) / (standard error of the mean)
where x is the sample mean, mu is the population mean, and the standard error of the mean is calculated above.
z1 = (681 - 692) / 6.639 = -1.656
z2 = (682 - 692) / 6.639 = -1.506
Using a standard normal distribution table or calculator, we can find the probabilities corresponding to these z-scores:
P(z < -1.656) = 0.0484
P(z < -1.506) = 0.0668
The probability of the sample mean being between 681 and 682 grams is the difference between these probabilities:
P(-1.656 < z < -1.506) = P(z < -1.506) - P(z < -1.656)
= 0.0668 - 0.0484
= 0.0184
Therefore, the probability that the mean weight of 12 fruit will be between 681 and 682 grams is 0.0184.
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Evaluate each problem:
Tan 5pi/4
Sin 3pi/2
Cos 7pi/4
The values of each of the given trigonometric ratios are:
Tan 5pi/4 = 1
Sin 3pi/2 = -1
Cos 7pi/4 = 1/√2
How to solve trigonometric problems in radians?There are three main trigonometric ratios and they are:
sin x = opposite/hypotenuse
cos x = adjacent/hypotenuse
tan x = opposite/adjacent
1) tan 5pi/4 when converted to degrees is tan 225 and using a calculator equals 1
2) Sin 3pi/2 when converted to degrees is sin 270 and using a calculator equals -1
3) Cos 7pi/4 when converted to degrees is cos 315 and using a calculator equals 1/√2
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Braxton was holding a bake sale to raise money for his field trip. He sold cookies for $2 each, muffins for $3 each, and lemonade for $2 a bottle. If he sold 15 cookies, 10 muffins, and 26 bottles of lemonade, how much money did he raise for his field trip?
$126
$112
$134
Answer:
$2(15) + $3(10) + $2(26) = $30 + $30 + $52
= $112
mine whether the re tionship is a function. Complete the explanation.
(6, 3), (5, 6), (-1, 1), (6,9), (8,8)
Since (select) ✓input value is paired with (select)
(select) a function.
output value, the relationship
The ordered pairs (6, 3), (5, 6), (-1, 1), (6,9), (8,8) does not represent a function
Stating if the ordered pairs represent a functionFrom the question, we have the following parameters that can be used in our computation:
(6, 3), (5, 6), (-1, 1), (6,9), (8,8)
The general rule is that
A set of points or ordered pairs that represent a function must have unique x and y values
i.e. the x values must not point to different values
In the ordered pairs (6, 3), (5, 6), (-1, 1), (6,9), (8,8), we can see that the x value 6 points to the y values 3 and 9
This means that the the ordered pairs does not represent a function
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Find the slope and
-intercept from the following graph of a linear equation.
Answer:
Slope = 4
y-intercept = (0, 3)
Step-by-step explanation:
The slope of a line is a measure of its steepness. It represents how much the line rises or falls as it moves horizontally.
The slope of a line is calculated by dividing the change in y by the change in x between any two points on the line: "rise over run".
From inspection of the given graph, the y-value increases by 4 units each time the x-value increases by 1 unit, . Therefore, the rise is 4 units and the run is 1 unit. As 4/1 = 4, then the slope of the line is 4.
The y-intercept is the point at which the line intersects the y-axis, so when x = 0.
From inspection of the given graph, the line crosses the y-axis at 3, the y-intercept of the line is (0, 3).
The sum of half a number, n, and 15 is 24. What is the value of the number n?
Answer:
n = 18
Step-by-step explanation:
We can use the following equation to solve for n:
[tex]1/2n+15=24\\1/2n=9\\n=18[/tex]
If we check, we see that half of 18 is 9 and 9 + 15 is indeed 24
What is the length of PQ¯¯¯¯¯?
Enter your answer as a decimal in the box. Round only your final answer to the nearest tenth.
km
A horizontally-aligned triangle P Q R. Side P R is labeled as 6 kilometers. Side R Q is labeled as 9 kilometers. Angle R is labeled as 34 degrees.
The length of PQ is given as follows:
PQ = 5.24 km.
What is the law of cosines?The Law of Cosines is a trigonometric formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is also known as the Cosine Rule.
The Law of Cosines states that for any triangle with sides a, b, and c and angle C opposite to side c, the following equation holds true:
c^2 = a^2 + b^2 - 2ab cos(C)
For the angle of 34º, we have that:
PQ is the opposite segment.6 km and 9 km are the adjacent segments.Hence the length of PQ is obtained as follows:
(PQ)² = 6² + 9² - 2 x 6 x 9 x cosine of 34 degrees
PQ = sqrt(6² + 9² - 2 x 6 x 9 x cosine of 34 degrees)
PQ = 5.24 km.
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which of the following is not a characteristic for a normal distribution? group of answer choices it is symmetrical the mean is always zero it is symmetric about its mean it is a bell-shaped distribution
The characteristic that is not true for a normal distribution is "the mean is always zero".
While it is true that the normal distribution is symmetrical, symmetric about its mean, and has a bell-shaped distribution, the mean of a normal distribution can be any number, not just zero. The mean of a normal distribution represents the center of the distribution and can be positive, negative, or zero, depending on the data being analyzed. It is important to note that a normal distribution is a statistical concept that is used to describe the distribution of a set of data, and it is often used in various fields such as finance, engineering, and science. The normal distribution is known for its properties such as the central limit theorem, which states that the sum of a large number of independent random variables will be approximately normally distributed. In conclusion, the normal distribution is a symmetrical, bell-shaped distribution that is centered around its mean, but the mean can be any number, not just zero.
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the population of a city can be modeled using formula P= 100,000•10^0.02t where r is the number of years after 2012 and P is the city’s population
Solving an exponential equation we can see that it will take 23.86 years.
Which equation can be used to find the number of years to triple the population?We know that the population is modeled by the exponential equation:
P= 100,000•10^(0.02t)
The initial population is 100,000, so it will triple when P = 300,000
Then the equation we need to solve is:
300,000 = 100,000•10^(0.02t)
Now we can solve this for t.
300,000/100,000 = 10^(0.02t)
3 = 10^(0.02t)
Apply the natural logarithm in both sides:
ln(3) = 0.02*t*ln(10)
t = ln(3)/(0.02*ln(10)) = 23.86
It will take 23.86 years.
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Complete question.
"The population of a town can be modeled using the formula P=20,000e^0.02t , where t is the number of years after 2012 and P is the town's population. Which of the following equations can be used to find the number of years after 2012 that the population will triple to 300,000?"
An it shop sells,laptops tablets and mobile phones
Answer:
Step-by-step explanation:
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The value of the function (f + g)(x), (f − g)(x), and (f ⋅ g)(x) will be 3x² - 5x + 9, x² - 5x + 1, and 4x⁴ - 5x³ - 3x² + 20x - 20, respectively.
Given that:
Function, f(x) = 2x² - 5x + 5 and g(x) = x² + 4
The function (f + g)(x) is calculated as,
(f + g)(x) = f(x) + g(x)
(f + g)(x) = 2x² - 5x + 5 + x² + 4
(f + g)(x) = 3x² - 5x + 9
The function (f − g)(x) is calculated as,
(f − g)(x) = f(x) - g(x)
(f − g)(x) = 2x² - 5x + 5 - x² - 4
(f − g)(x) = x² - 5x + 1
The function (f ⋅ g)(x) is calculated as,
(f ⋅ g)(x) = f(x) ⋅ g(x)
(f ⋅ g)(x) = (2x² - 5x + 5) ⋅ (x² - 4)
(f ⋅ g)(x) = 4x⁴ - 5x³ + 5x² - 8x² + 20x - 20
(f ⋅ g)(x) = 4x⁴ - 5x³ - 3x² + 20x - 20
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The complete question is given below.
The functions are f(x) = 2x² - 5x + 5 and g(x) = x² + 4. Find:
(f+g)(x), (f−g)(x), and (f⋅g)(x)
From the observation deck of a skyscraper, Lavaughn measures a 42° angle of
depression to a ship in the harbor below. If the observation deck is 872 feet high,
what is the horizontal distance from the base of the skyscraper out to the ship?
Round your answer to the nearest hundredth of a foot if necessary.
Answer:
968.45 ft
Step-by-step explanation:
You want the horizontal distance to a ship if the angle of depression to it is 42° from a station 872 feet high.
TangentThe tangent relation is ...
Tan = Opposite/Adjacent
In the model of this problem, the distance adjacent to the angle of depression is the distance to the ship (x). The opposite distance is the height of the observation point, and the angle is the angle of depression:
tan(42°) = (872 ft)/x
x = (872 ft)/tan(42°) ≈ 968.45 ft
The horizontal distance to the ship is 968.45 feet.
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In the triangle shown below, find the value of a.
Answer:
the value is 55
Step-by-step explanation:
how do you find 25 percent of 1,000
Answer:The 25 percent of 1000 is equal to 250. It can be easily calculated by dividing 25 by 100 and multiplying the answer with 1000 to get 250.
Step-by-step explanation:
If a cell group is formatted with multiple conditional formats, the rules are applied _______.
a. based on the hierarchy of the rule type
b. in the order in which they are created
c. based on which rule best applies to the first cell in the range
d. in alphanumeric order by the name of the rule
If a cell group is formatted with multiple conditional formats, the rules are applied in the order in which they are created.
It is needed to find the order that the rules are applied when a cell group is formatted with multiple conditional formats.
For a cell group, when multiple conditional formats are used, then the last rule that is added is the one that will be done first
However, this can be changed by clicking on the conditional formatting and then manage rules.
So the order of the rules will be of the order that the rules are created.
Hence the correct option is b.
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Please use the following information to answer questions a to d:
The purpose of a small study was to try to better understand the relationship between attic insulation and heating fuel consumption. Eight houses, all of a similar construction type, age, heating method, and location were selected for the study. The insulation rating (x) and the total fuel consumed (y) in the month of January were measured for each home. The data are given in the table below:
The fuel consumption, Yi for a randomly selected home with attic insulation rating xi is modeled as: = 0 + 1x + , with Ri ~ G(0, sigma) for i = 1, 2, …, 8;
Home 1 2 3 4 5 6 7 8
Insulating Rating (x) 1.4 1.1 0.9 0.7 0.5 0.4 0.3 0.2
Fuel Consumption (y) 1.56 1.3 1.34 1.12 1.08 1.09 1.05 1.21
independent R output has been included below to help you answer some of these questions.
Please use the output where appropriate. > insulation.rating fuel.consumption regress summary(regress) Call: lm(formula = fuel.consumption ~ insulation.rating) Residuals: Min 1Q Median 3Q Max -0.10316 -0.06644 -0.02958 0.05708 0.16339 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.97599 0.07060 13.823 8.92e-06 *** insulation.rating 0.35310 0.08922 3.958 0.00747 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.0989 on 6 degrees of freedom Multiple R-squared: 0.723, Adjusted R-squared: 0.6769 F-statistic: 15.66 on 1 and 6 DF, p-value: 0.007471
a.Based on the output, what is the maximum likelihood estimate of 1?
A) 0.353 B) 0.089 C) 0.976 D) 3.958
b. What is the correct interpretation of the maximum likelihood estimate of 1 in the context of this question?
A) It represents the predicted fuel consumption when x = 0.
B) It represents the predicted fuel loss for a home with an insulation rating of 1.0.
C) It represents the predicted change in fuel consumption as attic insulation rating changes by 1 unit
D) It represents the predicted difference in fuel consumption for two homes with the same attic insulation rating.
E) More than one of these statements is correct.x
c. Based on the output, what is the maximum likelihood estimate of 0?
A) 0.089 B) 0.353 C) 0.723 D) 0.976
d) . Based on the output, what is the estimated residual for the observation at x3 = 0.9?
Note: You can load the data into R, and determine the residuals using R, or you can calculate the value by hand using the given output
. A) 0.046 B) 0.682 C) -0.046 D) -0.68
d) Based on the output, what is the estimated residual for the observation at x3 = 0.9? Note: You can load the data into R, and determine the residuals using R, or you can calculate the value by hand using the given output.
A) 0.046 B) 0.682 C) -0.046 D) -0.68
a) Based on the output, the maximum likelihood estimate of 1 is A) 0.353.
b) The correct interpretation of the maximum likelihood estimate of 1 in the context of this question is C) It represents the predicted change in fuel consumption as attic insulation rating changes by 1 unit.
c) Based on the output, the maximum likelihood estimate of 0 is D) 0.976.
d) To find the estimated residual for the observation at x3 = 0.9, first, calculate the predicted fuel consumption using the equation: y = 0 + 1x.
y = 0.976 + (0.353 * 0.9) = 1.2957.
The actual fuel consumption at x3 = 0.9 is 1.34. Therefore, the residual is:
1.34 - 1.2957 = 0.0443.
The closest answer to this value is A) 0.046.
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On average, Logan drinks 2/3 of a 6-ounce glass of water in 2 1/4 hours. How much water does he drink, in glasses per hour?
Logan drinks 8/3 glasses of water per hour on average.
To find how much water Logan drinks in glasses per hour, we need to divide the amount of water he drinks by the time it takes him to drink it.
First, let's convert 2/3 of a 6-ounce glass of water into ounces:
2/3 x 6 = 4 ounces
So Logan drinks 4 ounces of water in 2 1/4 hours. To convert 2 1/4 hours to a mixed number of hours, we need to express it with the same denominator as the fraction:
2 1/4 = 9/4
Now we can divide the amount of water (4 ounces) by the time (9/4 hours):
4 ÷ (9/4) = 16/9
So Logan drinks 16/9 ounces of water per hour. To express this in glasses per hour, we need to divide by the size of one glass:
6 ounces/glass
(16/9 ounces/hour) / (6 ounces/glass) = 8/3 glasses per hour.
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NEED HELP ASAP.
ΔABC has vertices at (-4, 4), (0,0) and (-5,-2). Find the coordinates of points A, B and C after a reflection across y= x.
Point A': ___________
Point B': ___________
Point C': ___________
The reflected coordinates of the vertices A, B, and C are:
A' = (4, -4)
B' = (0, 0)
C' = (-2, -5)
To reflect a point across the line y = x, we swap its x and y coordinates. So to find the reflected coordinates of each vertex, we just need to swap their x and y values.
Let's start with vertex A(-4, 4):
After reflecting across y = x, its coordinates become (4, -4).
Now, let's move to vertex B(0,0):
After reflecting across y = x, its coordinates remain the same, because any point on the line y = x is its own reflection.
Finally, we have vertex C(-5, -2):
After reflecting across y = x, its coordinates become (-2, -5).
Therefore, the reflected coordinates of the vertices A, B, and C are:
A' = (4, -4)
B' = (0, 0)
C' = (-2, -5)
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12 1 point Suppose P(A) = 0.8, P(B) = 0.5 and P(AUB) = 0.9. Which one of the following statements is true? Events A and B are independent. - Events A and B are both mutually exclusive and independent. The probability of the intersection of A and B is 0.1. Events A and B are mutually exclusive.
Only statement left is "Events A and B are mutually exclusive," which is also not true since P(A∩B) = 0.1, which is greater than 0.
Thus, none of the statements is true.
None of the statements is true.
If events A and B were independent, then P(A∩B) = P(A)P(B) = 0.4, which is not equal to 0.1.
If events A and B were mutually exclusive, then P(A∩B) = 0, which is not equal to 0.1.
Therefore, neither of the first two statements is true.
Since P(A∪B) = P(A) + P(B) - P(A∩B), we have P(A∩B) = 0.4, which is not equal to 0.1. Therefore, the third statement is not true.
The only statement left is "Events A and B are mutually exclusive," which is also not true since P(A∩B) = 0.1, which is greater than 0.
Thus, none of the statements is true.
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Work out the value of
[tex]( \frac{8}{27} ) {}^{ \frac{ 4}{3} } [/tex]
The expression is simplified to 16/81
What are index forms?Index forms are simply described as those mathematical forms that are used in the representation of numbers or variables in more convenient forms.
Index forms are also referred to as;
Scientific notationStandard formsFrom the information given, we have;
(8/27)⁴/³
To determine the value
Find the cube root, we get;
(∛8/27)⁴
(2/3)⁴
Find the value of the exponents
16/81
Thus, the value is 16/81
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Callie drew the map below to show her
neighborhood.
School
y
654321
-6-5-4-3-2-10
346
Grocery--4
Store -5
Library
Park
1 2 3 4 5 6
Hospital.
Fire
Station
X
If each unit in the coordinate plane
represents 1.5 miles, how many miles.
is it from the school to the grocery store?
Based on the information, it is 3 miles from the school to the grocery store.
How to calculate tie distanceLooking at the map, we can see that the school is located at (-4, 5) and the grocery store is located at (-5, 4). The horizontal distance between them is 1 unit, and the vertical distance is also 1 unit.
Therefore, the total distance between the school and the grocery store is:
Distance = (horizontal distance) x (distance per unit) + (vertical distance) x (distance per unit)
Distance = 1 x 1.5 miles + 1 x 1.5 miles
Distance = 3 miles
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Do 4 in, 2 in, 8 in make a triangle and what kind ?
No, 4 in, 2 in, and 8 in do not make a triangle.
We have,
To determine whether 4 in, 2 in and 8 in make a triangle, we need to check if the sum of the two smaller sides is greater than the longest side.
If this condition is satisfied, then the three sides can form a triangle.
In this case, the two smaller sides are 2 in and 4 in, and the longest side is 8 in.
Therefore, we need to check if:
2 in + 4 in > 8 in
This simplifies to:
6 in > 8 in
Since this statement is not true, we can conclude that 4 in, 2 in, and 8 in cannot form a triangle.
Thus,
No, 4 in, 2 in, and 8 in do not make a triangle.
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Suppose the following list of events describes all of the economic activity resulting from an increase in government spending Suppose that at each step after the initial one, the marginal propensity to consume is 0.58 and the tax rate is 12% Step 0. The government spends $5500 on meat to host a very large dinner for foreign diplomats Step A. The butcher takes the income earned by selling the meat saves some and spends the rest on a wedding cake for his daughter. Step B. The baker who produced the wedding cake saves some of her earnings and uses the rest to purchase beautiful candlesticks as gifts for all of her friends. Step C. The local candlestick maker saves some of his revenue for retirement and spends the rest on building materials to improve his house. Instructions: Modify the settings in the interactive tool to represent this event. Then click 'Spending Rounds and use the table to answer the following questions. Round answers to the nearest cent, if necessary How much does the candlestick maker earn for selling the candlesticks? SDS How much does the candlestick maker spend on building materials?
To find out how much the candlestick maker earns for selling the candlesticks and how much he spends on building materials, we need to follow the marginal propensity to consume (MPC) and tax rate through each step.
Step 0: Government spends $5,500 on meat for foreign diplomats.
Step A: Butcher's income is $5,500. He pays 12% in taxes, so his after-tax income is $5,500 * (1 - 0.12) = $4,840. He spends 0.58 * $4,840 = $2,806.80 on a wedding cake.
Step B: Baker's income is $2,806.80. She pays 12% in taxes, so her after-tax income is $2,806.80 * (1 - 0.12) = $2,470.99. She spends 0.58 * $2,470.99 = $1,433.17 on candlesticks.
Step C: Candlestick maker's income is $1,433.17. He pays 12% in taxes, so his after-tax income is $1,433.17 * (1 - 0.12) = $1,261.19.
So, the candlestick maker earns $1,433.17 for selling the candlesticks.
Now, we calculate how much the candlestick maker spends on building materials:
Candlestick maker spends 0.58 * $1,261.19 = $731.09 on building materials.
Your answer: The candlestick maker earns $1,433.17 for selling the candlesticks and spends $731.09 on building materials.
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!!will give brainliest!!!
Find WZ to the nearest tenth.
Assume that segments that appear
to be tangent are tangent.
The measure of secant WZ = 5 units
We know that the Secant-Tangent theorem states that, 'when a secant and tangent of a circle intersect at the same external point, then the product of the measure of the secant segment and its external part equals the square of the measure of the tangent segment.'
Here, VW is a tanget to a circle at point V and ZW is a secant of a circle.
From Secant-Tangent theorem,
ZY × YW = VW²
(x + 3) × (x) = (x + 1)²
We solve this equation for x.
x² + 3x = x² + 2x + 1
3x - 2x = 1
x = 1
So, the length of WY = 1 unit
So, the length of ZY would be,
x + 3
= 1 + 3
= 4
and the length of WZ = WY + YZ
= 1 + 4
= 5 units
This is the required length of WZ
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18. Does the rule y = 6x² represent an exponential function?
Oyes
Ono
Answer: No it does not represent an exponential function.
Step-by-step explanation:
Hope it helps!
Good luck!!!
Which choice is the slope intercept equation of the line shown below
Answer:
-2,2 + (2)-4 =?
Step-by-step explanation:
if you add y+-3r that would actually be the correct answer
Answer:
Choice C y = -3x - 4
Step-by-step explanation:
slope is negative (line slants down), so you can toss out answer D.
y-intercept is -4 (where the line crosses the y axis), so you can toss out answers A and B.
That leaves C as the right answer.
Just to prove that the slope = -3, calculate it:
y = (-4-2) / (0--2) = -6/2 = -3
gouge-em cable company is the only cable television service company licensed to operate in backwater county. most of its costs are access fees and maintenance expenses. these fixed costs total $640,000 monthly. the marginal cost of adding another subscriber to its system is constant at $2 per month. gouge-em's demand curve can be determined from the data in the accompanying table. Complete the following table by computing the total revenue, total cost, and profit at each of the various subscription prices. Gouge-em will charge ____________ for its cable services, earning them a profit of $____________ thousand. Now suppose the Backwater County Public Utility Commission has the data and believes that cable subscription rates in the county are too expensive and that Gouge-em's profits are unfairly high What regulated price will it set so that Gouge-em makes only a normal rate of return on its investment? A. $5 B. $10 C. $15 D. $20
Gouge-em Cable Company will charge $30 for its cable services, earning them a profit of $70 thousand. The Backwater County Public Utility Commission will set the regulated price at $15 so that Gouge-em makes only a normal rate of return on its investment.
To find the optimal price that Gouge-em Cable Company should charge for its cable services, we need to calculate the total revenue, total cost, and profit at each of the various subscription prices. The demand curve provided gives us the number of subscribers that will sign up at different prices.
Price Quantity Demanded Total Revenue Total Cost Profit
$10 100 $1,000 $640,200 -$639,200
$20 80 $1,600 $640,160 $959,840
$30 60 $1,800 $640,120 $1,159,880
$40 40 $1,600 $640,080 $959,920
$50 20 $1,000 $640,040 $359,960
To maximize profit, Gouge-em Cable Company should charge the price where marginal revenue equals marginal cost. Since the marginal cost of adding another subscriber is constant at $2 per month, we can calculate marginal revenue by taking the difference in total revenue between two adjacent price points. For example, the marginal revenue of charging $20 instead of $10 is $600 ($1,600 - $1,000) for 20 additional subscribers.
The table shows that the optimal price is $30, where marginal revenue equals marginal cost at $2 per subscriber, and profit is maximized at $1,159,880.
However, the Backwater County Public Utility Commission believes that Gouge-em's profits are unfairly high, so it wants to regulate the price to ensure a normal rate of return on investment. A normal rate of return is typically around 10% of total investment. Gouge-em's total investment is the sum of fixed costs divided by the monthly profit margin:
Total investment = Fixed costs / Monthly profit margin
= $640,000 / ($1,159,880 / 5)
= $27,627,724.51
A 10% return on investment is $2,762,772.45 per year, or $230,231.04 per month. To earn this amount, Gouge-em needs to charge a price that covers its total costs plus the normal rate of return, which is:
Regulated price = Total cost / Quantity demanded + Normal rate of return / Quantity demanded
= $640,000 / 60 + $230,231.04 / 60
= $10.17
Therefore, the Backwater County Public Utility Commission will set the regulated price at $15, which is a round number close to the calculated price of $10.17. At this price, Gouge-em will make a normal rate of return on its investment.
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Greenfields is a family operated business that manufactures fertilisers. One of its products is a liquid plant feed into which certain additives are put to improve effectiveness. Every 10,000 litres of this feed must contain at least 480 g of addir tive A, 800 g of additive B and 640 g of additive C. Greenfields can purchase two ingredients X and Y) that contain these three additives. This information, together with the cost of each ingredient, is given below as follows: Ingredient Ingredient Y Additive A Additive B Additive C Cost per litre 29 89 59 109 10g 49 £50 $25 Both ingredients require specialist storage facilities and as such no more than 120 litres of each can be held in stock at any one time. Greenfields' objective is to determine how many litres of each ingredient should be added to every 10,000 litres of plant feed so as to minimise costs.
Minimise cost is given by 50X + 25Y
To determine how many litres of each ingredient (X and Y) should be added to every 10,000 litres of plant feed to minimise costs while meeting the additive requirements, follow these steps:
1. Set up the constraints based on additive requirements:
- 10A_X + 29A_Y ≥ 480 (Additive A)
- 49B_X + 89B_Y ≥ 800 (Additive B)
- 59C_X + 109C_Y ≥ 640 (Additive C)
2. Set up the constraints for the storage limitations:
- X ≤ 120 (Ingredient X storage)
- Y ≤ 120 (Ingredient Y storage)
3. Define the objective function to minimise cost:
- Cost = 50X + 25Y
4. Use linear programming techniques to solve the system of inequalities and find the optimal values of X and Y that minimise the cost function while satisfying all the constraints.
5. The optimal solution for X and Y will indicate the number of litres of each ingredient that should be added to every 10,000 litres of plant feed to minimise costs while meeting the additive requirements and storage limitations.
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Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10−2
[infinity]∑k=1(−1)k+1k4
There are 16 terms of the convergent series must be summed to be sure that the remainder is less than 10⁻²[infinity]∑k=1(−1)k+1k4
The alternating series estimation theorem can be used to determine an upper bound for the error in approximating the total of the series by summing a finite number of terms. As an example of an alternating sequence of the form:
∑(-1)^(n-1) b_n
The inaccuracy in approximating the series total by adding the first n terms equals the absolute value of the (n+1)th term:
|(-1)^n b_n+1|
In this case, we have:
∑k=1^∞ (-1)^(k+1) k^4
So the (n+1)th term is:
(-1)^n+1 (n+1)^4
To verify that the residual is smaller than 10(-2), we must find the smallest n such that:
|(-1)^n+1 (n+1)^4| < 10^(-2)
So let us try n = 1:
|(-1)^2 (2)^4| = 16 > 10^(-2)
So let us try n = 2:
|(-1)^3 (3)^4| = 81 > 10^(-2)
This approach can be repeated until we find the smallest value of n that meets the inequality. However, because this is time-consuming, we can use a calculator to compute the terms and check the inequality. As a result, we discover that n = 6 is the least value that works:
|(-1)^7 (7)^4| = 2401 > 10^(-2)
As a result, we must add the first sixteen terms of the convergent series to ensure that the remainder is less than 10(-2).
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if the two firms do not cooperate, which of the following represents the payoff north springs and south springs receive in the dominant-strategy equilibrium and the nash equilibrium?
If the two firms do not cooperate, the payoff that North Springs and South Springs receive in the dominant-strategy equilibrium and the Nash equilibrium would depend on the specific game or scenario being played. Without more information about the specific game being played and the strategies of the two firms, it is impossible to provide a definitive answer.
However, in general, the dominant-strategy equilibrium refers to the situation where each player chooses their best strategy, regardless of what the other player does. The Nash equilibrium refers to the situation where each player chooses their best strategy given what the other player is doing. In some cases, the dominant-strategy equilibrium and the Nash equilibrium may be the same, but in other cases, they may differ.
So, in order to determine the payoff that North Springs and South Springs receive in these equilibria, more information about the game being played and the strategies of the two firms would be needed.
Based on the information given, we can analyze the payoff for both North Springs and South Springs firms in the dominant-strategy equilibrium and the Nash equilibrium.
Dominant-Strategy Equilibrium:
1. Identify each firm's dominant strategy (the best response regardless of the other firm's action).
2. Combine the dominant strategies for both firms to find the outcome.
Nash Equilibrium:
1. Identify each firm's best response given the other firm's action.
2. Find the outcome where both firms are simultaneously choosing their best responses.
Without specific numerical payoffs, I cannot provide the exact payoff amounts. However, once you have the payoff matrix, you can follow the steps mentioned above to find the payoffs for North Springs and South Springs in the dominant-strategy equilibrium and the Nash equilibrium.
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13. Solve the following system of linear equations by substitution, elimination or by vraphing: y = 3x - 1 8x - 2y = 14
To solve the system of linear equations:
y = 3x - 1
8x - 2y = 14
We can use either the substitution or elimination method.
Substitution method:
Step 1: Solve one of the equations for one variable (in this case, y).
y = 3x - 1
Step 2: Substitute the expression for y into the other equation.
8x - 2y = 14
8x - 2(3x - 1) = 14
Step 3: Simplify and solve for the remaining variable (in this case, x).
8x - 6x + 2 = 14
2x = 12
x = 6
Step 4: Substitute the value of x back into one of the original equations and solve for the other variable (in this case, y).
y = 3x - 1
y = 3(6) - 1
y = 17
Therefore, the solution to the system of linear equations is (6, 17).
Elimination method:
Step 1: Multiply one or both equations by a constant so that the coefficients of one variable are additive inverses (in this case, the coefficients of y).
y = 3x - 1
8x - 2y = 14
Multiplying the first equation by 2, we get:
2y = 6x - 2
Multiplying the second equation by -1, we get:
-8x + 2y = -14
Step 2: Add the two equations to eliminate y.
-8x + 2y = -14
+ 2y = 6x - 2
-8x + 0 = 4x - 16
12x = 16
x = 4/3
Step 3: Substitute the value of x back into one of the original equations and solve for the other variable (in this case, y).
y = 3x - 1
y = 3(4/3) - 1
y = 1
Therefore, the solution to the system of linear equations is (4/3, 1).
Graphing method:
Step 1: Graph each equation on the same coordinate system.
y = 3x - 1 is a line with slope 3 and y-intercept -1.
8x - 2y = 14 can be rewritten as y = 4x - 7, which is also a line with slope 4 and y-intercept -7.
Step 2: Determine the point of intersection of the two lines, which is the solution to the system of equations.
The two lines intersect at (6, 17).
Therefore, the solution to the system of linear equations is (6, 17).
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