Answer:
B. {y | y = -7, -1, 0, 9}
Step-by-step explanation:
The range is the y values while the domain is the x values. In this case, we want to use the y values.
PLEASE HELP, ASAP!! QUESTION IN PICTURE!
Select all that apply
A. A
B. B
C. C
D. D
E. E
Answer:
B, D and E
Step-by-step explanation:
I just need 20 characters
The equations y = 2x - 5 and y = \sqrt{3x - 1} are graphed on the coordinate grid.
How many real solutions does the equation \sqrt{3x - 1} =2x - 5 have?
A. 0
B. 1
C. 2
D. cannot be determined from the graph
Answer:
1 real solution.
Step-by-step explanation:
The solution is the point where the 2 graphs intersect (close to the point
(4, 3).
The equation have one real solution.
the correct option is (B)
What are intersecting lines?Two or more lines which share exactly one common point are called intersecting lines.
Given:
y = 2x - 5 and y = √{3x - 1}
As seen from the graph the lines of both given equation is intersecting line.
So, the type of lines are intersecting lines.
and, intersecting lines give one real solution where the lines intersect.
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give simple working out
The size of the angle x is 330°
How to find the size of the angle x?Angle geometry is a branch of mathematics that deals with the study of angles and their properties. Angles are formed when two lines or line segments intersect or when a line segment meets a point.
Angle geometry involves the measurement of angles, the relationships between angles, and the properties of angles in different shapes and figures.
The sum angle at a point is equal to 360 degrees. Thus, we can say:
x + 30° = 360°
x = 360° - 30°
x = 330°
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The number of prime factors of 3×5×7+7 is
The number of prime factors of 3×5×7+7 is 3.
To find the number of prime factors, we need to calculate the given expression:
3×5×7+7 = 105+7 = 112.
The number 112 can be factored as 2^4 × 7.
In the first step, we factor out the common prime factor of 7 from both terms in the expression. This gives us 7(3×5+1). Next, we simplify the expression within the parentheses to get 7(15+1). This further simplifies to 7×16 = 112.
So, the prime factorization of 112 is 2^4 × 7. The prime factors are 2 and 7. Therefore, the number of prime factors of 3×5×7+7 is 3.
In summary, the expression 3×5×7+7 simplifies to 112, which has three prime factors: 2, 2, and 7. The factor of 2 appears four times in the prime factorization, but we count each unique prime factor only once. Thus, the number of prime factors is 3.
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Select the correct scientific notation form of this numeral using only 3 significant figures. 8,468 L
8,468 written in scientific notation is 8.47 x 10³.
What is the scientific notation?Scientific notation is used to compress large numbers in smaller numbers. In order to write a number in scientific notation, the number is written as a decimal number, between 1 and 10 and multiplied by a power of 10.
For example:
• The number : 1 x 10² is equivalent to 100
• The number 1 x 10³ is equivalent to 1000
8,468 becomes 8.47 x 10³
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Did we invent math or did we discover it?
Answer:
we invented it. it's a tool to help things make sense.
Answer:
Step-by-step explanation:
We invent it. it was invented by Archimedes.
A circular room has a floor of diameter 590cm. Estimate the 92.2 perimeter of the room in metres
Answer:
18.5 m
Step-by-step explanation:
The diameter in meters is 5.90, so the circumference of the room is ...
C = πd = 5.90π m ≈ 18.5 m
The perimeter of the room is about 18.5 m.
Find the t-value that would be used to construct a 95% confidence interval with a sample size n=24. a. 1.740 b. 2.110 c. 2.069 d. 1.714 4
The t-value that would be used to construct a 95% confidence interval with a sample size of n=24 is c. 2.069.
To explain why, consider the idea of a t-distribution. We utilize the t-distribution instead of the usual normal distribution when working with small sample sizes (less than 30) and unknown population standard deviations. The t-distribution is more variable than the usual normal distribution, and this difference is compensated for by using a t-value rather than a z-value.
The t-value we select is determined by two factors: the desired level of confidence and the degrees of freedom (df) for our sample. We have 23 degrees of freedom for a 95% confidence interval with n=24 (df=n-1). We can calculate the t-value for a 95% confidence interval with 23 df using a t-table or calculator. This implies we can be 95% certain that the real population means is inside our estimated confidence zone.
It's worth noting that as the sample size grows larger, the t-distribution approaches the regular normal distribution, and the t-value approaches the z-value. So, for large sample sizes (more than 30), the ordinary normal distribution and a z-value can be used instead of the t-distribution and a t-value.
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A car is traveling at a constant speed. It travels 31/2 miles in 51/4 minutes. How many minutes will it take the car to travel 70 miles?
I need to knoww tooooooo
Answer:105
Step-by-step explanation:
Jada has a new puppy that weighs 9 pounds. the vet says that the puppy is now at about 20% of its adult weight. what will be the adult weight of the puppy?
The adult weight of the puppy calculated using percentage is 45 pounds .
In mathematics, a value or ratio that can be expressed as a simple fraction of 100 is known as a percentage.
When determining a percentage of a number, we should first divide it by 100 before multiplying the result by 100.
As a result, the proportion refers to a component per 100. The word percent denotes a percentage of 100.
It is represented by the symbol "%." Percentages can also be written as decimals or fractions, like in 0.6%, 0.25%, etc.
Academic grades are calculated using percentages for every subject. Ram, for example, received a 78% on his qualifying exam.
The weight of the puppy is 9 pounds.
This is 20% of the adult weight.
Let the adult weight be x.
therefore 20% of x = 9
Calculating we get x = 45 pounds
Hence the adult weight of the puppy is 45 pounds.
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Which of the following rational functions has a horizontal asymptote at y = 2 and vertical asymptotes at x = 3 and x = –4?
a: y equals x squared over the quantity x squared plus x minus 12 end quantity
b: y equals x squared over the quantity x squared minus x minus 12 end quantity
c: y equals 2 times x squared over the quantity x squared plus x minus 12 end quantity
d: y equals 2 times x squared over the quantity x squared minus x minus 12 end quantity
Neither option (a) nor option (b) has a horizontal asymptote at y = 2, there is no correct answer to this question.
What is rational function?A rational function is a mathematical function that can be expressed as a ratio of two polynomial functions, where the denominator is not equal to zero.
To have vertical asymptotes at x = 3 and x = –4, the denominator of the rational function must have factors of (x – 3) and (x + 4), respectively.
Option (a) has a denominator of (x² + x – 12), which can be factored as (x – 3)(x + 4), satisfying the conditions for vertical asymptotes.
Option (b) has a denominator of (x² – x – 12), which can also be factored as (x – 3)(x + 4), satisfying the conditions for vertical asymptotes.
Therefore, the answer is either option (a) or option (b).
To determine which of these options has a horizontal asymptote at y = 2, we can perform long division or use the fact that the leading term of the rational function will determine the horizontal asymptote.
Dividing x² by (x² + x – 12), we get:
x² + x - 12 | x² + 0x + 0
- x² - x
Thus, the quotient is 1 and the remainder is x. So the rational function simplifies to:
y = x/(x² + x - 12)
The leading term of this rational function is x/x², which simplifies to 1/x. Therefore, it does not have a horizontal asymptote at y = 2.
Dividing x² by (x² – x – 12), we get:
x² - x - 12 | x² + 0x + 0
+ x² - x
Thus, the quotient is 1 and the remainder is x. So the rational function simplifies to:
y = x/(x² - x - 12)
The leading term of this rational function is x/x², which simplifies to 1/x. Therefore, it does not have a horizontal asymptote at y = 2.
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Develop a fictitious hypothesis under which ANOVA:
1. Development of a realistic hypothesis
2. Innovation regarding the data used/adopted/formulated and its relation to the proposed project. 3. Hypothesis testing using ANOVA. 4. Presentation of the results.
The fictitious Hypothesis represents the ,
The type of fertilizer used significantly affects the growth rate of plants.
Development of a realistic hypothesis,
This hypothesis is based on the idea that different types of fertilizers may have varying effects on the growth of plants.
It is a common concern in agriculture and gardening to determine the most effective fertilizer to maximize plant growth.
By formulating this hypothesis,
Investigate whether there are statistically significant differences in plant growth rates based on the type of fertilizer used.
Innovation regarding the data used/adopted/formulated and its relation to the proposed project,
In this study, aim to innovate by using a diverse range of fertilizers that are commonly used in agriculture and gardening.
Include both synthetic and organic fertilizers, as well as variations in nutrient compositions and application methods.
By incorporating a wide range of fertilizer types,
Explore the potential variations in plant growth response .
And identify any specific fertilizer formulations or strategies that result in superior growth outcomes.
Hypothesis testing using ANOVA,
To test the hypothesis,
Conduct an analysis of variance (ANOVA) on the data collected from multiple groups of plants treated with different fertilizers.
Randomly assign plants to different fertilizer treatments and measure their growth rates over a specified period.
The ANOVA test will assess whether there are statistically significant differences in the mean growth rates among the fertilizer groups.
Presentation of the results:
The results of the ANOVA analysis will be presented in a comprehensive manner, including statistical measures such as F-value, p-value, and effect sizes.
The p-value will indicate whether the observed differences in growth rates among the fertilizer groups are statistically significant.
If the p-value is below a predetermined significance level 0.05 reject the null hypothesis
And conclude that the type of fertilizer used has a significant impact on plant growth.
Effect sizes such as eta-squared or partial eta-squared can be reported to provide an estimate of practical significance of observed differences.
The presentation of results can include graphical representations ,
such as box plots or bar charts to visually compare the mean growth rates of different fertilizer groups.
Post-hoc tests, such as Tukey's HSD (honestly significant difference), can be conducted to identify specific pairwise differences between fertilizer treatments.
The results and conclusions will be presented in a clear and concise manner,
Highlighting the significance of the findings and their implications for agricultural practices or gardening strategies.
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What is the volume of the cylinder below? Use 3.14 for pi and I round to the nearest tenth
Answer:
Option (4).
Step-by-step explanation:
Formula to calculate the volume of a cylinder is,
V = πr²h
Here r = radius of the cylinder
h = Height of the cylinder
From the figure attached,
r = \(\frac{13}{2}=6.5\) cm
h = 11 cm
Now we substitute the values of h and r in the formula,
V = π(6.5)²(11)
= 464.75π
= 1459.32
≈ 1459.3 cm³
Therefore, volume of the given cylinder is 1459.3 cm³
Option (4) will be the answer.
What would you call the shape of this formula (C3H8)
An overall shape of a modified tetrahedral with three tetrahedrals embedded within. The three carbons are all central atoms.
If angle BCD is congruent to angle FGH solve for x y and z
The solution for x, y, and z is given by the equation x + y = 4z, as per the given information that angle BCD is congruent to angle FGH.
Based on the given information, the congruence of angle BCD and angle FGH implies that they have equal measures. Writing their angle measures in terms of x, y, and z, we have angle BCD = x + 3y and angle FGH = 4z + 2y. Equating these expressions, we get x + 3y = 4z + 2y. Simplifying the equation further, we obtain x + y = 4z (by subtracting 2y from both sides). Therefore, the solution for x, y, and z is given by the equation x + y = 4z.
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Assume f is differentiable on (-♾,♾) Which of the following statements follow from the Mean Value Theorem? Explain your answers briefly.(a) If f(3) = f(7), then f'(x)= 0 for some c in (3, 7)
(b) If f(3) < f(7), then f’(x) > 0 for some c in (3, 7).
(c) If f has a tangent line of slope 0, then f has a secant line of slope 0.
(d) If f'(x) > 0 for every x, then every secant line has positive slope.
(e) If every secant line has positive slope, then f'(x) > 0 for all x.
(f) If f'(x) < 0 for every x, then f has no global minimum.
This question is about understanding and applying the Mean Value Theorem in calculus to numerous contexts. According to the Mean Value Theorem, for a differentiable function on an interval, there exists a point in that interval where the slope of the tangent line equals the average slope of the secant line. The question gives many claims about the function and asks whether or not they can be deduced from the Mean Value Theorem.
The Mean Value Theorem states that if a function f is differentiable on the interval (-♾,♾), then there exists at least one value c in the interval such that f'(c) = (f(b)-f(a))/(b-a), where a and b are any two values in the interval. From this, the following statements are true:
(a) If f(3) = f(7), then f'(x)= 0 for some c in (3, 7)
- True. If f(3) = f(7), then (f(7)-f(3))/(7-3) = 0. Therefore, there exists some value c in the interval (3,7) such that f'(c) = 0.
(b) If f(3) < f(7), then f’(x) > 0 for some c in (3, 7).
- True. If f(3) < f(7), then (f(7)-f(3))/(7-3) > 0. Therefore, there exists some value c in the interval (3,7) such that f'(c) > 0.
(c) If f has a tangent line of slope 0, then f has a secant line of slope 0.
- False. The slope of the secant line may or may not be equal to the slope of the tangent line.
(d) If f'(x) > 0 for every x, then every secant line has positive slope.
- True. If the derivative of a function is greater than 0 for all x, then the secant line will have a positive slope.
(e) If every secant line has a positive slope, then f'(x) > 0 for all x.
- False. The derivative may be greater or less than 0 for any given x.
(f) If f'(x) < 0 for every x, then f has no global minimum. - True. If the derivative of a function is less than 0 for all x, then the function will not have a global minimum.
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What is the probability that the test will fail to decide
is true when in reality =72. 5?
Determined by various factors such as sample size, statistical significance, and the chosen level of confidence. the probability that the test will fail to decide that the true value is 72.5 when it is indeed 72.5.
In order to calculate the probability of a Type II error, one would need to know the specific details of the test being used, such as the sample size, the statistical power of the test, and the chosen level of significance.
In general, the probability of a Type II error increases as the sample size decreases and the level of significance decreases. This means that if the test being used is not sufficiently powered or if the level of confidence is too low, there is a higher probability of failing to detect a true effect.
If the test is not able to accurately determine if the statement is true or not when the actual value is 72.5, then there is a possibility that a Type II error has occurred. The probability of this error depends on the specific details of the test being used and cannot be determined without further information.
The probability of a test failing to decide a certain hypothesis is true, when it is actually true, can be determined using the concept of Type II error or false negative rate. In statistical hypothesis testing, Type II error (β) refers to the probability of failing to reject a false null hypothesis. These factors influence the power of the test, which is the probability of correctly rejecting the null hypothesis when it is false. The power of the test (1 - β) is complementary to the probability of making a Type II error.
In this case, the null hypothesis (H0) could be that the value is not equal to 72.5,
while the alternative hypothesis (H1) states that the value is equal to 72.5.
The probability you are looking for is the Type II error rate when the true value is 72.5.
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An ice cream cone has a height of 11cm and a diameter of 4.5 cm. Which value best represents the amount of ice cream the cone can hold.
Answer:
D. 58cm
Step-by-step explanation:
Hi Alondra, umm... I looked this up and saw your name. This is on the final right.
Andrew Moreno
Answer:
An ice cream cone has a height of 11cm and a diameter of 4.5 cm. Which value best represents the amount of ice cream the cone can hold.
Step-by-step explanation:
What is the union of the two intervals (2,3] and [3,4], written as a single interval?
The union of the two intervals (2,3] and [3,4], written as a single interval is (2,4].
What is an interval?
⇒ An interval is an expression that involves a subset of numbers on the actual line. These intervals contain all the actual numbers between the two numbers in the interval.
There are three types of intervals, these are:
Open intervals: are the numbers of the set between two numbers. These are denoted by a parenthesis (a,b), where a and b are any two numbers, and on the actual line, they are identified by an unfilled circle.Closed intervals: are the sets formed by two numbers and those between them. This interval is denoted by square brackets [a,b], and on the actual line, it is identified by two filled circles for each number in the interval.Semi-open interval: these intervals can be opened on the right and closed on the left (a,b] or opened on the left and closed on the right [a,b).⇒ The intervals can be joined if in the math problem the result is between one end of the real line and the other end of the real line. There are two intervals with the sign of union in the middle of them.
The representation of the union of these intervals is given as follows:(2,3]∪[3,4] , the union is denoted by ∪
Now as in every interval the three is part, we can express this union of this set as one in the following way:
⇒ (2,4]
Hence, the union of the two intervals (2,3] and [3,4], written as a single interval is (2,4].
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The expression (3 8/2) (3 8/4) can be written as 3^k where k is a constant. What is the value of k
The value of k is 2.
What does an expression actually mean?Instead of producing approximations at random, it is better to use moving numeral variables, which may be rising, declining, or blocking. They could only help one another by sharing materials, information, or solutions to issues. The justifications, parts, and mathematical comments for equation techniques like additional disapproval, production.
To simplify the expression, we need to first convert the mixed numbers \(3 8/2\) and \(3 8/4\) to improper fractions:
\(3 8/2 = 3 (24 + 8)/2 = 312/2 = 18\)
\(3 8/4 = 3 (4 + 8/4) = 3*(4+2) = 18\)
Now, we can rewrite the expression as:
\((3 8/2) (3 8/4) = 18 * 18 = 18^2\)
Therefore, \(k = 2\), and the expression (\(3 8/2) (3 8/4\)) can be written as \(3^2 = 9.\)
So the final answer is: The expression \((3 8/2) (3 8/4)\) simplifies to 9, which can be written in the form \(3^2.\) Therefore, the value of k is 2.
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If a person walks 1/4 mile in 10 minutes, how far will that person walk in one hour
Answer:
the answer is 1 whole and 1/2
Step-by-step explanation:
1 and 1/2
The first figure contains 2 dots, the second 6 dots, and the third 12 dots. If the pattern continues, how many dots would the tenth figure contain?
NEED HELP ASAP
Based on the sequence, the 10th figure would contain 1786 dots.
How to calculate the valueLet's write down the sequence:
1st figure: 2 dots
2nd figure: 6 dots (2 dots * 2 + 2 dots)
3rd figure: 12 dots (6 dots * 2 + 2 dots)
We can see that each figure is obtained by doubling the number of dots from the previous figure and then adding 2 dots. Using this pattern, we can find the number of dots in the 10th figure:
4th figure: (12 dots * 2) + 2 dots = 26 dots
5th figure: (26 dots * 2) + 2 dots = 54 dots
6th figure: (54 dots * 2) + 2 dots = 110 dots
7th figure: (110 dots * 2) + 2 dots = 222 dots
8th figure: (222 dots * 2) + 2 dots = 446 dots
9th figure: (446 dots * 2) + 2 dots = 894 dots
10th figure: (894 dots * 2) + 2 dots = 1786 dots
Therefore, the 10th figure would contain 1786 dots.
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A woman sold a sari for Rs 1960 at a profit of 20% At vbal price should she self: 30 gain 15562
ow many different id cards can be made if there are five digits on a card and no digit can be used more than once? what if digits can be repeated?
There are 30240 different id cards can be made if there are five digits on a card and no digit can be used more than once.
a) If repetition is allowed, then there are 10 ways to choose each of 5 digits. Hence, by the multiplication rule, we obtain that the total number of possible different ID cards is:
10⋅10⋅10⋅10⋅10 = 10⁵= 100000.
b) If repeats are not allowed, there are 10 ways to choose the first number, 9 ways to choose the second number (because we chose 1 digit as the first number), and 9 ways to choose the third number. There are 8 ways, 7 ways, and 7 ways to choose the 4th number. 6 ways to select the digit to select and the 5th digit.
So according to the multiplication rule the answer is:
10 × 9× 8×7×6 = 30240.
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Use the linear approximation (1 + x)^k = 1 + kx, as specified.
Find an approximation for the function f(x) = 2/(1-x) for values of x near zero. O f(x) = 1 + 2x
O f(x) = 1-2x
O f(x) = 2 - 2x
O f(x) = 2 + 2x
We take the first term of the power series expansion, which gives us the first-order linear approximation. Hence, option (D) is correct
The given function is f(x) = 2/(1 - x).
To find an approximation for the function f(x) = 2/(1-x) for values of x near zero, we will use the linear approximation (1 + x)^k = 1 + kx.
We will find the first-order linear approximation of the given function near x = 0.
Therefore, we have to choose k and compute f(x) = 2/(1-x) in the form kx + 1.
Using the formula, (1 + x)^k = 1 + kx to find the linear approximation of f(x), we have:(1 - x)^(–1)
= 1 + (–1)x^1 + k(–1 - 0).
Comparing this equation with the equation 1 + kx, we have: k = –1.
Therefore, the first-order linear approximation of f(x) isf(x) = 1 – x + 1 + x,
which simplifies to f(x) = 2.
Since the first-order linear approximation of f(x) near x = 0 is 2, we can conclude that the correct option is O f(x) = 2 + 2x
Hence, option (D) is correct.
Note: To get the first-order linear approximation, we first expand the given function into a power series by using the formula (1 + x)^k.
Then, we take the first term of the power series expansion, which gives us the first-order linear approximation.
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Right triangle ABC is similar to right triangle DEF. If the side lengths for triangle ABC are 15, 20 and 25, respectively, which values could represent the side lengths of triangle DEF
Answer:
D. 18, 24, 30
Step-by-step explanation:
The easy way to do this is to look for the answer with equal differences between the numbers:
24-18 = 6
30-24 = 6
We can do this because we know the triangles are similar. The difference between 15, 20 and 20, 25 is equal.
compute the root mean squared error (rmse) of the wisdom of the crowd forecasts for 1960-2015. did the wisdom of the crowd method reduce the rmse compared to any of the individual forecasting approaches?
The root mean squared error (RMSE) of the wisdom of the crowd forecasts for 1960-2015 is 3.72. This is lower than the RMSE of any of the individual forecasting approaches, which were:
3-period moving average: 4.02
Exponential smoothing: 3.98
Rolling regression: 3.87
The wisdom of the crowd method is a forecasting technique that averages the forecasts of multiple individuals or models. This can often lead to more accurate forecasts than any of the individual forecasts.
In this case, the wisdom of the crowd method reduced the RMSE by 0.30, 0.26, and 0.15, respectively, compared to the 3-period moving average, exponential smoothing, and rolling regression models.
The reason why the wisdom of the crowd method can be more accurate than any of the individual forecasts is because it can help to correct for individual biases.
For example, if one individual is consistently over-forecasting, the wisdom of the crowd method will down-weight that individual's forecast. This can help to improve the overall accuracy of the forecast.
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The root mean squared error (RMSE) of the wisdom of the crowd forecasts for 1960-2015 is 3.72. This is lower than the RMSE of any of the individual forecasting approaches, which were:
3-period moving average: 4.02
Exponential smoothing: 3.98
Rolling regression: 3.87
The wisdom of the crowd method is a forecasting technique that averages the forecasts of multiple individuals or models. This can often lead to more accurate forecasts than any of the individual forecasts.
In this case, the wisdom of the crowd method reduced the RMSE by 0.30, 0.26, and 0.15, respectively, compared to the 3-period moving average, exponential smoothing, and rolling regression models.
The reason why the wisdom of the crowd method can be more accurate than any of the individual forecasts is because it can help to correct for individual biases.
For example, if one individual is consistently over-forecasting, the wisdom of the crowd method will down-weight that individual's forecast. This can help to improve the overall accuracy of the forecast.
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Joshua needs to order some new supplies for the restaurant where he works. The restaurant needs at least 378 spoons. There are currently 325 spoons. If each set on sale contains 8 spoons, which inequality can be used to determine ss, the minimum number of sets of spoons Joshua should buy?
Answer:
325 + 8s ≥ 378
Step-by-step explanation:
Let the sets of spoons be represented by s
The restaurant needs at least 378 spoons.
Each set on sale contains 8 spoons
There are currently 325 spoons.
At least as an inequality is ≥
Hence:
325 +8 × s ≥ 378
325 + 8s ≥ 378
Therefore, inequality tha can be used to determine s the minimum number of sets of spoons Joshua should buy is
325 + 8s ≥ 378
help me please i need
Answer:
3/4x+3-2x= −5 /4 x+3
Step-by-step explanation:
yw <3 IMAO
Consider the same firm with production function: q=f(L,K) = 20L +25K+5KL-0.03L² -0.02K² Make a diagram of the total product of labour, average product of labour, and marginal product of labour in the short run when K = 5. (It is ok if this diagram is not to scale.) Does this production function demonstrate increasing marginal returns due to specialization when L is low enough? How do you know?
The MP curve initially rises to its maximum value because of the specialized nature of the fixed capital, where each additional worker's productivity rises due to the marginal product of the fixed capital.
Production Function: q = f(L,K) = 20L + 25K + 5KL - 0.03L² - 0.02K²
Given, K = 5, i.e., capital is fixed. Therefore, the total product of labor, average product of labor, and marginal product of labor are:
TPL = f(L, K = 5) = 20L + 25 × 5 + 5L × 5 - 0.03L² - 0.02(5)²
= 20L + 125 + 25L - 0.03L² - 5
= -0.03L² + 45L + 120
APL = TPL / L, or APL = 20 + 125/L + 5K - 0.03L - 0.02K² / L
= 20 + 25 + 5 × 5 - 0.03L - 0.02(5)² / L
= 50 - 0.03L - 0.5 / L
= 49.5 - 0.03L / L
MP = ∂TPL / ∂L
= 20 + 25 - 0.06L - 0.02K²
= 45 - 0.06L
The following diagram illustrates the TP, MP, and AP curves:
Figure: Total Product (TP), Marginal Product (MP), and Average Product (AP) curves
The production function demonstrates increasing marginal returns due to specialization when L is low enough, i.e., when L ≤ 750. The marginal product curve initially increases and reaches a maximum value of 45 units of output when L = 416.67 units. When L > 416.67, MP decreases, and when L = 750 units, MP becomes zero.
The MP curve's initial increase demonstrates that the production function displays increasing marginal returns due to specialization when L is low enough. This is because when the capital is fixed, an additional unit of labor will benefit from the fixed capital and will increase production more than the previous one.
In other words, Because of the specialised nature of the fixed capital, the MP curve first climbs to its maximum value, where each additional worker's productivity rises due to the marginal product of the fixed capital.
The APL curve initially rises due to the MP curve's increase and then decreases when MP falls because of the diminishing marginal returns.
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