Answer:
a. a × b > 0 ∀ a,b ∈ R : a,b < 0
b. a - a = 0 ∀ a ∈ R
Step-by-step explanation:
a. Let a and b be the numbers. Since it says product of two numbers is greater than zero, we write a × b > 0. Since a and b are real numbers, we write a,b ∈ R where ∈ denotes element of a set and R is the set of real numbers. We then use the connective ∀ which denotes "for all" to join a × b > 0 with a,b ∈ R. So, we write a × b > 0 ∀ a,b ∈ R. Since a and b are negative, we write a,b < 0. We now use the connective : which denotes "such that" to combine a × b > 0 ∀ a,b ∈ R with a,b < 0 to give a × b > 0 ∀ a,b ∈ R : a,b < 0. So, the expression is
a × b > 0 ∀ a,b ∈ R : a,b < 0
b. Let a be the number. Since we are looking for a difference, we write a - a. Since it is equal to zero, we write a - a = 0. Since a is an element of real numbers,R, we write a ∈ R, where ∈ denotes "element of". So, a ∈ R denotes a is an element of real numbers R. We combine these two expressions with the connective ∀ which denotes "for all" to give a - a = 0 ∀ a ∈ R. So, the expression is
a - a = 0 ∀ a ∈ R
The mathematical statements given are translated into logical expressions with predicates, quantifiers, logical connectives, and mathematical operators. The first statement is written as ∀x∀y ((x < 0 ∧ y < 0) → P(x, y)), and the second as ∀x (D(x, x)).
The mathematical statements can be expressed using predicates, quantifiers, logical connectives, and mathematical operators as follows:
For the statement 'The product of two negative real numbers is positive':
Let the predicate P(x, y) represent 'the product of x and y is positive', where x and y are real numbers. Then, the statement can be written as:
∀x∀y ((x < 0 ∧ y < 0) → P(x, y))
This translates to: 'For all real numbers x and y, if x and y are both negative, then the product of x and y is positive.'.
For the statement 'The difference of a real number and itself is zero':
Let D(x, y) be a predicate that states 'the difference of x and y is zero'. Then, the statement can be formulated as:
∀x (D(x, x))
Which reads as: 'For all real numbers x, the difference of x and itself is zero.'.
Therefore, The first statement is written as ∀x∀y ((x < 0 ∧ y < 0) → P(x, y)), and the second as ∀x (D(x, x)).
Write an expression for the number or floors the building can have for a given building height. Tell what variable in your expression represents. (Each floor will be 12 feet tall)
Final answer:
To calculate the number of floors a building can have for a given height, use the formula 'Number of Floors = h / 12', where 'h' is the total building height in feet and each floor is 12 feet tall.
Explanation:
To write an expression for the number of floors a building can have for a given building height, where each floor is 12 feet tall, we use the following formula:
Number of Floors = Total Building Height (in feet) / Height of One Floor (in feet)
Let 'h' represent the total height of the building in feet. Then the expression becomes:
Number of Floors = h / 12
For example, if a building has a height of 384 feet, the number of floors can be calculated as 384 feet / 12 feet per floor = 32 floors.
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Suppose a school has 20 classes: 16 with 25 students in each, three with 100 students in each, and one with 300 students for a total of 1000 students. (a) Suppose one of the 20 classes is chosen at random, let Y be the number of students in that class. Write out the p.m.f. for Y and use it to calculate the average class size at this school. (b) Suppose that one of the 1000 students is chosen at random. Let X be the size of the class to which the student belongs. Write out the p.m.f. of X and find E[X]. (c) Fill in the following: On average, a student at this school is in a class with students. On average, a class at this school has students. Comment
Answer:
Step-by-step explanation:
Given that a school has 20 classes: 16 with 25 students in each, three with 100 students in each, and one with 300 students for a total of 1000 students.
a) Y = no of students in the class out of 20 selected at random
Y can take values as
Y 25 100 300 Total
freq 16 3 1 20
P(Y) 16/20 3/20 1/20 1
y*p 20 15 15 50
E(Y) = 50
b) X = size of student from which one student is taken at random
X 25 100 300 Total
No of classes 16 3 1 20
Total students
(X*no of classes) 400 300 300 1000
P(x) 0.4 0.3 0.3 1
X*P 160 90 90 240
E(x) =240
c) On an average, a student at this school is in a class with 240 students. On average, a class at this school has 50 students.
i.e. expectation of number of students in a class is 50 while expectation of a student having students in the class is 240
Christy purchased 6.75 pounds of red licorice and 2.37 pounds of black licorice. How much licorice does she need to put in a bag if she divides the total amount into 10 equal-sized bags?
Answer: The amount she puts in a bag = 0.912 pound.
Step-by-step explanation:
Given : Amount of red licorice purchased by Christy = 6.75 pounds
Amount of black licorice purchased by Christy = 2.37 pounds
Total amount of licorice purchased by Christy = 6.75 + 2.37 pounds
= 9.12 pounds
Then, if she divides the total amount into 10 equal-sized bags , the amount she put in a bag = (Total amount of licorice purchased) ÷ 10
= 9.12 ÷ 10 = 0.912 pound
Hence, the amount she puts in a bag = 0.912 pound.
Final answer:
To find how much licorice Christy needs to put into each of 10 bags, add the weights of red and black licorice and then divide by 10. She will need to put 0.912 pounds of licorice into each bag.
Explanation:
Christy purchased 6.75 pounds of red licorice and 2.37 pounds of black licorice. To find the total amount of licorice, we add the weight of the red and black licorice together:
Total weight of licorice = [tex]6.75 pounds + 2.37 pounds = 9.12 pounds.[/tex]
If she divides the total amount into 10 equal-sized bags, we divide the total weight by 10:
Weight per bag = [tex]9.12 pounds / 10 bags = 0.912 pounds per bag.[/tex]
Aaron and Blake both wrote down equation. Their equation has the same slope, But Aaron's y-intercept was negative and Blake's y-intercept was positive. How many solutions for their system of equations have? Explain your reasoning
Answer:
Parallel lines do not intersect therefore the system of equations cannot have any solution. Therefore there no solutions to the given system of equations.
Step-by-step explanation:
Aaron and Blake both wrote down equation. Their equation has the same slope, But Aaron's y-intercept was negative and Blake's y-intercept was positive.
Therefore both equations will represent lines that are parallel.
Parallel lines do not intersect therefore the system of equations cannot have any solution.
Peaceful Travel Agency offers vacation packages. Each vacation package includes a city, a month, and an airline. The agency has cities, month, and airlines to choose from. How many different vacation packages do they offer?
Answer:12 packages.
Step-by-step explanation:
The agency has 2 cities,1 month and 6 airlines to choose from.(completion of the question)
Total package of the agency = 2cities×1 month × 6 airlines=12 packages.
Officials project that between 201020102010 and 205020502050, the population of a certain city will drastically change. The model below gives the projection of the city's population , PPP, in thousands, with respect to time, ttt, in years, where 201020102010 corresponds to t=0t=0t, equals, 0. \quad P = 175 + \dfrac{11}{2}tP=175+ 2 11 tP, equals, 175, plus, start fraction, 11, divided by, 2, end fraction, t What does the 175175175 mean in the equation?
Answer:
175 means the city's population is 175 thousands in 2010.
Step-by-step explanation:
The model below gives the projection of the city's population , P, in thousands, with respect to time, t, in years, where 2010 corresponds to t=0.
The given equation is
[tex]P=175+\dfrac{11}{2}t[/tex]
We need to interpret 175 for the given model.
Substitute t=0 in the above equation.
[tex]P=175+\dfrac{11}{2}(0)[/tex]
[tex]P=175[/tex]
At t=0 the value of P is 175 it means the city's population is 175 thousands in 2010.
The questions on a multiple choice test each have 4 answer choices. Describe a model that you could use to simulate the outcome of guessing tje correct answers to a 50 question test
Answer:
The outcome of guessing the correct answers to a 50 question test is 12.5.
Step-by-step explanation:
Let us first consider one question.
Since it has 4 options so we can say that;
Total number of possible outcomes = 4
Since 1 answer is correct so we can say that;
Favorable outcome = 1
Now we can say that;
Probability for choosing correct answer can be calculated by Favorable outcome divided by Total number of possible outcomes.
framing in equation form we get;
[tex]P(Correct\ Answer) = \frac{1}{4}[/tex]
Now Given:
Total number of question = 50
So we can say that;
Total number of correct answer out of 50 is equal to Probability for choosing correct answer multiplied by Total number of question.
framing in equation form we get;
Guessing number of correct answer out of 50 = [tex]\frac14 \times 50 = 12.5[/tex]
Hence The outcome of guessing the correct answers to a 50 question test is 12.5.
You can use a binomial distribution model to simulate guessing the correct answers on a 50-question test with 4 answer choices per question. The probability of getting at least 30 correct answers.
Explanation:To simulate the outcome of guessing the correct answers to a 50 question test, you can use a binomial distribution model. A binomial distribution represents the probability of a certain number of successful outcomes in a fixed number of independent trials. In this case, the success would be guessing the correct answer.
Each question has 4 answer choices, so the probability of guessing the correct answer for each question is 1/4. The number of successful guesses in a 50-question test can follow a binomial distribution with parameters n = 50 (number of trials) and p = 1/4 (probability of success).
Using this model, you can calculate probabilities of various outcomes, such as the probability of getting exactly 20 correct answers by guessing
Hence, the probability of getting at least 30 correct answers.
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Suppose Rick has 40 ft of fencing with which he can build a rectangular garden.Letxrepresent the length of the garden and letyrepresent the width.(a) Write and inequality representing the fact that the total perimeter of thegarden is at most 40 ft.(b) Sketch part of the solution set for this inequality that represents all possiblevalues for the length and with of the garden. (Hint:Note that both the lengthand the width must be positive.)
Answer:
20 >= x + y
Step-by-step explanation:
Given:
- The length of garden = x
- The width of the garden = y
- Total fence available = 40 ft
- Rectangular garden
Find:
(a) Write and inequality representing the fact that the total perimeter of thegarden is at most 40 ft.
(b) Sketch part of the solution set for this inequality that represents all possiblevalues for the length and with of the garden.
Solution:
- The perimeter of the rectangular garden is P at most 40 ft:
P >= 2*x + 2*y
40 >= 2*x + 2*y
20 >= x + y
- The sketch of the graph will be all points in the shaded region denoted by the inequality as follows:
y =< 20 - x
- See the triangular shaded region.
Answer:
Step-by-step explanation:
Answer:
Step-by-step explanation:
Since Rick has 40ft of fencing
Then, the perimeter cannot be more than 40ft if he decided to lay the block on a single layer and not on each other
Then if the length is x
And the breadth is y
Perimeter of a rectangle is 2(l+b)
Therefore,
Perimeter is less than or equal to 40ft
2(l+b) ≤ 40
b. 2(l+b)≤ 40
Then divide both side by 2
l+b≤ 20.
Then l ≤ 20-b
Also, b ≤ 20-l
Check attachment for graph
A quality control engineer at a potato chip company tests the bag filling machine by weighing bags of potato chips. Not every bag contains exactly the same weight. But if more than 15% of bags are over-filled then they stop production to fix the machine.
They define over-filled to be more than 1 ounce above the weight on the package. The engineer weighs 100 bags and finds that 21 of them are over-filled.
He plans to test the hypotheses: H0: p = 0.15 versus Ha: p > 0.15 (where p is the true proportion of overfilled bags).
What is the test statistic?
Z = 1.68
Z = -1.68
Z = 4
Z = -1.47
Answer:
Option A) 1.68
Step-by-step explanation:
We are given the following in the question:
Sample size, n = 100
p = 15% = 0.15
Number of bags overfilled , x = 21
First, we design the null and the alternate hypothesis
[tex]H_{0}: p = 0.15\\H_A: p > 0.15[/tex]
Formula:
[tex]\hat{p} = \dfrac{x}{n} = \dfrac{21}{100} = 0.21[/tex]
[tex]z = \dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}[/tex]
Putting values, we get,
[tex]z = \dfrac{0.21-0.15}{\sqrt{\dfrac{0.15(1-0.15)}{100}}}\\\\z = 1.68[/tex]
Thus, the correct answer is
Option A) 1.68
To test the hypothesis H0: p = 0.15 versus Ha: p > 0.15, the test statistic is Z = 1.68.
Explanation:To test the hypothesis H0: p = 0.15 versus Ha: p > 0.15, where p is the true proportion of overfilled bags, we can use the test statistic Z. The formula for the test statistic is Z = (P' - p) / √(p(1-p) / n), where P' is the sample proportion, p is the hypothesized proportion, and n is the sample size.
In this case, the sample proportion P' = 21/100 = 0.21, the hypothesized proportion p = 0.15, and the sample size n = 100.
Plugging these values into the formula, we get Z = (0.21 - 0.15) / √(0.15(1-0.15) / 100) = 1.68.
Therefore, the test statistic is Z = 1.68.
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A number that indicates the degree and direction of the relationship between variables is called a(n) _____.
Answer: The answer is Correlation because its a connection between 2 or more things, or interdependence of variable quantities. Plus its in the definition, it says the process of establishing a relationship or connection between 2 or more measures.
When dragons on planet Pern lay eggs, the eggs are either green or yellow. The biologists have observed over the years that 32% of the eggs are yellow, and the rest green. Next spring the lead scientist has permission to randomly select 60 of the dragon eggs to incubate. Consider all the possible samples of 60 dragon eggs.What is the usual number of yellow eggs in samples of 60 eggs? (Give answers as SENSIBLE whole numbers.)minimum usual number of yellow eggs =maximum usual number of yellow eggs
Answer:
19
Step-by-step explanation:
If the scientist chooses 60 random eggs then the overall probability of selecting yellow eggs is 32%. So, there's a chance of he randomly collecting 60 eggs out of which 19 could be yellow.
(60/100) x 32 ~ 19...
The usual number of yellow dragon eggs expected in a sample of 60, given a 32% probability, is 19 when rounded to the nearest whole number.
The question at hand involves finding the expected number of yellow eggs in a sample size of 60, given a known probability of a yellow egg occurrence.
Since we know that 32% of the dragon eggs on planet Pern are yellow, we can calculate the usual number of yellow eggs in the samples by multiplying the sample size (60 eggs) by the probability of getting a yellow egg (0.32).
Therefore, the expected or usual number of yellow eggs in the sample would be 60 × 0.32 = 19.2.
Considering that we cannot have a fraction of an egg, the usual number of yellow eggs in samples of 60 eggs would typically be rounded to the nearest whole number, which is 19.
The side of the base of a square prism is decreasing at a rate of 7 kilometers per minute and the height of the prism is increasing at a rate of 10 kilometers per minute. At a certain instant, the base's side is 4 kilometers and the height is 9 kilometers.
The volume of the square prism with a side of base decreasing at a rate of 7 km/min and height increasing at 10 km/min is decreasing at a rate of 344 cubic km/min.
Explanation:In this problem, we're dealing with rates of change and the geometric properties of a square prism. A square prism can be characterized by the side length of its base (we'll call this length 's') and its height (h). The volume (V) of a square prism is given by the equation V = s^2 * h. We're told that the side length s is decreasing at a rate of 7 kilometers per minute (ds/dt = -7 km/min) and the height h is increasing at a rate of 10 kilometers per minute (dh/dt = 10 km/min).
To find the rate at which the volume is changing with respect to time, we can take the derivative of the volume equation with respect to time. Thus, dV/dt = 2*s*ds/dt*h + s^2*dh/dt. Substituting the given values, when s = 4 km and h = 9 km, we find: dV/dt = 2*4*(-7)*9 + 4^2*10 = -504 + 160 = -344 cubic kilometers per minute. Therefore, at the given instant, the volume of the square prism is decreasing at a rate of 344 cubic kilometers per minute.
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The rate of change of the surface area of the prism at that instant is -204 square kilometers per minute. Correct Option is Option C.
To solve this problem, let's first start by identifying the formula for the surface area (SA) of a square prism, which is given as [tex]2s^2 + 4sh[/tex] , where s is the side of the base and h is the height.
Now, let's find the rate of change of the surface area with respect to time, dSA/dt. We'll use the chain rule to differentiate the surface area formula:
1. Differentiate the given surface area formula:
[tex]SA = 2s^2 + 4sh[/tex]
[tex]d(SA)/dt = d(2s^2)/dt + d(4sh)/dt[/tex]
2. Apply the chain rule:
d(SA)/dt = 2 * 2s * (ds/dt) + 4 * (s * (dh/dt) + h * (ds/dt))
3. Plug in the given values: s = 4 km, ds/dt = -7 km/min, h = 9 km, dh/dt = 10 km/min:
d(SA)/dt = 4 * 4 * (-7) + 4 * (4 * 10 + 9 * (-7))
d(SA)/dt = -112 + 4 * (40 - 63)
d(SA)/dt = -112 + 4 * (-23)
d(SA)/dt = -112 - 92
d(SA)/dt = -204
The rate of change of the surface area of the prism at that instant is -204 square kilometers per minute. Correct Option is Option C.
Complete Question:- The side of the base of a square prism is decreasing at a rate of 7 kilometers per minute and the height of the prism is increasing at a rate of 10 kilometers per minute. At a certain instant, the base's side is 4 kilometers and the height is 9 kilometers. What is the rate of change of the surface area of the prism at that instant (in square kilometers per minute)? Choose 1 answer: A 204 B 148 C -204 D -148 The surface area of a square prism with base side 8 and height h is [tex]2s^2+4sh.[/tex]
In a survey of 545 human resource professionals, each was asked about the importance of the age of a job applicant. The survey subjects were randomly selected by pollsters from a reputable market research firm.
Answer:
The sampling method described in this question appears to be sound
Step-by-step explanation:
It appears to be sound because the data are not biased in any way, since every member and set of members has an equal chance of being included in the sample selection. Also, random samples are usually fairly representative since they don't favor certain members.
The point A (8, 2) is reflected over the line x = 5, and then is reflected over the line x = -1. What are the coordinates of A'?
Answer:
A'(8, -10)
Step-by-step explanation:
Firstly we have to know the distance from the point A to the line, the point is (8, 2) and the line is x = 5 .
the point has two components, at x and y
the component in y will give us the distance we have on the axis of the X thas is 2.
T he line is 5 units from the axis.
Now to know the distance we have to subtract
2 - 5 = -3
Here is the point with respect to the line x = 5
For the reflected over the line we just need to change the sign, now we got +3 . To this we add the value of the line that is 5
3 + 5 = 8.
So the reflected point over the line x = 5 is (8, 8)
We do the same with the line x = -1.
(8, 8) and x = -1
..
8 - (-1) = 9
..
(-9) + (-1) = -10
(8, -10)
(9x90)+(8x11)+(4x5) if someone can help please
Answer:
918
Step-by-step explanation:
Use distributive property.
(810) + (88) + (20) = 918
Answer:
918
Step-by-step explanation: hope it was helpful
Express tan(23°−21°) in terms of tangents of 23∘and 21∘ You do NOT need to type in the degree symbol. Be sure to PREVIEW your answer before submitting!
Answer:
tan(23° -21°) = (tan(23°) -tan(21°))/(1 +tan(23°)tan(21°))
Step-by-step explanation:
The formula for the tangent of the difference of angles is ...
tan(a-b) = (tan(a) -tan(b))/(1 +tan(a)tan(b))
Filling in the values a=23° and b=21°, you get the formula shown above.
The expression for tan(23°−21°) in terms of the tangents of 23° and 21° is: tan(23°−21°) = (tan 23° - tan 21°) / (1 + tan 23° * tan 21°).
To express tan(23°−21°) in terms of the tangents of 23° and 21°, we can use the angle subtraction formula for tangent, which is:
tan(α - β) = (tan α - tan β) / (1 + tan α * tan β)
Applying this to tan(23°−21°), we get:
tan(23°−21°) = (tan 23° - tan 21°) / (1 + tan 23° * tan 21°)
We can't directly calculate the values from the given table since 23° and 21° are not listed. However, we understand the formula required to express tan(23°−21°) in terms of the tangents of these two angles.
Mary purchased an annuity that pays her $500 per month for the rest of her life. She paid $70,000 for the annuity. Based on IRS annuity tables, Mary's life expectancy is 16 years. How much of the first $500 payment will Mary include in her gross income (round to two decimals)?
Answer: $135.42
Step-by-step explanation:
Based on IRS tables, Mary is expected to receive 192 (16 years x 12 months) annuity payments. Her investment in the annuity is $70,000 and her return of capital for each annuity payment is $70,000/192 = $364.58. The return of capital portion of each annuity payment is not taxable (not included in gross income). Mary must include the excess received ($500.00 – 364.58) of $135.42 in her gross income.
To calculate how much of the first $500 payment Mary will include in her gross income, we need to determine the exclusion ratio. The exclusion ratio is the ratio of the investment in the annuity to the expected return. In this case, Mary will include $36.45 of the first $500 payment in her gross income.
Explanation:To determine how much of the first $500 payment Mary will include in her gross income, we need to calculate the exclusion ratio. The exclusion ratio is the ratio of the investment in the annuity to the expected return. In this case, Mary paid $70,000 for the annuity and her life expectancy is 16 years, so the exclusion ratio is $70,000 / ($500/month * 12 months/year * 16 years) = 0.0729. To determine the amount she will include in her gross income, we multiply the monthly payment by the exclusion ratio: $500 * 0.0729 = $36.45. Therefore, Mary will include $36.45 of the first $500 payment in her gross income.
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If cos(θ) = 6/8 and θ is in the IV quadrant, then fine:
(a) tan(θ)cot(θ)
(b) csc(θ)tan(θ)
(c) sin^2(θ) + cos^2(θ)
Answer:
a) 1
b) [tex]\frac{4}{3}[/tex]
c) = 1
Step-by-step explanation:
We are given the following in the question:
[tex]\cos \theta = \dfrac{6}{8}[/tex]
θ is in the IV quadrant.
[tex]\sin^2 \theta + \cos^2 \theta = 1\\\\\sin \theta = \sqrt{1-\dfrac{36}{64}} = -\dfrac{2\sqrt7}{8}\\\\\tan \theta = \dfrac{\sin \theta}{\cos \theta} = -\dfrac{2\sqrt7}{6}\\\\\csc \theta = \dfrac{1}{\sin \theta} = -\dfrac{8}{2\sqrt7}[/tex]
Evaluate the following:
a)
[tex]\tan \theta\times \cot \theta =\tan \theta\times\dfrac{1}{\tan \theta} = 1[/tex]
b)
[tex]\csc \theta\times \tan \theta\\\\= -\dfrac{8}{2\sqrt7}\times -\dfrac{2\sqrt7}{6} = \dfrac{4}{3}[/tex]
c)
[tex]\sin^2 \theta + \cos^2 \theta = 1\\\text{using the trignometric identity}[/tex]
The admission fee at an amusement park is $1.50 for children and $4.00 for adults. On a certain day, 304 people entered the park, and the admission fees collected totaled $856. Let x be the number of children that were admitted and let y be the number of adults that were admitted.?
Answer:
x = 144 children
y = 160 adults
Step-by-step explanation:
We have parameters already defined. Number of children that entered is x while number of admitted adults is y.
Since the total number of people admitted was 304, this means:
x + y = 304
Total fees for that day was 856
Mathematically:
1.5x + 4y = 856
From the first equation, we can say y = 304 - x
We then substitute this into the second equation:
1.5x + 4(304 - x) = 856
1.5x + 1216 - 4x = 856
1216-856 = 4x-1.5x
2.5x = 360
x = 360/2.5 = 144
Since y = 304 - 144
y = 160
The number of children admitted was 144 and the number of adult is 160
The Commerce Department reported receiving the following applications for the Malcolm Baldrige National Quality Award: 23 from large manufacturing firms, 18 from large service firms, and 30 from small businesses.
1. Is type of business a qualitative or quantitative variable?
2. What percentage of the applications came from small businesses (to 1 decimal)?
Answer:
Categorical42.25%Step-by-step explanation:
CATEGORICAL, because the type of business will not be numerical.30 of the 71 (23+18+30) businesses are small businesses.30/71 = 0.4225 = 42.25%
How many kilocalories are in a chicken quesadilla that has 28 grams of protein, 40 grams of carbohydrate, and 30 grams of fat? Group of answer choices
Final answer:
The chicken quesadilla contains a total of 542 kilocalories, which is derived by summing the caloric contributions from protein, carbohydrate, and fat.
Explanation:
To calculate the number of kilocalories in a chicken quesadilla with 28 grams of protein, 40 grams of carbohydrate, and 30 grams of fat, we use the known caloric values for each macronutrient. Protein and carbohydrate each have 4 Calories per gram, while fat has 9 Calories per gram.
Protein: 28 grams × 4 Calories/gram = 112 Calories
Carbohydrate: 40 grams × 4 Calories/gram = 160 Calories
Fat: 30 grams × 9 Calories/gram = 270 Calories
Adding up these amounts, we have:
112 Calories (from protein) + 160 Calories (from carbohydrate) + 270 Calories (from fat) = 542 Calories
To convert Calories into kilocalories, we remember that 1 kilocalorie = 1 Calorie. Therefore, the chicken quesadilla contains 542 kilocalories.
You and your friend are going to some candy. You eat 3/4 of a box of candy. Your friend eats 1/2 as much as you do. How much of a box of candy does your friend eat?
Answer:
[tex]\frac{3}{8}[/tex] of a box of candy your friend eat.
Step-by-step explanation:
Given:
You and your friend are going to some candy.
You eat 3/4 of a box of candy.
Your friend eats 1/2 as much as you do.
Now, to find that how much of a box of candy does your friend eat.
Quantity of a box of candy you eat = [tex]\frac{3}{4}[/tex]
Quantity of candy your friend eat = [tex]\frac{1}{2} \ of\ \frac{3}{4}[/tex]
Now, to get how much the candy your friend eat we multiply 1/2 and 3/4:
= [tex]\frac{1}{2} \times \frac{3}{4}[/tex]
= [tex]\frac{3}{8}[/tex] .
Therefore, [tex]\frac{3}{8}[/tex] of a box of candy your friend eat.
On average, employees at a particular company historically missed 4 days of work per year. Then there was a management change. Now the average is 7 days. What is the percent of increase in the average number of missed days?
Answer:
75% increase
Step-by-step explanation:
Given:
Number of missed days of work before the management change = 4
Number of missed days of work after the management change = 7
So, increase in the number of missed days of work is given by subtracting the old number from the new number and is equal to:
Increase in missed number of days = 7 - 4 = 3 days
Now, the percent of this increase is calculated by dividing the increase by the old number and then multiplying the result by 100.
Percent of increase is given as:
[tex]Percent\ of\ increase=\frac{Increased\ value}{Old\ value}\times 100\\\\Percent\ of\ increase=\frac{3}{4}\times 100\\\\Percent\ of\ increase=0.75\times 100\\\\Percent\ of\ increase=75\%[/tex]
So, there is a 75% increase in the number of missed days of work after the change in the management.
3. Using techniques from Calculus, show directly that the maximum value of a 1-D Gaussian distribution occurs at the point x = μ.
Answer:
For a scaler variable, the Gaussian distribution has a probability density function of
p(x |µ, σ² ) = N(x; µ, σ² ) = 1 / 2π×[tex]e^{\frac{-(x-u)^{2}}{2s^{2} } }[/tex]
The term will have a maximum value at the top of the slope of the 1-D Gaussian distribution curve that is when exp(0) =1 or when x = µ
Step-by-step explanation:
Gaussian distributions have similar shape, with the mean controlling the location and the variance controls the dispersion
From the graph of the probability distribution function it is seen that the the peak is the point at which the slope = 0, where µ = 0 and σ² = 1 then solution for the peak = exponential function = 0 or x = µ
Please explain if you can
Answer:
Therefore,
[tex]x=9\\\\y=7[/tex]
Step-by-step explanation:
Given:
D,E,F are the midpoints of side AB, BC,and AC such that
AB = 4x - 18
BC = 2x - 4
DF = x
FE = y
To Find:
x = ?
y = ?
Solution:
Midpoint Theorem:
The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.
E and F are Midpoint of Sides BC and AC, then
[tex]FE=\dfrac{1}{2}AB[/tex] ......By Midpoint Theorem
Substituting the values we get
[tex]x=\dfrac{1}{2}(4x-18)=2x-9\\\\2x-x=9\\\\x=9[/tex]
Similarly,
D and Fare Midpoint of Sides AB and AC, then
[tex]DF=\dfrac{1}{2}BC[/tex] ......By Midpoint Theorem
Substituting the values we get
[tex]y=\dfrac{1}{2}(2x-4)=x-2\\\\Substitute\ x=9\\\\y=9-2=7[/tex]
Therefore,
[tex]x=9\\\\y=7[/tex]
A rectangle has width w inches and height h, where the width is twice the height. Both w and h are functions of time,t, measured in seconds. If A represents the area of the rectangle, what is the rate of change of A with respect to t at the instant where the width is 4 inches and the height is increasing at the rate of 2 inches per second?
Answer:
dA/dt = 16 square inches per second
Step-by-step explanation:
Width of rectangle is w
Height of rectangle is h
width = twice height
w = 2h
Area = wh = (2h)*h
A = 2h^2
Differentiating the equation with respect to time
dA/dt = 2+2h dh/dt
dA/dt = 4h dh/dt
According to the given situation when width is 4 inches
h = w/2
h = 2
Rate of change of A is
dA/dt = wh dh/dt
dA/dt = 4(2)(2)
dA/dt = 16 square inches per second
The rate of change of the area of the rectangle with respect to time at the specified instant is 16 square inches per second.
Explanation:The question asks for the rate of change of the area of a rectangle with respect to time, where the rectangle's width is twice its height, both width and height are functions of time and the rate at which height is increasing is given.
The area A of the rectangle is given by the formula A = w*h. Substituting w = 2h into the formula gives A = 2h². The rate of change of the area with respect to time (dA/dt) can be found by differentiating A with respect to time. By the chain rule this gives dA/dt = 2*2h(dh/dt) = 4h*dh/dt.
At the given instant, where the width is 4 inches (so the height is 2 inches) and the rate of change of the height is 2 inches per second, we have dA/dt = 4*2*2 = 16 square inches per second. Therefore, the rate of change of the area of the rectangle with respect to time at that instant is 16 square inches per second.
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A car with an initial cost of $23,000 is decreasing in value at a rate of 8% each year. Write the exponential decay function described in this situation. Then use your function to determine when the value of the car will be $15,000, to the nearest year.
Answer:
Step-by-step explanation:
We would apply the formula for exponential decay which is expressed as
A = P(1 - r/n)^ nt
Where
A represents the value after t years.
n represents the period for which the decrease in value is calculated
t represents the number of years.
P represents the value population.
r represents rate of decrease.
From the information given,
P = 23000
r = 8% = 8/100 = 0.08
n = 1
Therefore, the exponential decay function described in this situation is
A = 23000(1 - 0.08/n)1)^ 1 × t
A = 23000(0.92)^t
If A = 15000, then
15000 = 23000(0.92)^t
0.92^t = 15000/23000 = 0.6522
Taking log of both sides to base 10
Log 0.92^t = log 0.6522
tlog 0.92 = log 0.6522
- 0.036t = - 0.1856
t = - 0.1856/- 0.036
t = 5 years to the nearest year
The value of the car will be $15,000 after approximately 4 years.
To model the situation with an exponential decay function, we use the formula:
[tex]\[ V(t) = P \times (1 - r)^t \][/tex]
Given that the initial cost [tex]P[/tex] is $23,000 and the annual depreciation rate r is 8%, or 0.08 in decimal form, we can write the exponential decay function as:
[tex]\[ V(t) = 23000 \times (1 - 0.08)^t \][/tex]
[tex]\[ V(t) = 23000 \times (0.92)^t \][/tex]
To find out when the value of the car will be $15,000, we set [tex]V(t)[/tex] equal to $15,000 and solve for t.
[tex]\[ \frac{15000}{23000} = (0.92)^t \][/tex]
[tex]\[ 0.652173913043478 = (0.92)^t \][/tex]
Using a calculator, we find:
[tex]\[ t \approx \frac{\ln(0.652173913043478)}{\ln(0.92)} \approx 4.037 \][/tex]
Since we are looking for the time in whole years, we round to the nearest year:
[tex]\[ t \approx 4 \][/tex]
Therefore, the value of the car will be $15,000 after approximately 4 years.
The volume of a cylinder is increasing at a rate of 10π cubic meters per hour.
The height of the cylinder is fixed at 5 meters.
At a certain instant, the volume is 80π cubic meters.
What is the rate of change of the surface area of the cylinder at that instant (in square meters per hour)?
Answer:
13π/2 m²/h
Step-by-step explanation:
Volume of a cylinder is:
V = πr²h
If h is a constant, then taking derivative of V with respect to time:
dV/dt = 2πrh dr/dt
Surface area of a cylinder is:
A = 2πr² + 2πrh
Taking derivative with respect to time:
dA/dt = (4πr + 2πh) dr/dt
Given that dV/dt = 10π, V = 80π, and h = 5, we need to find dA/dt. But first, we need to find r and dr/dt.
V = πr²h
80π = πr² (5)
r = 4
dV/dt = 2πrh dr/dt
10π = 2π (4) (5) dr/dt
dr/dt = 1/4
dA/dt = (4πr + 2πh) dr/dt
dA/dt = (4π (4) + 2π (5)) (1/4)
dA/dt = 13π/2
The surface area of the cylinder is increasing at 13π/2 m²/h.
The required surface area of the cylinder is [tex]\frac{13}{2}[/tex] m²/h.
Given that,
The volume of the cylinder increase with the rate of 10π cubic meters per hour.
The height of the cylinder is fixed at 5 meters.
At a certain instant, the volume is 80π cubic meters.
We have to determine,
What is the rate of change of the surface area of the cylinder at that instant.
According to the question,
Height of cylinder = 5m
Volume of cylinder increase with rate = 10π cubic meters per hour.
At certain instant volume become = 80π cubic meters per hour.
Volume of cylinder is given as,
V = πr²h
Where h is a constant,
[tex]v = \pi r^{2} h\\\\80\pi = \pi r^{2}. (5)\\\\\frac{80\pi }{5\pi } = r^{2} \\\\r^{2} = 16\\\\r = 4[/tex]
Then, taking derivative of V with respect to time:
[tex]\frac{dv}{dt} = 2\pi rh.\frac{dr}{dt} \\\\[/tex]
Where, dv\dt = 10π, V = 80π, and h = 5,
Then,
[tex]10\pi = 2\pi (4)(5). \frac{dr}{dt} \\\\10\pi = 40\pi \frac{dr}{dt} \\\\\frac{10\pi }{40\pi } = \frac{dr}{dt} \\\\\\\frac{dr}{dt} = \frac{1}{4}[/tex]
Then,
Surface area of a cylinder is define as,
[tex]A = 2\pi r^{2} + 2\pi rh\\\\\ \frac{da}{dt} = (4\pi r + 2\pi h)\frac{dr}{dt} \\\\\frac{da}{dt} = (4\pi (4) + 2\pi (5))\times\frac{1}{4} \\\\\frac{da}{dt} = \frac{26\pi }{4} \\\\\frac{da}{dt} = \frac{13\pi }{2}[/tex]
Hence, The required surface area of the cylinder is [tex]\frac{13}{2}[/tex] m²/h.
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A spherical balloon holding 35 lbm of air on earth has a diameter of 10 ft. Determine (a) the specific volume, in unit ft3/lbm, and (b) the weight, in lbf, of the air within the balloon.
Answer:
a) Specific volume of the air in balloon is [tex]14.96 ft^3/lbm[/tex]
b)The weight of the air within the balloon is 1,126.09 lbf.
Step-by-step explanation:
Mass of air, m = 35 lbm
Volume of the air = V
Diameter of balloon = d = 10 ft
radius of the balloon = r= 0.5 d = 5 ft
Volume of balloon = V
[tex]V=\frac{4}{3}\pr r^3[/tex]
[tex]V=\frac{4}{3}\times 3.14\times (5 ft)^3[/tex]
Specific volume of the air in balloon = S
[tex]S=\frac{V}{m}=\frac{\frac{4}{3}\times 3.14\times (5 ft)^3}{35 lbm}[/tex]
[tex]S=14.96 ft^3/lbm[/tex]
Specific volume of the air in balloon is [tex]14.96 ft^3/lbm[/tex]
[tex]lbf=32.174 lbm ft/s^2[/tex]
Weight of the air = W
Acceleration due to gravity = [tex]32.174 lbm ft/s^2[/tex]
Weight = m\times g[/tex]
[tex]W=35 lbm\times 32.174 lbm ft/s^2[/tex]
[tex]W=1,126.09 lbf[/tex]
The weight of the air within the balloon is 1,126.09 lbf.
A construction worker leans his ladder against a building making a 60 degree angle with the ground. If the ladder is 20 feet long, how far away is the base or the ladder from the building? Round to the nearest tenth
Answer: the distance from the base of the ladder from the building is 10 feet.
Step-by-step explanation:
The ladder forms a right angle triangle with the wall of the building and the ground.
The length of the ladder represents the hypotenuse of the right angle triangle.
The distance, h of the base of the ladder from the building represents the adjacent side of the triangle.
To determine h, we would apply
the cosine trigonometric ratio.
Cos θ = adjacent side/hypotenuse. Therefore,
Cos 60 = h/20
h = 20Cos60 = 20 × 0.5
h = 10 feet
Final answer:
To solve for the distance from the base of the ladder to the building, the cosine function is used because the scenario forms a right-angled triangle with the ground and the wall. By the equation cos(60°) = Base / Hypotenuse, with the ladder being the hypotenuse, the base is calculated to be 10 feet from the building.
Explanation:
The question involves using trigonometry to determine the distance from the base of the ladder to the building. When a ladder makes a 60° angle with the ground and its length is given (20 feet), this forms a right-angled triangle with the ground and the building's wall. To find the distance from the base of the ladder to the building, we can use the cosine function, which relates the adjacent side to the hypotenuse in a right-angled triangle.
Using the equation:
cos(60°) = Base / Hypotenuse
Here, the hypotenuse is the length of the ladder which is 20 feet. Therefore:
cos(60°) = Base / 20 feet
cos(60°) equals 0.5 when calculating in degrees. Consequently:
0.5 = Base / 20 feet
Multiplying both sides by 20 gives:
Base = 20 feet * 0.5
Base = 10 feet
Thus, the base of the ladder is 10 feet from the building.