Let α be one root of the quadratic equation x² + sx + F = 0. If β is the other root, then we have: Therefore, the condition is that the sum of roots of the quadratic equation x² + sx + F = 0 must be zero.
Let α be one root of the quadratic equation x² + sx + F = 0. If β is the other root, then we have:
α + β = -s (sum of roots)
αβ = F (product of roots)
We are given that α is five times β, i.e., α = 5β. Substituting this in the first equation, we get:
5β + β = -s
6β = -s
β = -s/6
Using this value of β, we can find the corresponding value of α:
α = 5β = -5s/6
So, the condition for one root to be five times the other is:
α = 5β or -5s/6 = 5(-s/6)
s = 0
Therefore, the condition is that the sum of roots of the quadratic equation x² + sx + F = 0 must be zero.
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A shade of paint, Orange Mango, can be made by mixing red paint and
yellow paint in the ratio 2: 5. Teddy has 20 litres of red paint and
40 litres of yellow paint.
Work out the maximum volume of Orange Mango paint that Teddy can
make.
Give your answer in litres.
A shade of paint, Orange Mango, can be made by mixing red paint and yellow paint in a ratio of 2:5. Teddy has 20 litres of red paint and 40 litres of yellow paint.The maximum volume of Orange Mango paint that Teddy can make is 24 litres.
The most quantity of Orange Mango paint that Teddy can make, we need to discover the proscribing thing that stops him from the usage of all of his red and yellow paint. In this method, we need to evaluate the ratio of pink-yellow paint that he has with the ratio of crimson-to-yellow paint that he desires. Here is how we try this:
The ratio of crimson to yellow paint that he has is 20:40, which may be simplified with the aid of dividing each side by means of 20 to get 1:2.
The ratio of crimson to yellow paint that he needs is 2:5, which can be simplified by dividing each aspect by using the finest commonplace element of 2 and 5, that's 1, to get 2:5.
To compare those ratios, we are able to pass-multiply them and notice which aspect is greater. In this approach, we multiply the numerator of one ratio via the denominator of the opposite ratio and compare the effects. Here is how we try this:
1/2 compared to 2/5
(1×5) compared to (2×2)
5 in comparison to 4
We can see that five is larger than 4, this means that the ratio of pink to yellow paint that he has is greater than the ratio of purple to yellow paint that he desires. This means that he has extra crimson paint than he desires, and consequently, purple paint is the restricting component.
To find out how much pink paint he can use, we need to discover how an awful lot yellow paint he wishes for every litre of purple paint. We can do that by way of dividing the numerator and denominator of the ratio of pink to yellow paint that he needs with the aid of the numerator of that ratio. Here is how we do this:
2/5 ÷ 2/1 = 2/5 × 1/2 = 1/5
This manner that for every litre of purple paint, he needs 0.2 litres of yellow paint. Since he has 20 litres of pink paint, he can use it all and multiply it with the aid of 0.2 to discover how much yellow paint he wishes. Here is how we do that:
20 × 0.2 = 4
This means that he needs 4 litres of yellow paint. Since he has 40 litres of yellow paint, he has more than enough and may use the handiest 4 litres.
To discover the most volume of Orange Mango paint that Teddy could make, we need to add up the volumes of purple and yellow paint that he makes use of. Here is how we do this:
20 + 4 = 24
With this approach, he could make 24 litres of Orange Mango paint.
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The maximum volume of Orange Mango paint that Teddy can make is 56 liters.
We have,
The ratio of red paint to yellow paint required to make Orange Mango paint is 2:5.
This means for every 2 liters of red paint, 5 liters of yellow paint are needed.
Teddy has 20 liters of red paint and 40 liters of yellow paint.
To determine the maximum volume of Orange Mango paint Teddy can make, we need to find which paint he has in excess and how much of the other paint he needs to use to create the mixture.
Teddy has 20 liters of red paint, which is equivalent to 20/2.5 = 8 units of the mixture (since 2+5=7 units are needed to make the mixture).
Teddy has 40 liters of yellow paint, which is equivalent to 40/2.5 = 16 units of the mixture.
Since Teddy has more yellow paint, the amount of Orange Mango paint he can make is limited by the amount of red paint he has.
Thus,
Teddy can make a maximum of 8 units of Orange Mango paint, which is equivalent to 8 x 7 = 56 liters of the mixture.
Therefore,
The maximum volume of Orange Mango paint that Teddy can make is 56 liters.
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Which of the following is an accurate definition for the power of a statistical test ?
A the probability of supporting true null hypothesis
B the probability of supporting a false null hypothesis
C the probability of rejecting a false null hypothesis
D the probability of rejecting a true null hypothesis
Which of the following is an accurate definition for the power of a statistical test is C. The power of a statistical test refers to the probability of rejecting a false null hypothesis. This means that the test is able to correctly identify when there is a significant difference between two groups or conditions.
statistical tests are used to determine whether the results of an experiment are likely due to chance or whether there is a real effect. The null hypothesis states that there is no significant difference, while the alternative hypothesis states that there is. The power of a statistical test refers to the ability of the test to correctly reject the null hypothesis when it is false, and thus support the alternative hypothesis.
, the power of a statistical test is the probability of correctly rejecting a false null hypothesis, which indicates that there is a significant difference between two groups or conditions.
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a bag contains 5 red marbles, 6 blue marbles, 10 white marbles and 9 yellow marbles. you are asked to draw 5 marbles from the bag without replacement. what is the probability of drawing 3 or more white marbles?
the probability of drawing 3 or more white marbles when 5 marbles are drawn from the bag without replacement.
To calculate this probability, we can use the formula for the hypergeometric distribution, which is:
P(X ≥ 3) = [C(10,3) * C(10,2)] / C(30,5) + [C(10,4) * C(10,1)] / C(30,5) + [C(10,5)] / C(30,5)
Where C(n,r) is the number of ways to choose r items from a set of n items.
formula is that we need to consider three different cases: drawing exactly 3 white marbles, drawing exactly 4 white marbles, and drawing all 5 white marbles. For each case, we calculate the probability of drawing the required number of white marbles and the probability of drawing the remaining marbles (non-white marbles), and then multiply them together. Finally, we add up the probabilities for all three cases to get the total probability of drawing 3 or more white marbles.
Using the formula, we get:
P(X ≥ 3) = [(C(10,3) * C(20,2)) + (C(10,4) * C(20,1)) + C(10,5)] / C(30,5)
= [(120 * 190) + (210 * 20) + 252] / 142506
= 0.209
Therefore, the probability of drawing 3 or more white marbles when 5 marbles are drawn from the bag without replacement is 0.209.
using the hypergeometric distribution formula, we can calculate the probability of drawing 3 or more white marbles from a bag containing 5 red marbles, 6 blue marbles, 10 white marbles, and 9 yellow marbles when 5 marbles are drawn without replacement. The probability is 0.209.
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Un cubo tiene un área total de 25312
pulgadas cuadradas. ¿Cuál es el área de una cara del cubo en pulgadas cuadradas?
The area of the face of the cube given is 4225 in².
Given that a cube has a total surface area of 25312 in², we need to find the area of one face of the cube,
So,
TSA of a cube = 6 × side²
So,
6 × side² = 25312
side² = 4225
side ≈ 65
Hence the cube has a side length of 65 inches,
Now we know that the face of a cube is in shape of square so the area of the face = side²
= 4225 in²
Hence the area of the face of the cube given is 4225 in².
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The translated question is
A cube has a total area of 25312
square inches. What is the area of a cube face in square inches?
A direct variation includes the points (-20,-40) and (7,n) find n
A direct variation includes the points (-20,-40) and (7,n) ,The value of n is 14..
in a direct variation, two variables are related in a way that can be expressed as y = kx, where k is a constant of proportionality. in this case, we are given two points that are related by a direct variation:
(-20,-40) and (7,n)
to find the value of n, we need to first determine the value of k. we can do this by using the formula for a direct variation:
y = kx
using the first point (-20,-40), we can substitute x = -20 and y = -40 to get:
-40 = k(-20)
solving for k, we get:
k = -40/-20 = 2
now that we know k, we can use it to find n using the second point (7,n):
n = kx
n = 2(7)
n = 14
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Which form of payment can Courtney lose and NOT get back?
Answer: The form of payment that Courtney can lose and not get back would be cash.
Step-by-step explanation:
The reason why cash couldn't be returned back because it doesn't have a piece of U.S money wouldn't have a specific certified owner that they can be returned to.
Other forms of payment like credit cards, debit cards, and checks have specific bank information with the card holder's name on it so if Courtney actually lost one of those, she would be most likely to receive it back without any complicated verification process.
Therefore, Courtney would not be able to receive her cash back if she accidentally loses it. Hope this helps you with your Imagine Math homework or assignment!
-From Certified 5th Grade Honors Student
At a car rental office, 63% of the customers are men, and 37% are women. What is the probability that the customer will be a man who requests either a compact car or a luxury car?
The solution is: 3.7%, is the probability that the next customer will be a woman who requests a convertible.
Here, we have,
given that,
At a car rental office, 63% of the customers are men and 37% are women.
If 25% of people request a compact car, 37% request a full-sized, 10% request a convertible,16% request an SUV, and 12% request a luxury car
As per given
P(woman)= 37%
P(convertible) = 10%
So combined probability is:
P = 37%*10%
= 37%*0.1
= 3.7%
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complete question:
At a car rental office, 63% of the customers are men and 37% are women. If 25% of people request a compact car, 37% request a full-sized, 10% request a convertible,16% request an SUV, and 12% request a luxury car, what is the probability that the next customer will be a woman who requests a convertible?
A fish is hovering at –320 feet in relation to sea level. Then it descends 2 feet per second for 8 seconds. Brenda says that the new position of the fish in relation to sea level is represented by the equation below.
2 times 8 + (negative 320) = 336
Which explains whether Brenda is correct?
Brenda is correct because her answer represents 336 feet below sea level.
Brenda is correct because Negative 16 + (negative 320) = 336.
Brenda is not correct because the new position is 2 times 8 + (negative 320) = Negative 304.
Brenda is not correct because the equation should be Negative 2 times 8 + (negative 320) = negative 336.
The question presents a scenario where a fish is hovering at a certain depth below sea level, and then descends at a certain rate for a certain duration. So, Brenda is correct because her answer represents the new position of the fish in relation to sea level after the descent.
Brenda has provided an equation that represents the new position of the fish in relation to sea level after the descent. The equation is 2 times 8 + (negative 320) = 336. To determine if Brenda is correct, we need to understand the meaning of each term in the equation. The first term, 2 times 8, represents the distance the fish has descended, which is 16 feet (2 feet per second for 8 seconds).
The second term, negative 320, represents the initial depth of the fish below sea level. When we add these two terms together, we get the new depth of the fish below sea level, which is 336 feet.
Therefore, Brenda is correct because her answer represents the new position of the fish in relation to sea level after the descent. It is important to understand the meaning of each term in an equation to correctly interpret its result.
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think of a number. double the number. add $200$. divide the answer by $4$. subtract one-half the original number. what is the value of the result?
Finally, we need to subtract one-half the original number, which is $\frac{x}{2}$. So the final expression is:$$\frac{x+100}{2} - \frac{x}{2} = 50$$
1. Let's call the original number x.
2. Double the number: 2x
3. Add 200: 2x + 200
4. Divide the answer by 4: (2x + 200) / 4
5. Subtract one-half the original number: ((2x + 200) / 4) - (x / 2)
Now let's simplify the expression:
((2x + 200) / 4) - (x / 2)
= (2x/4 + 200/4) - (x / 2)
= (x/2 + 50) - (x / 2)
= x/2 - x/2 + 50
= 0 + 50
So, the value of the result is 50.
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in 2010, by 40 years of age, _____ percent of individuals had ever been married.
According to the U.S. Census Bureau, in 2010, by the age of 40, approximately 85% of individuals had ever been married.
This statistic is based on data from the American Community Survey, which is conducted by the U.S. Census Bureau. The survey includes questions about marital status and provides information on the percentage of individuals who have ever been married by various ages. The statistic that 85% of individuals had ever been married by the age of 40 suggests that marriage is still a common life experience in the United States. However, it's important to note that this statistic may be influenced by a variety of factors, including changes in societal attitudes towards marriage, cultural and religious practices, and economic factors. Additionally, it's important to consider that this statistic may not be representative of all populations or demographic groups, as marriage rates can vary significantly based on factors such as race, ethnicity, and income.
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1.1 1.2 1.3 1.4 1.5 1.6 prudence wants to paint the front of the house. She has two identical windows as well as a circular vent near the roof. Gable 1,2 m 0.6 m 1,8 m 0.5m 500 cm Vent Window 2,6 m Use Formulae, to answer the questions that follow. Area = L x B Area = 2 x r²(π = 3,142) Area = ²× b × h Convert 500 cm to m Calculate the area of one window. Calculate the area of the vent. Calculate the area of the rectangular part of the wall. Calculate the area of the gable. Prudence states that the area of the Wall that must pe painted is 14,42 m². Is this statement valid? Set your work out clearly. (PLEASE NOTE THAT THE VENT AND THE WINDOWS WILL NOT BE PAINTED)
The Area of the gable is 0.9 m²
Prudence's statement is not valid.
1. Convert 500 cm to meters:
Since 1 meter is equal to 100 centimeters,
So, 500 cm ÷ 100 = 5 meters
2. The formula for the area of a rectangle is Area = Length x Width.
Given the dimensions:
Length = 1.2 m
Width = 1.4 m
Area of one window = 1.2 m x 1.4 m = 1.68 m²
3. Calculate the area of the vent:
The formula for the area of a circle is Area = π x r².
Radius = (2.6 m) / 2 = 1.3 m
Area of the vent = 3.142 x (1.3 m)² ≈ 5.309 m²
4. Length = 1.1 m
Width = 1.6 m
Area of the rectangular part of the wall = 1.1 m x 1.6 m
= 1.76 m²
5. Calculate the area of the gable:
Given the dimensions:
Base = 0.6 m
Height = 1.8 m
Area of the gable = 0.5 m x 1.8 m = 0.9 m²
Prudence states that the area of the wall that must be painted is 14.42 m².
So, Area of the rectangular part of the wall + Area of the gable
= 1.76 m² + 0.9 m² = 2.66 m²
Therefore, Prudence's statement is not valid.
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If U is the set of the numbers on a 6-sided die and B is the set of numbers on a 6-sided die that are greater than 3, find B
The calculated elements in the set B is {4, 5, 6}
Finding the elements in the set BFrom the question, we have the following parameters that can be used in our computation:
U is the set of the numbers on a 6-sided die B is the set of numbers on a 6-sided die that are greater than 3The above means that
U = {1, 2, 3, 4, 5, 6}
The numbers greater than 3 in the above set are
Elements = 4, 5 and 6
This means that the elements in the set B is {4, 5, 6}
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pleasee help the method is either substitution, equal values, or elimination and PLEASE explain step by step
y = 0.25x + 3
x = -4y + 8
Answer:
x = -2 and y = 2.5
Step-by-step explanation:
using elimination method:
y = 0.25x + 3 call this equation '1'
x = -4y + 8
bring 4y on its own
4y = -x + 8 call this equation '2'
multiply '1' by 4:
4y = x + 12 call this equation '3'
add '2' and '3'
8y = (-x + x) + 20
8y = 20
y = 20/8 = 2.5.
go back to '1'
we have y = 0.25x + 3
2.5 = 0.25x + 3
subtract 3 from both sides
2.5 - 3 = 0.25x
-0.5 = 0.25x
multiply both sides by 4
-2 = x
x = -2
now put both y =2.5 and x = -2 into '2' to see if all is fine
4y = -x + 8
4(2.5) = -(-2) + 8
10 = +2 + 8
10 = 10
so x = -2 and y = 2.5
the researchers noticed that the total number of registered users appears to be increasing at about a constant rate. if this pattern continues, which of the following best approximates the total number of registered users, in millions, in year 12 (two years after the last entry in the table) ? responses 30.6 30.6 31.2 31.2 31.8 31.8 32.4
The best approximate for the total number of registered users in year 12 would be B. 31. 2 million.
How to find the approximate ?First, we can find the differences over the years in the total number of registered users.
Year 4 = 26. 9 - 26. 5 = 0. 4
Year 6 = 28.0 - 27. 4 = 0. 6
Year 8 = 29. 1 - 28. 6 = 0. 5
Year 10 = 30. 2 - 29. 6 = 0. 6
This shows that the largest increase has been by 0. 6 in a year. Assuming that there are 0. 6 increases over two years till Year 12, the total number becomes :
= 30. 2 + 0. 6 + 0. 6
= 31. 4
The closest option that is lower is 31. 2 million users so this is the best approximate.
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What is the area of the polygon given below?
19
12
2
2
3
Answer: 271 square units
prove that any graph of minimum degree at least three contains a cycle of even length.
Answer:
a cycle is a sequence of non-repeated vertices and the degree of a graph is the number of neighbors the vertex has.
17. Nikki rides her scooter a distance of
64.75 feet in 7 seconds. Assume the
relationship between distance and
time is proportional.
Part A
Write an equation to represent the
distance traveled, y, after x seconds.
Express your answer in the form
y = kx.
An equation to represent the distance traveled, y, after x seconds is y = 9.25x.
What is a proportional relationship?In Mathematics, a proportional relationship produces equivalent ratios and it can be modeled or represented by the following mathematical equation:
y = kx
Where:
k is the constant of proportionality.y represent the distance.x represent the time.Next, we would determine the constant of proportionality (k) based on the data points provided as follows:
Constant of proportionality, k = y/x
Constant of proportionality, k = 64.75/7
Constant of proportionality, k = 9.25.
Therefore, the required equation is given by;
y = kx
y = 9.25x
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The figure shows quadrilateral PQRS. What additional information do you need in order to conclude that △SPR≅△QRP by the ASA Triangle Congruence Theorem? Explain.
The additional information for the figure is
angle SRP ≅ angle QPRHow to find the additional informationFirst lets look at ASA congruence theorem
The ASA Triangle Congruence Theorem also known as the Angle-Side-Angle Congruence Theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
In the figure the included side is PR and one to the angle required that is given is angle SPR ≅ angle PRQ
The additional information will be angle SRP ≅ angle QPR
hence we have that
Angle = angle SPR ≅ angle PRQ
Side = PR ≅ RP
Angle = angle SRP ≅ angle QPR
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Choose the INCORRECT statement below.
For a car moving at a constant speed, the distance traveled varies directly with the time spent driving. If such a car travels 330 miles in 6 hours, how long does it take to travel 385 miles?
It takes 7 hours to travel 385 miles at a constant speed.
To solve this problem, we need to use the direct variation formula, which states that distance traveled is directly proportional to time spent driving. This means that if we double the time spent driving, the distance traveled will also double.
Using this formula, we can set up a proportion to find the answer. We know that the car travels 330 miles in 6 hours, so we can write:
330/6 = 385/x
Where x is the time it takes to travel 385 miles.
To solve for x, we can cross-multiply and simplify:
330x = 2310
x = 7
It's important to note that this problem assumes that the car maintains a constant speed throughout the entire trip. If the car speeds up or slows down at any point, the distance traveled and time spent driving will not follow a direct variation relationship.
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probability of choosing 3 men and 1 woman at random for a 4 person committee if there are 10 men and 6 women to choose from?
The probability of choosing 3 men and 1 woman at random for a 4 person committee can be calculated using the combination formula.
We want to choose 3 men from a pool of 10 men, which can be done in 10 choose 3 ways. We also want to choose 1 woman from a pool of 6 women, which can be done in 6 choose 1 ways. The total number of ways to choose a 4 person committee from a pool of 16 people is 16 choose 4. Therefore, the probability of choosing 3 men and 1 woman at random for a 4 person committee is:
(10 choose 3) * (6 choose 1) / (16 choose 4) = 120 * 6 / 1820 = 0.3956
So, the probability of choosing 3 men and 1 woman at random for a 4 person committee is approximately 0.3956 or 39.56%. This means that if we randomly choose a 4 person committee from a pool of 10 men and 6 women, there is a 39.56% chance that the committee will consist of 3 men and 1 woman.
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Hi can someone who is great at math please help me with this math question. I’m really struggling with it. Find the area of the middle square. And show the steps
The area of the middle square should be = 36.
How to calculate the area of the middle square?To calculate the area of the middle square, the formula for the area of a square should be used an sit is given below:
Area of a square = a²
where;
a = Length of sides
But,the length of sides of a square are equal therefore;
Area of square = 6×6 = 36
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Y’all please help I need an explanation and an answer
8 cups = 64cm
From the catalog, we know that a stack of 2 cups is 16 cm tall, and a stack of 4 cups is 20 cm tall.
To find the height of a stack of 8 cups, we can use proportionality.
Let x be the height, in cm, of a stack of 8 cups. Then, we can set up the following proportion:
2 cups / 16 cm = 8 cups / x cm
Cross-multiplying, we get:
2 cups * x cm = 16 cm * 8 cups
Simplifying, we get:
2x = 128
x = 64 cm
Therefore, the height of a stack of 8 cups is 64 cm.
a magazine editor uses a data informed approach to determine which freelance writers should be offered a more permanent position. they take a pool of recommended freelance writers and determine that at least 80% of their work has been accepted for publication over a 2 year period. the editor considers 50 candidates for hire. of the 50 candidates, 20 of them have not been accepted for publication at an acceptable rate. with this data we test the following hypotheses at a 5% significance level: h0: the proportion of accepted publications of writers being recommended for hire is equal to .8 ha: the proportion of accepted publications of writers being recommended for hire is less than .8 the p-value is .01. what should the editor conclude?
The test is conducted at a 5% significance level, and the p-value is 0.01. Since the p-value (0.01) is less than the significance level (0.05), the editor should reject the null hypothesis (H0) and conclude that the proportion of accepted publications of writers being recommended for hire is less than 0.8.
Based on the data and the hypothesis testing, the magazine editor should conclude that the proportion of accepted publications of writers being recommended for hire is less than 0.8. This is because the p-value is 0.01, which is less than the significance level of 0.05. This means that there is strong evidence to reject the null hypothesis (h0) and accept the alternative hypothesis (ha). Therefore, the editor should not offer a more permanent position to the 20 candidates who have not been accepted for publication at an acceptable rate. Instead, they should focus on the remaining 30 candidates who have a higher proportion of accepted publications and are more likely to meet the standards of the magazine. By using a data-informed approach, the editor can make more objective and effective hiring decisions and ensure that the quality of the magazine remains high.
The magazine editor is using a data-informed approach to evaluate freelance writers for a more permanent position. They consider 50 candidates, and out of those, 20 have not been accepted for publication at an acceptable rate (80% acceptance over 2 years). The hypotheses being tested are:
H0: The proportion of accepted publications of writers being recommended for hire is equal to 0.8.
Ha: The proportion of accepted publications of writers being recommended for hire is less than 0.8.
This suggests that the editor may need to refine their selection process to improve the quality of the freelance candidates they consider for permanent positions.
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Can someone help on this please ? Thank you:)
Answer:
[tex] {2}^{ \frac{2}{7} } \times {2}^{ \frac{2}{7} } = {2}^{ \frac{2}{7} + \frac{2}{7} } [/tex]
[tex] {2}^{ \frac{2}{7} } \times {2}^{ \frac{2}{7} } = {2}^{ \frac{4}{7} } [/tex]
Is it growth or decay?
The function y = 14(1.49)ˣ is categorized as growth with 49% percentage increase
Categorizing the functions as decay or growthFrom the question, we have the following parameters that can be used in our computation:
y = 14(1.49)ˣ
An exponential function is represented as
y = abˣ
If b > 1, then it is a growth function
Otherwise, it a decay function
Using the above as a guide, we have the following:
y = 14(1.49)ˣ is a growth function because 1.49 is greater than 1
The percentage increase is then calculated as
percentage increase = 1.49 - 1
percentage increase = 0.49
Express as percentage
percentage increase = 49%
Hence, the percentage increase is 49%
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A map of an amusement park is shown on the coordinate plane with the approximate location of several rides.
coordinate plane with points at negative 14 comma 1 labeled Woozy Wheel, negative 6 comma 2 labeled Bumper Boats, negative 2 comma negative 4 labeled Roller Rail, negative 2 comma negative 6 labeled Trolley Train, 2 comma negative 3 labeled Silly Slide, and 6 comma 11 labeled Parachute Plunge
Determine the distance between the Woozy Wheel and the Roller Rail.
119 units
11 units
169 units
13 units
The distance between the Woozy Wheel and the Roller Rail is 13 units.
To determine the distance between the Woozy Wheel and the Roller Rail, we can use the distance formula in the coordinate plane.
The coordinates of the Woozy Wheel are (-14, 1), and the coordinates of the Roller Rail are (-2, -4).
Using the distance formula, the distance (d) between two points (x₁, y₁) and (x₂, y₂) is given by:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
Substituting the coordinates of the Woozy Wheel and the Roller Rail into the formula:
d = √((-2 - (-14))² + (-4 - 1)²)
= √(12² + (-5)²)
= √(144 + 25)
= √169
= 13
Therefore, the distance between the Woozy Wheel and the Roller Rail is 13 units.
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A curve is described by the parametric equations x = t^2 + 2t and y = t^3 + t^2. An equation of the line tangent to the curve at the point determined by t = 1 is...?
To find the equation of the tangent line, we need to find the derivative of both x and y with respect to t and evaluate them at t = 1 to get the slope of the tangent line. The equation of the line tangent to the curve at the point determined by t = 1 is y = (5/4)(x - 3) + 2.
dx/dt = 2t + 2
dy/dt = 3t^2 + 2t
At t = 1, dx/dt = 4 and dy/dt = 5
So the slope of the tangent line is m = dy/dx = (dy/dt)/(dx/dt) = 5/4
Now we need to find the point on the curve where t = 1.
x = t^2 + 2t = 1^2 + 2(1) = 3
y = t^3 + t^2 = 1^3 + 1^2 = 2
Therefore, the point on the curve where t = 1 is (3, 2)
Using the point-slope form of a line, we can write the equation of the tangent line as:
y - 2 = (5/4)(x - 3)
Simplifying:
y = (5/4)x - (7/2)
So the equation of the line tangent to the curve at the point determined by t = 1 is y = (5/4)x - (7/2).
A curve is described by the parametric equations x = t^2 + 2t and y = t^3 + t^2. An equation of the line tangent to the curve at the point determined by t = 1 is given by the formula y = m(x - x1) + y1, where m is the slope, and (x1, y1) is the point of tangency.
First, find the point of tangency (x1, y1) by plugging t = 1 into the parametric equations:
x1 = (1)^2 + 2(1) = 3
y1 = (1)^3 + (1)^2 = 2
Next, find the derivatives of x and y with respect to t:
dx/dt = 2t + 2
dy/dt = 3t^2 + 2t
Now, find the slope m by dividing dy/dt by dx/dt at t = 1:
m = (dy/dt) / (dx/dt) = (3(1)^2 + 2(1)) / (2(1) + 2) = (5/4)
Finally, plug the slope m and point (x1, y1) into the equation of the tangent line:
y = (5/4)(x - 3) + 2
So, the equation of the line tangent to the curve at the point determined by t = 1 is y = (5/4)(x - 3) + 2.
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C
2 mm
9 mm
What is the length of the hypotenuse? If
necessary, round to the nearest tenth.
C =
millimeters
The value of the length of the hypotenuse is,
⇒ c = √85
Since, Three vertices and three angles totaling 180 degrees make up a triangle, a three-sided polygon with three sides.
And, Two rays (half-lines) that share a terminal make up an angle. The rays serve as the angle's sides, occasionally serving as the angle's legs and occasionally serving as its arms, while the latter is referred to as the vertex of the angle.
We have to given that;
In triangle ,
Perpendicular side = 9 mm
Base = 2 mm
Since, We know that;
The Pythagorean theorem states that the square of the hypotenuse of a right triangle equals the sum of the squares of its two opposite sides.
Hence, By Pythagoras theorem we get;
⇒ Hypotenuse² = Perpendicular² + Base²
⇒ c² = 9² + 2²
⇒ c² = 81 + 4
⇒ c = √85
Thus, The length of the hypotenuse is,
⇒ c = √85
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a teacher of saba school wants to estimate the distance students travel to school. the sample mean and population standard deviation are found to be 8 km and 2.25 km, respectively. how large a sample should be selected to be 99% confident that the sample mean differs from the population mean by /- 0.5 km (e) only?
The sample size should be selected as 100 in order to be 99% confident that the sample mean differs from the population mean by ±0.5 km (or within a range of 0.5 km).
To determine the sample size needed to estimate the distance students travel to school with a 99% confidence level and a margin of error of ±0.5 km, we can use the formula for sample size calculation:
n = (Z * σ / E)^2
where:
n = required sample size
Z = Z-score corresponding to the desired confidence level (in this case, 99% confidence level, which corresponds to Z = 2.576)
σ = population standard deviation
E = margin of error
Plugging in the given values:
n = (2.576 * 2.25 / 0.5)^2
Calculating this equation gives us the required sample size:
n ≈ 99.98
Since we cannot have a fractional sample size, we round up to the nearest whole number.
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