It is not always possible for the hunter to ensure that the distance between her and the rabbit is at most 100 after $10⁹$ rounds. The distance between the hunter and the rabbit can increase by a factor of 2 in each round, making it impossible for the hunter to catch the rabbit in some scenarios.
It is not always possible for the hunter to ensure that the distance between her and the rabbit is at most 100 after $10⁹$ rounds, regardless of how the rabbit moves or what points are reported by the tracking device. This is because the distance between the hunter and the rabbit can increase by a factor of 2 in each round, which means that after $10⁹$ rounds, the distance between them could be as much as $2⁽¹⁰⁾⁹$ times the initial distance. This value could be much larger than 100, making it impossible for the hunter to catch the rabbit. Therefore, there are scenarios in which the hunter cannot catch the rabbit within $10⁹$ rounds.
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2. Find the number of ways in which 9 different computer games can be shared out between Wendy, Jim and Hannah so that each person receives an odd number of computer games.
Using combinatorics, there are 1926 ways to distribute 9 different computer games among Wendy, Jim, and Hannah so that each person receives an odd number of games.
What is the number of ways the game is shared between themIf each person receives an odd number of games, then the number of games must be 1, 3, 5, or 7. There are four odd numbers to choose from.
We can use combinatorics to find the odd number in each turn.
Let's consider each odd number in turn.
If each person gets 1 game, then we are looking for the number of ways to choose 3 games out of 9 to distribute. This is simply the number of combinations of 3 things taken from a set of 9 things, which is 9C3, or (9!)/(3!6!) = 84.
If each person gets 3 games, then we are looking for the number of ways to choose 3 games out of 9 to distribute to one person. Once those games have been chosen, we are left with 6 games to distribute to the other two people. The number of ways to choose 3 games out of 9 is again 84, but we must multiply this by the number of ways to choose 3 games out of the remaining 6. This is 6C3, or (6!)/(3!3!) = 20. Therefore, the total number of ways to distribute 9 games with each person getting 3 games is 84*20 = 1680.
If each person gets 5 games, then we are looking for the number of ways to choose 5 games out of 9 to distribute to one person. Once those games have been chosen, we are left with 4 games to distribute to the other two people. The number of ways to choose 5 games out of 9 is 126, and the number of ways to choose 4 games out of the remaining 4 is 1. Therefore, the total number of ways to distribute 9 games with each person getting 5 games is 126*1 = 126.
If each person gets 7 games, then we are looking for the number of ways to choose 7 games out of 9 to distribute to one person. Once those games have been chosen, we are left with 2 games to distribute to the other two people. The number of ways to choose 7 games out of 9 is 36, and the number of ways to choose 2 games out of the remaining 2 is 1. Therefore, the total number of ways to distribute 9 games with each person getting 7 games is 36*1 = 36.
Adding up the number of ways to distribute the games in each case, we get 84 + 1680 + 126 + 36 = 1926.
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Gerald has markers and pens in his backpack. He has a total of 78. If he has 11 more markers and pens, how many markers does he have?
(Systems of equation)
The required number of markers does Gerald has 56 markers.
How to find markers does he have?Let's represent the number of markers that Gerald has with the variable "m", and the number of pens he has with the variable "p". We can set up two equations based on the information given:
According to question:m + p = 78 (equation 1)
(m + 11) + (p + 11) = 78 (equation 2)
In equation 2, we add 11 to both the number of markers and the number of pens, since Gerald has 11 more of each. We can simplify equation 2 by combining like terms:
m + p + 22 = 78
m + p = 56 (equation 3)
Now we have two equations with two variables. We can use either substitution or elimination to solve for one of the variables. Let's use substitution and solve for "m" in terms of "p" using equation 1:
m + p = 78
m = 78 - p
Now we can substitute this expression for "m" into equation 3:
m + p = 56
(78 - p) + p = 56
Simplifying the equation, we get:
78 - p + p = 56
78 = 56 + p
p = 22
Therefore, Gerald has 22 pens. To find the number of markers he has, we can substitute this value of "p" into equation 1:
m + 22 = 78
m = 56
So, Gerald has 56 markers.
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5) There are 7,213 applicants Only one half of the applicants which is about will be accepted in the program for the astroment progogiem
In a case whereby there are 7213 applicants for the astronaut program only one half of the applicants which is about 3607 applicants will be accepted into the program.
How can the number of the applicants be determined?The concept that will be used here is multiplication. One of the four fundamental mathematical operations, along with addition, subtraction, and division, is multiplication. Multiply in mathematics refers to the continual addition of sets of identical size.
The symbols for multiplication are cross (), asterisk (*), and dot (). You seem to utilize the cross a lot when you write in your notebooks. In both algebra and computer languages, the asterisk and dot are symbols (higher mathematics).
Total applicants =7213 applicants
Then one half of the applicants = 1/2 of 7213
=7213/2
= 3607 applicants
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complete question:
There are 7213 applicants for the astronaut program only one half of the applicants which is about ______will be accepted into the program
Find the cost per pound of a house blend of coffee that is made from 10 lb of Central American coffee that costs $9 per pound and 40 lb of South American coffee that costs $5.50 per pound.
The cost per pound of a house blend of coffee is $ 5 per pound.
What is the cost price?
The precise value that corresponds to the unit price paid is represented by the cost price. In some stock market theories, this value is utilized to establish the value of stock holdings and is used as a key determinant of profitability.
Here, we have
Weight of Central American coffee = 10 lb
Cost of Central American coffee = $7 per pound
Total value of Central American coffee = weight × cost = 10 ×7 = $ 70
Weight of South American coffee = 40 lb
Cost of South American coffee = $ 4.50 per pound
Total value of Central American coffee = weight × cost = 40 ×4.50 = $ 180
Total weight of house blend coffee = Weight of Central American coffee + weight of South American coffee
= 10 +40 = 50 lb
The total value of house blend coffee = value of Central American coffee + value of South American coffee
= $ 70 + $ 180 = $ 250
Cost per pound of house blend coffee = ( total value of house blend coffee) / (total weight of house blend coffee)
= 250 / 50 = $ 5 per pound
Hence, the cost per pound of a house blend of coffee is $ 5 per pound.
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Calculate the correct dosage to be administered. Order: medication K 15 mg, intravenously
Supply: Medication K 10 mg/ 0. 5 ml
Give: ? ml
The correct dosage to be administered is 5 ml.
To calculate the correct dosage to be administered, we can use the following formula:
Dose = (Desired dose / Stock strength) x Stock volume
In this case, the desired dose is 15 mg and the stock strength is 10 mg/0.5 ml. Therefore:
Dose = (15 mg / 10 mg/0.5 ml) x Stock volume
Simplifying this expression, we get:
Dose = 3 x Stock volume
To find the stock volume needed, we can rearrange this formula as follows:
Stock volume = Dose / 3
Substituting the values we have, we get:
Stock volume = 15 mg / 3 = 5 ml
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A group consisting of 32 aggressive zombies quadruples in size every hour. Which equation matches the number of zombies after 6 hours?
Answer:
131,072 Zombies:
the equation: 32 x 4^6 = 131,072
the equation in scientific notation form: 3.2 x 4^7 = 131,072
Step-by-step explanation:
The amount of zombies starts at 32. It quadruples per hour, so we need to multiply the amount by 4 six times(once for every hour).
We can use an exponent to represent this in an equation:
32 x 4^6 = 131,072
Or, if we put it in scientific notation,
3.2 X 4^7 = 131,072.
The equation which represents the number of zombies is;
= [tex]1.31072[/tex]×[tex]10^{5}[/tex]
What is an equation?A mathematical statement known as an equation is made up of two expressions joined together by the equal sign.
Given that
The amount of zombies = 32.
And It quadruples per hour,
Therefore,
Multiply the amount by 4 six times (as it occurs once for every hour).
The the exponent form to represent this equation:
32 x 4^6 = 131,072
Hence, in scientific notation the equation is,
= [tex]1.31072[/tex]×[tex]10^{5}[/tex]
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a sample of 900 computer chips revealed that 43% of the chips do not fail in the first 1000 hours of their use. the company's promotional literature states that 40% of the chips do not fail in the first 1000 hours of their use. the quality control manager wants to test the claim that the actual percentage that do not fail is different from the stated percentage. is there enough evidence at the 0.05 level to support the manager's claim?
Answer:
I think a is the right answer
Help please Hurry
Based on the data, which of the following best describes the experimental probability of the arrow landing on a black or white section of the spinner? (1 point)
• a
O b
• c
• d
The spinner is equally likely to land on black or white.
The spinner is 3 times more likely to land on white.
The information cannot be concluded from the data.
The spinner is 3 times more likely to land on black.
Answer:
it's d. answer d. 57/76 is the probability of landing on black. that is roughly 3 times as much as the probability of landing in white 19/76. 19 X 3 = 60 which is roughly around the ballpark of 57.
The axis of symmetry for the graph of a parabola is x = 3. If one point on the graph of the parabola is (0, -5), then another point on the graph is:
A. (6,-5)
B. (0, 2)
C. (3.-5)
D. (-5.0)
Answer:
A
Step-by-step explanation:
For any given point on a parabola,
(x,y)
If we know the axis of symmetry x=a.
We can find the other point
If the given point x value is less than the axis of symmetry
(x+2a,y)
if the given point y value is greater than axis of symmetry
(x-2a,y)
Here, since 0<3, we use the first formula to find the new point
(0+2(3),-5)=(6,-5)
So the point is A.
who will be in your sample? when, where, and how will you survey those people? describe the procedure you will use to target respondents.
My sample will include people who are older. I will survey those people through an online survey that I will send out email or post
I will use targeted ads on social media platforms ,I will also use search engine optimization (SEO)to ensure that my survey is seen by the right people. The survey will be conducted over a two weeks. In order to ensure that I get a representative sample of people, I will use demographic and psychographic targeting. This will include targeting people based on their age, gender, location, interests, and other relevant characteristics. My sample will include people who are older. I will survey those people through an online survey that I will send out email or post
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How do you prove that between any two irrationals there is a rational number and vice versa?
To prove that there exists a rational number between any two irrational numbers, we can use the fact that there exists a rational number between any two real numbers.
Let a and b be two irrational numbers with the property that a b. Then a and b are not rational by definition, which means they cannot be written in the form p/q, where p and q are integers and q is not equal to zero.
We can define a new number c = (a + b) / 2 because a b. This number is a rational number because it can be expressed as a fraction in lowest terms as (a + b) / 2.
As a result, we have demonstrated that a rational number c exists between any two irrational numbers a and b.
A similar argument can be used to demonstrate that there exists an irrational number between any two rational numbers.
Consider a and b to be two rational numbers such that a b. We can then define a new number c = (a + b) / 2.
If c is also rational, we're done because we can just use c as our desired number between a and b. However, if c is irrational, we have found our desired number between a and b, which is also irrational.
Assume that a and b are both even integers to see why c must be irrational in some cases. Then their average c is also even, implying that c can be written as p/q, where p and q are integers and q is not equal to zero, with both p and q being even. However, because p and q are both even, this contradicts the fact that they must have no common factors.
As a result, we have demonstrated that there exists an irrational number c between any two rational numbers a and b.
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Anthony makes 6 1/8 gallons of lemonade for a family picnic. His family drinks 2 1/4 gallons of the lemonade. Which of these containers is the SMALLEST Anthony can use to store the remaining lemonade?
Answer:
3 and 7/8 gallons remaining
Step-by-step explanation:
You did not give the container options, but hopefully this will help
Which of the following services will be provided by a full-service broker but not by a discount broker?
I. Research of potential investment opportunities
II. Purchase and sale of stock at your request
III. Recommendation of investments
a.
I and III
b.
II only
c.
III only
d.
I, II, and III
Please select the best answer from the choices provided
A
B
C
D
Option A i.e. the first and the third service will be provided by a full-service broker but not by a discount broker.
What is meant by a broker?
An impartial party whose services are often used in several businesses is a broker. The main duty of a broker is to connect buyers and sellers; as a result, the broker acts as a neutral intermediary between a buyer and a seller.
The services which will be provided by a full-service broker but not by a discount broker are
Option I. Research of potential investment opportunities
Option III. Recommendation of investments
For any form of investment, research and recommendations are key and important components. Before you make any investments, you should conduct adequate research.
Hence, option A i.e. the first and the third service will be provided by a full-service broker but not by a discount broker.
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Please help me by finding the degree measure of angle x!
On solving the provided question, we cans ay that in this triangle by Pythagorean theorem [tex]A^2 = B^2 + C^2[/tex] => AC = [tex]\sqrt{115}[/tex]
What is triangle?A triangle is a polygon since it has three sides and three vertices. It is one of the basic geometric shapes. The name given to a triangle containing the vertices A, B, and C is Triangle ABC. A unique plane and triangle in Euclidean geometry are discovered when the three points are not collinear. Three sides and three corners define a triangle as a polygon. The triangle's corners are defined as the locations where the three sides converge. 180 degrees is the result of multiplying three triangle angles.
here, in this triangle by Pythagorean theorem
[tex]A^2 = B^2 + C^2[/tex]
AC = [tex]\sqrt{14^2 - 9^2}[/tex]
AC = [tex]\sqrt{196 - 81}[/tex]
AC = [tex]\sqrt{115}[/tex]
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Which pair of ratios can form a true proportion? I would usually give 45 points but i have 35 soo ye..
A. 9/12, 5/8
B. 25/20, 16/21
C. 6/27, 2/9
D. 7/12, 10/15
Pair of ratios can form a true proportion I would usually give 45 points but i have 35 soo ye 25/20, 16/21.
What is ratios?When b does not equal 0, an ordered pair of numbers a and b, represented as a / b, is said to be a ratio. Two ratios are set to be equal in an equation called a proportion. You might express the ratio as 1: 3 for instance, if there is 1 boy and 3 girls (for every one boy there are 3 girls) A ratio in mathematics displays the multiplicative relationship between two numbers. For instance, if there are eight oranges and six lemons in a dish of fruit, the ratio of oranges to lemons is eight to six. Lemons to oranges are divided 6:8 and oranges to all other fruits are divided 8:14 in a similar fashion.To learn more about ratios refer to:
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tyrone drew a logo and a dilation of the same logo on the coordinate plane. describe the dilation. then write a coordinate rule to represent it.
Dilation in coordinate geometry refers to the process of increasing or stretching the size of the originally drawn geometrical figure to a new figure by multiplying the co-ordinates of the original figure with a scale factor and obtain new co-ordinates to plot and form on the graph.
In the given figure, we observe that the original co-ordinates of the logo drew by Tyrone are:
A (-2,0) ; B (-2,3) ; C(2,2) ; D(2,-1)
From the graph, we see that the co-ordinates are multiplied by a scale factor of 5/2 and hence turn to new co-ordinates:
A'(-5,0) ; B'(-5, 15/2); C'(5,5) ; D'(5, -5/2)
Thus, we see that the rule followed to form the dilation of the logo on the coordinate plane was (5/2x, 5/2y).
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The bag of pretzels contains four pounds. Two pounds were eaten on Monday, and 6 ounces were eaten on Tuesday. How many ounces of pretzels remain in the bag? (1 pound = 16 oz. )
Answer:
26 oz
Step-by-step explanation:
4 lb x 16 oz/lb = 64 oz
64 oz - (2 lb)(16 oz/lb) - 6 oz = 26 oz are left
what is the solution to 4+5ex+2=11?
The solution to the equation is (a) x = ln(7/5) - 2
How to determine the solution to the equationFrom the question, we have the following parameters that can be used in our computation:
4 + 5e^(x + 2) = 11
Evaluate the like terms in the equation
So, we have the following representation
5e^(x + 2) = 7
Divide through the equation by 5
This gives
e^(x + 2) = 7/5
Take the natural logarithm of both sides
x + 2 = ln(7/5)
So, we have
x = ln(7/5) - 2
Hence, the solution is x = ln(7/5) - 2
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A rectangular prism is 3 yards long, 2 yards wide, and 12 yards high. What is the surface area of the rectangular prism?
The total surface area of the rectangular prism is 24 * 2 + 36 * 2 + 6 * 2 = 72 + 72 + 12 = 156 square yards.
The surface area of a rectangular prism is given by the sum of the areas of its six faces. To find the surface area of a rectangular prism that is 3 yards long, 2 yards wide, and 12 yards high, we need to find the area of each of its four rectangular faces and then add these areas together.
First, we find the area of the two faces that are 2 yards wide and 12 yards high. Each of these faces has an area of 2 * 12 = 24 square yards.
Next, we find the area of the two faces that are 3 yards long and 12 yards high. Each of these faces has an area of 3 * 12 = 36 square yards.
Finally, we find the area of the two faces that are 3 yards long and 2 yards wide. Each of these faces has an area of 3 * 2 = 6 square yards.
So, the total surface area of the rectangular prism is 24 * 2 + 36 * 2 + 6 * 2 = 72 + 72 + 12 = 156 square yards.
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Example 16: Find the greatest number that will divide 113, 135 and 160 leaving remainders 5, 3 and 4, respectively.
Answer:
12
Step-by-step explanation:
If 113 divided by a number leaves a remainder of 5, then we need to find the prime factorization of 108.
If 135 divided by a number leaves a remainder of 3, then we need to find the prime factorization of 132.
If 160 divided by a number leaves a remainder of 4, then we need to find the prime factorization of 156.
108 = 2² × 3³
132 = 2² × 3 × 11
156 = 2² × 3 × 13
The GCF of 108, 132, and 156 is common factors with lower exponent.
GCF = 2² × 3 = 12
Answer: 12
jyothi is the human resource manager in a company. For the post of 2 data analyst, she has the resume of 3 men and 2 women
1) What will be the probability that no man will be selected
2)What is the probability that both men will get selected
3)If she gets one more resume of a woman, what is the probability that only one man gets selected
Answer: The probability that no man will be selected is the number of ways that only women can be selected divided by the total number of combinations of candidates for the 2 data analyst positions. There are 2 ways to choose 2 women from the 2 available women, and there are C(4,2) = 6 ways to choose 2 candidates from the 3 men and 2 women, so the probability that no man will be selected is 2/6 = 1/3.
The probability that both men will get selected is the number of ways to choose 2 men from the 3 available men divided by the total number of combinations of candidates for the 2 data analyst positions. There are C(3,2) = 3 ways to choose 2 men from the 3 available men, and there are C(5,2) = 10 ways to choose 2 candidates from the 3 men and 2 women, so the probability that both men will get selected is 3/10 = 3/5.
If Jyothi gets one more resume of a woman, there will be a total of 3 men and 3 women. The probability that only one man gets selected is the number of ways to choose 1 man and 1 woman from the 3 men and 3 women divided by the total number of combinations of candidates for the 2 data analyst positions. There are C(3,1) * C(3,1) = 9 ways to choose 1 man and 1 woman, and there are C(6,2) = 15 ways to choose 2 candidates from the 3 men and 3 women, so the probability that only one man gets selected is 9/15 = 3/5.
Step-by-step explanation:
what do you get when you remove all the perfect squares of √39y^ 9?
For removing all the perfect squares of √(39y^9), we need to factor out any perfect squares from the expression under the square root.
What is a Perfect Square and how to remove all the perfect squares?To get rid of all the perfect squares in (39y9), factor out any perfect squares in the expression under the square root.
First, by factoring 39 and y9, we can simplify the expression under the square root:
[tex]√(39y^9) = √(313y^8y) = √(313*(y^4)^2*y)[/tex]
Because (y4)2 is a perfect square, we can take it out from under the square root:
[tex]√(39y^9) = √(313(y^4)^2y) = (y^4)√(313*y)[/tex]
By removing all of the perfect squares from (39y9), we get:
[tex](y^4)√(313y)[/tex]
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what is 5.4772 rounded to the nearest whole number
Answer:
5
Step-by-step explanation:
5.4772
To round to the nearest whole number, look at the number to the right of it (4) if 5 or greater round the 5 up to '6' o/w leave it as '5'
the proper way to construct a stem-and-ieaf display for the data set {62,67,68,73, 73, 79,91,94,95,97} is to
The proper way to construct a stem-and-leaf plot to display the data set {62,67,68,73, 73, 79,91,94,95,97} is to include a stem labeled ‘8’ and enter no leaf on the stem. So, the correct choice is option (b).
A stem and leaf graph is like a special table where each data value is separated into a "stem" (the first number) and a "leaf" (usually the last number). Leaves are sorted in ascending order from left to right. We have a data set with values {62,67,68,73, 73, 79,91,94,95,97}. Now, to construct the stem- leaf plot for our data.
Stem | Leaf
6 | 2 7
7 | 3 3 9
9 | 1 4 5 7
For stem and leaf diagram if some values are not present for some stem, then no leaf should be added to that stem. If we skip that stem than it would distort the shape of distribution of the data.
stem | leaf
6 | 2 7
7 | 3 3 9
8 |
9 | 1 4 5 7
Hence, include a stem labeled 8 and enter no leaves on the stem.
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Complete question:
The proper way to construct a stem-and-leaf display for the data set {62, 67, 68, 73, 73, 79, 91, 94, 95, 97} is to
a)exclude a stem labeled ‘8
b)include a stem labeled ‘8’ and enter no leaves on the stem.
c) include a stem labeled ‘(8)’ and enter no leaves on the stem.
d) include a stem labeled ‘8’ and enter one leaf value of ‘0’ on the stem.
Please Help. My test is almost over and I think I will fail if I do not get this question answered!!!
The value of x the width of the solar panel is 18.5.
What is Pythagoras Theorem?If ABC is a triangle with AC as the hypotenuse and angle B with 90 degrees then we have Pythagoras theorem:
|AC|^2 = |AB|^2 + |BC|^2
where |AB| = length of line segment AB. (AB and BC are rest of the two sides of that triangle ABC, AC being the hypotenuse).
We are given that;
Length of base of panel= 40feet
Other length = 12feet
Now,
Applying the pythagoras theorum;
|AC|^2 = |AB|^2 + |BC|^2
|AC|^2= 12^2 + 14^2
|AC|^2= 144+196
AC= 18.5
Therefore, by Pythagoras theorem x will be equal to 18.5.
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Someone help me out on this, please!
Find the equation of the ellipse if the major and minor axes have lengths 2 and 5, the major axis is vertical, and the center of the ellipse is (-1,-2)
Answer: The equation of an ellipse with major axis of length 2, minor axis of length 5, center at (-1, -2), and major axis aligned with the vertical axis, can be given by:
((x + 1)/2)^2 + ((y + 2)/5)^2 = 1
The standard form of an ellipse is:
(x^2)/a^2 + (y^2)/b^2 = 1,
where a and b are the lengths of the major and minor axes, respectively. To convert the equation to this form, we can divide both the x and y terms by the corresponding axis lengths, and shift the center of the ellipse to the origin.
Step-by-step explanation:
Explain how you would find the value of x.
Find the value of x
please help!!
The measure of the angle x in the figure is 55 degrees
How to find the value of x.From the question, we have the following parameters that can be used in our computation:
The polygon and the external angles
The sum of the exterior angles of a polygon is 360°.
Using the above as a guide, we have the following:
5 * x + x + 30 = 360
Evaluate the like terms
So, we have
6x = 330
Divide by 6
x = 55
Hence, the value of x is 55
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HELP QUICKLY
what is x
2.5/x = 10/12
PLS HELP ASAP
Answer:
x = 3
Step-by-step explanation:
2.5/x = 10/12
Multiply both sides by x
2.5 = 10/12x
Divide both sides by 10/12
3 = x
Solve the value for k:
[tex]\displaystyle{\lim_{x\to \infty} \left(1+\dfrac{65}{x}\right)^{3x} = e^k}[/tex]
The value of k in the exponential function is -3.
What is the value of k in the exponential function?
Taking the natural logarithm of both sides of the equation, we have:
ln [ (1 + 65/x)^(3x) ] = ln(e^k)
Using the properties of logarithms, we can simplify the left side of the equation:
3x ln(1 + 65/x) = k
Taking the limit as x approaches infinity, we have:
lim (x → ∞) 3x ln(1 + 65/x)
= lim (x → ∞) [ln(1 + 65/x) / (1/x)] × 3
Using L'Hopital's rule, we can evaluate the limit:
lim (x → ∞) [ln(1 + 65/x) / (1/x)]
= lim (x → ∞) [-65 / (x(1+65/x))]
= -65/65
= -1
Therefore, we have:
lim (x → ∞) 3x ln(1 + 65/x)
= lim (x → ∞) [ln(1 + 65/x) / (1/x)] × 3
= -3
Thus, the exponential equation simplifies to:
-3 = k
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We have the expression:
[tex]\displaystyle{\lim_{x\to \infty} \left(1+\dfrac{65}{x}\right)^{3x} = e^k} \\[/tex]
As [tex]\displaystyle{x\to \infty} \\[/tex], the term [tex]\displaystyle{\dfrac{65}{x}} \\[/tex] approaches 0. Therefore, we can rewrite the expression as:
[tex]\displaystyle{\lim_{x\to \infty} \left(1+\dfrac{65}{x}\right)^{x \cdot \dfrac{3x}{65}} = e^k} \\[/tex]
Simplifying the exponent, we get:
[tex]\displaystyle{\lim_{x\to \infty} \left(1+\dfrac{65}{x}\right)^{\dfrac{3x^2}{65}} = e^k} \\[/tex]
Now, as [tex]\displaystyle{x\to \infty} \\[/tex], the expression [tex]\displaystyle{\left(1+\dfrac{65}{x}\right)^{\dfrac{3x^2}{65}}} \\[/tex] approaches [tex]\displaystyle{e^{-3}} \\[/tex].
Therefore, we have:
[tex]\displaystyle{e^k = e^{-3}} \\[/tex]
Comparing the exponents, we can conclude that:
[tex]\displaystyle{k = -3} \\[/tex]
Hence, the correct value of k is indeed -3.
Find the value of each trigonometric ratios
Tan. X
It is important to note that this calculation provides the value of tan(x) in terms of the cosine of the angle x, which is dependent on the specific triangle that is being considered.
Given the triangle with sides ZY = 48, ZX = 50, and XY = 14, we can use trigonometry to find the value of tan(x).
First, we can use the Pythagorean theorem to find the value of YX, the length of the side opposite angle x:
[tex]YX = \sqrt{(ZY^2 + ZX^2 - 2 * ZY * ZX * cos(x))[/tex]
We know that ZY = 48 and ZX = 50, so we can substitute these values into the equation:
[tex]YX = \sqrt{(48^2 + 50^2 - 2 * 48 * 50 * cos(x))[/tex]
[tex]YX = \sqrt{(2304 + 2500 - 2400 * cos(x))[/tex]
Next, we can use the definition of tangent to find the value of tan(x):
[tex]tan(x) = YX/ZX = \sqrt{(2304 + 2500 - 2400 * cos(x)) / 50[/tex]
It is important to note that this calculation provides the value of tan(x) in terms of the cosine of the angle x, which is dependent on the specific triangle that is being considered. The exact value of tan(x) will depend on the specific angle x in the triangle.
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