Answer:
351.7
Step-by-step explanation:
Volume of a cylinder:
V = πr²h
V = 3.14(4²)(7)
V = 3.14(16)(7)
V = 50.24(7)
V = 351.68
Rounded to the nearest tenth:
351.7
The diagonal of a rectangular room is 13 ft long. One wall measures 7ft longer than the adjacent wall. Find the dimensions of the room.
Let x be the length of the shorter wall of the rectangular room and y be the length of the longer wall of the rectangular room. Since the longer wall is 7 ft longer than the shorter wall, we can write the equation y = x + 7. We can also express the diagonal of the rectangular room using the Pythagorean theorem as x^2 + y^2 = 13^2. We can solve for x and y by substituting the equation y = x + 7 into the equation x^2 + y^2 = 13^2 and then solving for x. Doing this, we get x^2 + (x + 7)^2 = 13^2. Expanding the square on the right side of the equation and then rearranging the terms, we get x^2 + 2x^2 + 14x + 49 = 169. Combining like terms, we get 3x^2 + 14x - 120 = 0. This quadratic equation can be factored as (x - 8)(3x + 15) = 0. Since the length of a side of a rectangle must be positive, we can ignore the solution x = -15/3. So, the length of the shorter wall of the rectangular room is x = 8 ft. The length of the longer wall can be found by substituting this value into the equation y = x + 7, giving us y = 8 + 7 = 15 ft. Therefore, the dimensions of the rectangular room are 8 ft by 15 ft.
The measures of the angles of a triangle are shown in the figure below. Solve for x.
to
76⁰
41
The measures of the angles of a triangle are 76⁰, 41⁰, and 63⁰.
What is Angle Sum Property?
The sum of all angles of a triangle is equal to the angle of a straight line i.e. 180°. If we have a triangle ABC, then the Sum of angles A , B, and angle C will be 180 ° and the value of the exterior angle is equal to the sum of two interior opposite angle.
We have,
The measures of the angles of a triangle are:
x,
76⁰,
41⁰.
so, we have to calculate the value of the x⁰:
The sum of all the angles in the triangle is 180⁰
so,
to calculate the measure of x put the all values in equation such that using the above rule:
x + 76⁰ + 41⁰ = 180⁰
x = 180⁰ - 76⁰ - 41⁰
x = 63⁰
Hence, the measures of the angles of a triangle are 76⁰, 41⁰, and 63⁰.
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which angle corresponds with 3
∠8 is the corresponds with ∠3.
Hence, option B is the correct answer.
What is Corresponding pairs ?Congruent angles are those that correspond. Corresponding pairs are all angles that are positioned in relation to the parallel and transversal lines in the same way. When two parallel lines are intersected by another line, comparable angles are the angles that are created in matching corners or corresponding corners with the transversal (i.e. the transversal).
we know that
m∠3=m∠8 --------> by corresponding angles
m∠7=m∠8 -------> by vertical angles
so
m∠3=m∠7 -------> by alternate interior angles
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NOTE: The given question is incomplete on the portal. Here is the complete question.
QUESTION: Which angle corresponds to ∠3?
A) ∠6
B) ∠8
C) ∠1
D) ∠7
A hot air balloon travels 2016 miles in 96 hours. The balloon travels the same number of miles each hour. How many miles does the balloon travel in 1 hour
Answer:
21 m/ph
Step-by-step explanation:
2016 divided by 96
The balloon travels 21 miles in one hour.
What is division?In mathematics, division is one kind of operation. The phrases or numbers in this process are split into the same number of parts.
Given, a hot air balloon travels 2016 miles in 96 hours.
And the balloon travels the same number of miles each hour.
To find the same numbers of miles per hour:
We use the division of 2016 by 96.
That means,
2016 / 96
= 21
Therefore, the travel time of the balloon is 21 miles per hour.
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The fare charged for a rideshare service is a function of
the distance traveled. However, the fare differs
according to the time of day, availability, and other
variables. The distance and fares for 10 rides are
shown in the table. The equation of the least-squares
regression line is ý = 5.20 +2.33x, where y is the
predicted fare and x is the distance.
Distance
(Miles)
1
3
5
8
10
12
15
16
56
Mark this and return
Fare
(Dollars)
1.91
13.68
16.52
24.15
24.79
39.87
40.24
53.84
What is the residual for the rideshare cost with a
distance of 16 miles?
2.33
5.21
11.35
42.49
Answer: 11.35
Step-by-step explanation: The residual for the rideshare cost with a distance of 16 miles can be calculated using the equation of the least-squares regression line and the observed fare for that distance. The residual is defined as the difference between the observed value and the predicted value.
To find the predicted fare for a distance of 16 miles, we can substitute 16 for x in the equation of the least-squares regression line: ý = 5.20 + 2.33x. This gives us ý = 5.20 + 2.33 * 16 = 42.49.
The observed fare for a distance of 16 miles is 53.84, so the residual is 53.84 - 42.49 = 11.35. Therefore, the answer is 11.35.
Weekly wages at a certain factory are
normally distributed with a mean of
$400 and a standard deviation of $50.
Find the probability that a worker
selected at random makes between
$500 and $550.
Answer:
The probability that a worker selected at random makes between $500 and $550 is approximately 0.24.
Step-by-step explanation:
To calculate this probability, we can use the standard normal distribution, which is a normal distribution with a mean of 0 and a standard deviation of 1. We can convert the weekly wage amounts to standard units by subtracting the mean and dividing by the standard deviation.
For wages between $500 and $550, the corresponding standard units are:
$500 - $400 = $100 / $50 = 2
$550 - $400 = $150 / $50 = 3
We can use a standard normal table or a calculator to find the probability that a value chosen at random from the standard normal distribution falls within the range of 2 to 3 standard units. This probability is approximately 0.24.
Since the weekly wages at the factory are normally distributed with a mean of $400 and a standard deviation of $50, this probability is also the probability that a worker selected at random makes between $500 and $550.
Maya has 120 caramel apples to sell. Each caramel apple is covered with one topping.
• of the caramel apples are covered with peanuts.
• are covered with chocolate chips.
• are covered with coconut.
• The rest are covered with sprinkles.
How many caramel apples are covered with sprinkles?
The number of caramel apples that are covered with sprinkles is; 20
How to solve algebra word problems?We are given;
Total number of caramel apples to sell = 120
1/5 of the caramel apples are covered with peanuts.
1/3 of the caramel apples are covered with chocolate chips.
3/10 of the caramel apples are covered with coconut.
The rest are covered with sprinkles.
Let 'n' be the number of caramel apples are covered with sprinkles. Thus, we can develop the equation as;
120 = (1/5 * 120) + (1/3 * 120) + (3/10 * 120) + n
120 = 24 + 40 + 36 + n
n = 120 - 100
n = 20
Which means that 20 are those covered with sprinkles
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Complete question is;
Maya has 120 caramel apples to sell. Each caramel apple is covered with one topping.
1/5 of the caramel apples are covered with peanuts.
1/3 of the caramel apples are covered with chocolate chips.
3/10 of the caramel apples are covered with coconut.
The rest are covered with sprinkles.
How many caramel apples are covered with sprinkles?
Suppose previous research suggests that the mean length of all adult Anacondas is 13 feet with a standard deviation of 2.4 feet. Let W be the random variable that stands for length of adult Anacondas, so E(W)=13, SD(W)=2.4. You are planning on collecting a random sample of 50 adult Anacondas. Consider the RV Bar-W, which is the mean of the 50 sampled Anacondas. 98% of samples will have the realized value of Bar-W less than what value? Which of the answers reasonably approximates the requested value of the sample mean with justification?a. "Bar-w"=14.67 since we can use a normal approximation by the CLTb. "Bar-w"=17.93 since we can use a normal approximation by the CLTc. "Bar-w"=13.70 since we can use a normal approximation by the CLTd. "Bar-w"=12.30 since we can use a normal approximation by the CLTe.A normal approximation is inappropriate
The answers reasonably approximates the requested value of the sample mean with justification is c. "Bar-w"=13.70 since we can use a normal approximation by the CLT.
Since the sample size is large (n=50), we can use the central limit theorem (CLT) to approximate the sampling distribution of the sample mean with a normal distribution.
The mean of the sampling distribution of the sample mean is equal to the mean of the population, which is 13 feet in this case. The standard deviation of the sampling distribution of the sample mean is equal to the standard deviation of the population divided by the square root of the sample size. This is known as the standard error of the mean, and is denoted by SE(Bar-W).
In this case, SE(Bar-W) = 2.4/sqrt(50) = 0.48.
To find the 98th percentile of a normal distribution with mean 13 and standard deviation 0.48, we can use a standard normal table or a calculator to find that the 98th percentile is approximately 2.05.
Therefore, the value that 98% of samples will have the realized value of Bar-W less than is approximately 13 + 2.05 * 0.48 = 13.70. This means that the correct answer is (c) "Bar-w"=13.70 since we can use a normal approximation by the CLT.
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If the slope of the regression line is calculated to be 2.5 and the intercept 16 then the value of Y when X is 4 is:
Select one or more:
a.2.5
b.66.5
c.26
d.16
In order to calculate linear regression, we apply the formula Y = B + X. The current access level for 16 plus is 2.5. As a result, Option A is the right response. We get 26 from 16 + 10.
How do you interpret the slope and intercept of a regression line?The slope and intercept of a line show how steep it is, and the intercept shows where the line crosses an axis. It is possible to calculate an average rate of change using the slope and intercept, which describe the linear relationship between two variables.
An equation of the form Y = a + bX, where X is the explanatory variable and Y is the dependent variable, represents a linear regression line. A line's intercept (the value of y when x = 0) is equal to the slope of the line, which is b.
The regression slope intercept equation, [tex]y' = b_{0} + b_{1} x[/tex], where "[tex]b_{0}[/tex]" is the y-intercept and "[tex]b_{1} x[/tex]" is the slope, is actually just an algebraic version of the regression equation. Once the equation for linear regression has been discovered,
Therefore the correct answer is option a ) 2.5
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Two athletes start from the same point and move on a closed track of 600m. If they run in the same
direction at speeds 20m/s and 30m/s, when will they cross each other?
Mark one or more correct options
(A) 45 sec
(B) 1 min
(C) 1 min 30 sec
(D) 3 min
(E) 5 min
The solution is Option B , Option D , Option E
The athletes with speeds 20m/s and 30m/s will meet at every minute
What is Speed?
Speed is defined as the rate of change of position of an object in any direction. Speed is measured as the ratio of distance to the time in which the distance was covered. Speed is a scalar quantity as it has only direction and no magnitude
Speed = Distance / Time
Given data ,
Let the equation be represented as A
Now , the value of A is
The distance of the track is D = 600 m
Let the first athlete be A
Let the second athlete be B
The speed of A = 20 m/s
The speed of B = 30 m/s
The time taken by A to complete 600 m = 600 / 20 = 30 seconds
The time taken by B to complete 600 m = 600 / 30 = 20 seconds
Now , the time at which A and B will cross each other is the LCM of their respective time
So , The LCM of 20 and 30 is = 60 seconds
Therefore , the athletes will meet together at every 60 seconds or every 1 minute
Hence , the athletes will meet at every minute
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I need help with the word problem
Answer:
BAG 1 IS BETTER. IT IS BETTER TO PAY $18 FOR 12 POUNDS OF CAT FOOD.
Step-by-step explanation:
To find the unit cost, we need to know how much one pound of cat food cost for each bag
Bag 1
It said you get 12 pounds for $18
Divide $18 by 12 to find how much ONE pound is.
18 ÷ 12 = 1.5
ONE POUND COST $1.5
This is the unit cost
Let's do the same for the other bag of cat food!
Bag 2
Divide $24 by 15
24 ÷ 15 = 1.6
ONE POUND COST $1.6
This is the unit cost
It is better to pay for $1.5 per pound because it is LESS than $1.6
You will save money this way.
BAG 1 IS BETTER. IT IS BETTER TO PAY $18 FOR 12 POUNDS OF CAT FOOD.
Suppose two dice are rolled. Let X be the random variable measuring the sum of the two numbers rolled.
(a) Find the probability mass function for X.
(b) Find the expected value E(X).
(c) Find the variance V(X).
The expected value E(X) is 7 and the value of variance of X is 5.8333.
Two dice are rolled. Let X be the random variable measuring the sum of the two numbers rolled.
a) The probability of mass function is obtained below:
The possible outcomes in each of the dice are 1 to 6. Therefore, the possible outcomes when two dice is rolled is 36
The sample space, s for fair dice (red die and blue die) is given below:
N(s) = 36
From the given information, two dice are rolled let X be the random variable measuring the sum of the two numbers rolled.
b) The expected value is calculated below:
The probability mass function of X is,
The required mean is,
E(x) = ∑xP(X=x)
[tex]=[2[/tex]×[tex]\frac{1}{36}+3[/tex]×[tex]\frac{2}{36}[/tex]+....+11×[tex]\frac{2}{36}+12[/tex]×[tex](\frac{1}{36} )[/tex]]
=[0.0556+0.1667+0.3333+0.5556+0.8333+1.1111+1.000+0.8333+0.6111+0.3333
=7
C) The variance V(X) is calculated below:
The probability mass function of X is,
The required variance is,
V(x)= ∑[tex](X-x)^{2}P(x) = 5.8333[/tex]
Therefore, the expected value E(X) is 7 and the value of variance of X is 5.8333.
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Cómo se determina el condominio de la función cuadrática.
The domain is the set of all values in the set (- ∞, ∞).
What is a mathematical function, equation and expression?Function : In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function
Expression : A mathematical expression is made up of terms (constants and variables) separated by mathematical operators
Equation : A mathematical equation is used to equate two expressions.
Given is a quadratic function as follows -
ax² + bx + c
For any quadratic equation (with vertex at origin), the value of [y] or the range is either the set of all positive values from 0 to infinity when the quadratic function open upwards and the set of all negative values from 0 to -infinity when the quadratic function open downwards. The domain is the set of all values in the set (- ∞, ∞).
Therefore, the domain is the set of all values in the set (- ∞, ∞).
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{Question in english language is as follows -
How the domain of the quadratic function is determined.}
Which of the following research situations would be most likely to use an independent-measures design?
a) examine the development of vocabulary as a group of children mature from age 2 to age 3
b) examine the long-term effectiveness of a stop-smoking treatment by interviewing subjects 2 months and 7 months after the treatment ends
c) compare the mathematics skills for 9th grade boys versus 9th grade girls*
d) compare the blood-pressure readings before medication and after medication for a group a patients with high blood pressure
Option C, The research situation that would be most likely to use an independent-measures design is to compare the mathematics skills of 9th-grade boys versus 9th-grade girls.
An independent-measures design is used when there are two or more groups that are being compared and each individual only belongs to one group. In this research situation, the two groups being compared are 9th-grade boys and 9th-grade girls. Each individual in the study belongs to one group or the other, and they are not part of both groups. This makes it an ideal situation for using an independent-measures design.
Option a) examine the development of vocabulary as a group of children mature from age 2 to age 3 would likely use a within-subjects design, as the same group of children is being measured at two different time points.
Option b) Examining the long-term effectiveness of a stop-smoking treatment by interviewing subjects 2 months and 7 months after the treatment ends would also likely use a within-subjects design, as the same group of subjects is being measured at two different time points.
Option d) compare the blood-pressure readings before medication and after medication for a group of patients with high blood pressure would likely use a repeated-measures design, as the same group of patients is being measured at two different time points and the order of the measurements (before or after medication) is being controlled.
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What rigid motion maps the solid-line figure onto the dotted-line figure?
A reflection
B. rotation
C. translation
Answer:
A. Reflection
Step-by-step explanation:
I believe the answer would be reflection
pls whats the anwer im stuck 4+5(x-1)=34
Answer: 7
Step-by-step explanation:
To solve this equation, we first need to use the order of operations to simplify the expression on the left-hand side of the equation. The order of operations tells us that we should perform the operations within parentheses first, followed by any exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right).
In this case, we have a set of parentheses with the term 5(x-1) inside. This means we need to calculate the value of 5(x-1) first. We can do this by multiplying 5 and (x-1), which gives us 5x-5.
Next, we need to add 4 to 5x-5 to get the final expression on the left-hand side of the equation. This gives us 4+5x-5, which simplifies to 5x-1.
Now we have the equation 5x-1=34. To solve this equation for the value of x, we need to isolate the x term on one side of the equation by moving all the other terms to the other side. We can do this by adding 1 to both sides of the equation, which gives us 5x = 35.
Finally, we can divide both sides of the equation by 5 to find the value of x. This gives us x = 35/5 = 7. Therefore, the value of x that makes the equation true is 7.
Answer:7
Step-by-step explanation:
4+5(x-1)=34
Distribute: 4+5(x - 1) = 34
4+5x - 5=34
Subtract the numbers: 4 + 5x - 5=34
-1 + 5x =34
Rearrange terms: -1 + 5x =34
5x - 1 = 34
Solution: x=7
Hope this helps!!
Which points are included in the graph of the ceiling function f(x) = ⌈x⌉? Select all that apply.
A. (−0.5, −1)
B. (−1.5, −2)
C. (−3.5, −3)
D. (−3.5, −4)
E. (3.001, 4)
F. (−3.001, −3)
Answer:
C, E, F
Step-by-step explanation:
You want to know which points match the ceiling function among ...
A. (−0.5, −1)B. (−1.5, −2)C. (−3.5, −3)D. (−3.5, −4)E. (3.001, 4)F. (−3.001, −3)CeilingThe ceiling function is also known as the least integer function. It returns the smallest integer greater than or equal to its argument value. It is essentially a "round to next higher" function, with any integer input remaining unchanged.
The attachment shows the input and output of the ceiling function for the values on the answer choice list. C, E, F are points on the function's graph.
X-3y=12 what does it look like on the graph
The graph of x-3y = 12 is given.
What is graph?A graph is a visual representation or diagram that displays facts or values in an organized manner in mathematics. The points on a graph are typically used to depict the relationships between two or more things.
The equation is
x-3y = 12.
The graph of this line intersects the x-axis and y-axis.
And end behavior is infinity on both positive and negative.
And the graph is a straight line graph.
Therefore, behavior of x -3y = 12 is given in the graph.
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Business Organization
1. James invests $10,000 in a partnership with 3 other people. One of those people also invested
$10,000 and the other two invested $90,000 each. What percent of the business does James
own?
(please help me if I don't get help I won't graduate)
James owns just 5% of the company after the investment.
What is a percentage?Percentage can be calculated by dividing the value by the total value, and then multiplying the result by 100.
Percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%".
Percentage can be calculated by dividing the value by the total value, and then multiplying the result by 100. It is given by:
Percentage = (value / total value) * 100%
How to find the Percentage owned by James;
James' investment: $10,000
The other 3 people invested;
x = $10,000
y = $10,000
z = $10,000
Total investment made was = $200,000
James' Percentage = [tex]\frac{10000}{200000\\}[/tex] x 100
James' Percentage = 5%
So, we can say that James owns 5% of the total investment in the business.
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Which expression is equivalent to (8×10−2) (2.4×10−3)?
The expression that is equivalent to the given expression, (8×10−2) (2.4×10−3), is 1.92 × 10⁻⁴
Determining an equivalent expressionFrom the question, we are to determine the expression that is equivalent to the given expression
From the given information,
The given expression is
(8×10−2) (2.4×10−3)
First, we will write this expression properly
The given expression written properly is
(8 × 10⁻²)(2.4 × 10⁻³)
Now, we will evaluate the expression
Evaluating the expression
(8 × 10⁻²)(2.4 × 10⁻³)
8 × 2.4 × 10⁻² × 10⁻³
19.2 × 10⁻⁵
= 1.92 × 10⁻⁴
Hence, the expression is 1.92 × 10⁻⁴
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write the prime factorization the foll numbers
A.144
Answer: 2 times 2 times 2 times 2 times 3 times 3
Step-by-step explanation:
Question 3 (2 points)
For the following, find the arc length:
(Use π=3.14 and round your final answer to the hundredths.)
The length of the arc is 10.47 units
How to determine the length of the arc?From the question, we have the following parameters that can be used in our computation:
Radius = 10
Angle = π/3
The length of the arc can be calculated using
Arc length = Radius * Angle
Substitute the known values in the above equation, so, we have the following representation
Arc length = 10 * π/3
Evaluate
Arc length = 10.47
Hence, the arc length is 10.47
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1. The graph of a quadratic function is called a(an)
Copy and complete the table of values for the function.
2.
y=-1/3 x²
The graph of a quadratic function is called a parabola
The complete table is
x = -6, -3, 0, 3 6
y = -12 -3 0 -3 -12
How to complete the table of values?From the question, we have the following equation that can be used in our computation:
y=-1/3 x²
From the table of values , we have the following x values
x = -6, -3, 0, 3 and 6
Substitute x = -6, -3, 0, 3 and 6 in y=-1/3 x²
So, we have the following representation
y = -1/3 (-6)² = -12
y = -1/3 (-3)² = -3
y = -1/3 (0)² = 0
y = -1/3 (3)² = -3
y = -1/3 (6)² = -12
So, the complete table of values is
x = -6, -3, 0, 3 6
y = -12 -3 0 -3 -12
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You are playing a new video game. The table shows the proportional relationship between the number of levels completed and the time it took you to complete them.
Number of Levels 3 5
Time (hours) ? 2.5
How many minutes does it take you to complete 3 levels?
105 minutes
90 minutes
60 minutes
50 minutes
PLS HELP
Answer:
the answer is 90 mins option b
Step-by-step explanation:
Answer:
the answer is 90 minutes or B
hope this helps
mark me brainliest
Step-by-step explanation:
use the function f and the given real number a to find (f -1)'(a). (hint: see example 5. if an answer does not exist, enter dne.) f(x)
The function y= f(x)= [tex]x^{3}+3x-1[/tex] and a= -5 the real number is [tex]\frac{1}{6}[/tex].
y= f(x)= [tex]x^{3}+3x-1[/tex] and a= -5.
The slope of inverse functions are reciprocals at their corresponding points that is,
[tex]f^{-1}'(a)=\frac{1}{f'(b)}[/tex]
Where, [tex]f^{-1}'(a) = b[/tex] and f(b) = a
Now, determine the value of b for a= -5 using f(x)=[tex]x^{3}+3x-1[/tex]
f(b) = a
[tex]b^{3}+3b-1=-5[/tex]
[tex]b^{3}+3b= -4[/tex]
[tex]b = -1[/tex]
Therefore, [tex]f^{-1}'(a) = \frac{1}{f'(b)}[/tex]
[tex]f^{-1}'(-5)=\frac{1}{f'(-1)}[/tex]
Now, find f'(x) and evaluate it at
x= -1
f(x)=[tex]x^{3}+3x-1[/tex]
[tex]f'(x)= 3x^{2}+3[/tex]
f'(x)= [tex]3(x^{2} +1)[/tex]
then, f'(-1)= 3(1+1) = 6
Therefore, [tex]f^{-1}(-5)=\frac{1}{f^{-1}(-1)}=\frac{1}{6}[/tex].
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Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
an =
n sqr root 35 + 3n
lim nâ[infinity] an =
This sequence is not convergent because the nth term does not approach a single limit as n approaches infinity. The limit does not exist, so the answer is DNE.
This sequence does not converge to a single limit because the nth term of the sequence, an, has a coefficient of n, which means that the value of an increases without bound as n increases. This means that the value of an does not approach a single limit as n approaches infinity, and so the limit does not exist. Therefore, the answer for the limit as n approaches infinity is DNE.
This sequence does not converge to a single limit because the nth term of the sequence, an, has a coefficient of n, which means that the value of an increases without bound as n increases. As n increases, the value of an increases, and so the value of an does not approach a single limit as n approaches infinity. This means that the limit does not exist, and so the answer for the limit as n approaches infinity is DNE.
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Could I get some help with this? Thanks! Here’s the question to go along with it.
For each diagram, calculate the value of x. Show your work and include an explanation of what you used (definitions and theorems) to solve the problem. If not possible, state why.
Answer:
x = 16 1/3°
Step-by-step explanation:
You have parallel lines with consecutive interior angles marked 7x-28° and 5x+12°, and you want to find the value of x.
Consecutive interior anglesAt parallel lines, consecutive interior angles, or same-side interior angles, are supplementary. This means the total of the two marked angles is 180°.
7x -28° +5x +12° = 180°
12x -16° = 180° . . . . . . . . . . simplify
12x = 196° . . . . . . . . . add 16°
x = (49/3)° = 16 1/3° . . . . . . . divide by 12
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Additional comment
∠3 = 86 1/3°
∠5 = 93 2/3°
find the maximum area of the rectangle inscribed in the triangle formed by the x x-axis, the y y-axis and the line y
A shape's highest possible area is called its maximum area.The size of the rectangle that will result in the greatest area is 4 by 2 units.
What is meant by rectangle?Four sides, four corners, and four 90° right angles make up the closed, 2-D shape of a rectangle. A rectangle has parallel, equal sides on either side. Rectangles are two-dimensional shapes, and as such, they have length and breadth as their defining characteristics. The rectangle's shorter side is its width, while its longer side is its length.How to calculate a rectangle's area. The size of a rectangle. A = l × b. Once the length and breadth are known for any rectangle, the area may be determined. The area of the rectangle is calculated as a square-unit dimension by multiplying length and width.The equation is given as:
x + 2y - 8 = 0
Rewrite as:
2y = 8 - x
Divide both sides by 2,
y = [tex]\frac{8 - x}{2}[/tex]
y = 4 - [tex]\frac{x}{2}[/tex]
The area (A) of the rectangle is:
A = xy
Substitute
y = 4 - [tex]\frac{x}{2}[/tex]
A = x[tex](4 - \frac{x}{2})[/tex]
Expand
A = 4x - [tex]\frac{x^{2} }{2}[/tex]
Differentiate
A' = 4 - x
Set the derivative to 0,
4 - x = 0
Collect like terms
x = 4
Recall that:
y = 4 - [tex]\frac{x}{2}[/tex]
Substitute 4 for x,
y = 4 - [tex]\frac{4}{2}[/tex]
y = 4 - 2
y = 2
So, we have:
x = 4 and y = 2
Hence, the dimension of the rectangle that will produce the maximum area is 4 units by 2 units.
The complete question is:
A rectangle is inscribed in the region bounded by the x-axis, the y-axis, and the graph of x+2y-8=0. Approximate the dimensions of the rectangle that will produce the maximum area.
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Use Green's Theorem to evaluate the line integral along the given positively oriented curve.
integral.gif C (3y + 7e^sqrt(x)) dx + (8x + 5 cos y^2) dy
C is the boundary of the region enclosed by the parabolas y = x2 and x = y2
On solving the provided question, From the question, our region is defined by: lower bound: y= [tex]x^2[/tex] and upper bound: y = [tex]\sqrt{x}[/tex]
what is integration?Integrals are mathematical representations of numbers and functions that express notions such as volume, area, displacement, and other outcomes of the combination of little data. Finding integrals is the term used to describe the procedure.
By green's theorem -
[tex]\int\limits^{}_{} {} \, \int\limits^{}_{a} {5dA} \, = 5/3[/tex]
First, the integral given in this exercise corresponds to:
[tex]\int\limits^{}_{C} {((3y + 7e\sqrt{x}dx) +( 8x+ 5cos(y^2)) } \, dy[/tex]
Greens Theorem given as,
[tex]\int\limits^{}_{C} {(P(x,y)dx +Q(x,y)dy)} \, = \int\limits^{}_{} {x} \, \int\limits^{}_a {(-\beta /\beta _{y} )P(x,y) + (\beta /\beta _{y} )Q(x,y)} \, dA[/tex]
and we have -
P(x,y) = 3y + [tex]7e^{\sqrt{x}}[/tex]
Q(x,y) = 8x + 5cos([tex]y^2[/tex])
And,
[tex]\int\limits^{}_{C} {(P(x,y)dx +Q(x,y)dy)} \, = \int\limits^{}_{} {x} \, \int\limits^{}_a {5dA} \, dA[/tex]
From the question, our region is defined by:
lower bound: y= [tex]x^2[/tex]
upper bound: y = [tex]\sqrt{x}[/tex]
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f the temperature is 50 degrees Fahrenheit, what is the temperature in Celsius?
Answer:
the answer is 10 degrees Celsius
50 Degrees Fahrenheit is 10 degrees Celsius.
Here is the formula
(50°F − 32) × 5/9 = 10°C