Answer: -1/2
Step-by-step explanation:
[tex]\lim_{x \to \pi} \frac{\sin x}{2\sin x \cos x}\\\\=\lim_{x \to \pi} \frac{1}{2\cos x}\\\\=\frac{1}{2\cos \pi}\\\\=-\frac{1}{2}[/tex]
Answer:
[tex]-\frac{1}{2}[/tex]
Step-by-step explanation:
If we try direct substitution we get: [tex]lim_{x\to\pi}h(x)=\frac{sin(\pi)}{sin(2\pi)}[/tex]
The radian [tex]2\pi[/tex] is a full rotation from 0 radians, in other words it's equivalent to 0 radians.
Using the unit circle as a reference you'll remember when the angle is 0 radians, the y-value is zero, so: [tex]sin(2\pi)=0[/tex]
When the angle is: [tex]\pi[/tex] it's at a half rotation where the y-value is still zero on the unit circle.
So you get the indeterminate form of: [tex]\frac{0}{0}[/tex]
We can use L'Hôpital's rule which states: [tex]lim_{x\to c}\frac{f(x)}{g(x)}=lim_{x\to c}\frac{f'(x)}{g'(x)}[/tex] if the original limit is in indeterminate form such as the one above.
Taking the derivative of sin(x) we simply use the property that: [tex]\frac{d}{dx}[sin(x)]=cos(x)[/tex]
For the sin(2x) we can use the chain rule which essentially states: [tex]h(x)=f(g(x))\\h(x)=f'(g(x))*g'(x)[/tex]
It deals with composite equations, and in this case we can represent the composite function as:
[tex]f(x)=sin(g(x))\\g(x)=2x[/tex]
Using the chain rule we get: [tex]cos(2x)*2[/tex]
So now we have the limit: [tex]lim_{x\to \pi}\frac{cos(x)}{2cos(2x)}[/tex]
at pi radians the unit circle is at a half rotation where the x/y coordinates are: [tex](-1, 0)[/tex]
so cos(pi) = -1
at 2 * pi radians were basically back at 0 radians where the x/y coordinates are: [tex](1, 0)[/tex]
so cos(2 * pi) = 1
Now plugging in these values we get the fraction: [tex]\frac{-1}{2*1}[/tex] which just simplifies to: [tex]-\frac{1}{2}[/tex] which turns out to be the limit!
A two-digit number is 3 more than 4 times the sum of its digits. If 18 is added to the number, the digits are reversed. Find the number.
Answer:
Let the required number be ab.
Sum of digits = a + b ........ ( 1 )
10 a + b = 4 ( a + b )
10 a + b + 18 = 10 b + a
=> 9 a + 18 = 9 b
=> a + 2 = b
10 a + a + 2 = 4 ( a + a + 2 )
=> 11 a + 2 = 8 a + 8
=> 3 a = 6
=> a = 3
=> a + 2 = 5
=>ab = 35......... [ A ]
So the number becomes 35.
ANSWER :
The original number is 35.
hope it helps you
• (two-digit number) = 3 + 4(sum of its digits)
• (two-digit number) + 18 = Digit gets reverse
To Find :• The number.
Concept :-We will first assume the number on one's place be [tex]x[/tex] and ten's place be [tex]y[/tex] [why? because the given conditions are about the digits of number]
-Two-digit number then formed will be 10y + x and its reverse will be 10x + y[why? well let's take an example like a number 42. This number can be written in expanded form as 10×4+2×1 here 4 is number at ten's place (y) and 2 at one's place(x) It's reverse will be 24 it can be written in expanded form as 10×2+4×1 here 2 is number at ten's place (y) and 4 at one's place(x)]
-Then we will use given conditions to form the equation and will solve the equations to get the answer.
So now let's get started with our solution! :D
Solution :Let the one's and ten's digit of the number be [tex]x[/tex] and [tex]y[/tex] respectively.
According to first condition,
=> (10y + x) = 3 + 4(x + y)
=> 10y + x = 3 + 4x + 4y
=> 10y - 4y = 3 + 4x - x
=> 6y = 3 + 3x
[tex] \implies \sf \frac{6y}{3} = \frac{3}{3} + \frac{3x}{3} [/tex]
=> 2y - 1 = x ___(equation 1)
According to second condition,
=> (10y + x) + 18 = 10x + y
=> 10y - y + 18 = 10x - x
=> 9y + 18 = 9x
[tex] \implies \sf \frac{9y}{9} + \frac{18}{9} = \frac{9x}{9} [/tex]
=> y + 2 = x ___(equation 2)
Solving equation 1 and 2
y + 2 = x (equation 2)
y + 2 = 2y - 1 [From equation 1]
3 = 2y - y
3 = y (ten's place)
Now put value of y = 3 in equation 2
y + 2 = x (equation 2)
3 + 2 = x
5 = x (one's place)
So, the number formed is 35.
How can I find the slope of the line and what can the slope mean?
If we take two ordered pairs from the table, we can find the slope with the following equation:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]in this case, we can take the points (1,1/2) and (2,1) to get the following using the formula above:
[tex]\begin{gathered} m=\frac{1-\frac{1}{2}}{2-1}=\frac{\frac{1}{2}}{1}=\frac{1}{2} \\ \Rightarrow m=\frac{1}{2} \end{gathered}[/tex]we have that the slope is m = 1/2, and this means that for each 2 gallons of distilled water, Rodnika will add 1 gallon of sea salt
solve for x. round to the nearrest hundredth if necessary
Answer:
(b) 5.69
Step-by-step explanation:
You want the side opposite a 24° angle in a right triangle with a hypotenuse of 14 units.
SineThe sine relation is ...
Sin = Opposite/Hypotenuse
sin(24°) = x/14
Solving for x, we get ...
x = 14·sin(24°) ≈ 5.69
Point M is the midpoint of AB . AM=2x+9, and AB=8x−50.
What is the length of AM?
Using the midpoint theorem, the length of AM is 43
Calculating Length of a lineFrom the question, we are to determine the length of AM
From the given information,
Point M is the midpoint of AB
Thus,
From the midpoint theorem
We can write that
AM + MB = AB
AM = MB
AM + AM = AB
2AM = AB
Also, from the given information,
AM = 2x + 9
and
AB = 8x - 50
Therefore,
From 2AM = AB
2(2x + 9) = 8x - 50
Solve for x
2(2x + 9) = 8x - 50
4x + 18 = 8x - 50
18 + 50 = 8x - 4x
68 = 4x
x = 68/4
x = 17
But,
AM = 2x + 9
Substituting the value of x
Thus,
AM = 2(17) + 9
AM = 34 + 9
AM = 43
This means the length of AM is 43
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Suppose that 20% of the residents of Illinois have the flu and that 75% of the patients going to a doctor for testing have the flu. If a practioneer in Illinois uses a test that is 60% accurate...
a. What is the probability that a patient receives a false-negative result?
b. What is the probability that a patient receives a true-negative result?
c. What is the probability of a false-negative result given that the result is negative?
The probabilities in this problem are given as follows:
A. False - negative: 0.08 = 8%.
B. True - negative: 0.48 = 48%.
C. False-negative given that the result is negative: 0.1429 = 14.29%.
What is Conditional Probability?Conditional probability is the probability of one event happening, considering a previous event. The formula is:
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]
In which the parameters of the formula are given as follows:
P(B|A) is the probability of event B happening, given that event A happened.[tex]P(A \cap B)[/tex] is the probability of both events A and B happening.P(A) is the probability of event A happening.The probability of a false negative is composed as follows:
20% have the flu.Of those, 40% have negative results.Hence:
P(FN) = 0.2 x 0.4 = 0.08 = 8%.
The probability of a true negative is composed as follows:
80% do not have the flu.Of those, 60% have negative results.Hence:
P(TN) = 0.8 x 0.6 = 0.48 = 48%.
The probability of negative results is:
P(N) = P(FN) + P(TN) = 0.08 + 0.48 = 0.56.
Hence the conditional probability is given by:
P(FN|N) = 0.08/0.56 = 0.1429 = 14.29%.
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Select the three equations that pass through the points (–4, –16) and (5, 2):
y + 4 = 2(x – 16)
y – 2 = 2(x – 5)
y = 2x – 8
y + 16 = 2(x + 4)
The equation of line passing through (-4, -16) and (5, 2) are y = 2x – 8, y - 2 = 2(x - 5) and y + 16 = 2(x + 4)
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Equations are classified based on degree (value of highest exponents) as linear, quadratic, cubic and so on. Variables can be dependent or independent. Dependent variables depend on other variable while an independent variable do not depend.
The standard form for linear equation is:
y = mx + b
Where m is the slope and b is the y intercept
The equation of line passing through (-4, -16) and (5, 2) is:
y - (-16) = [(2-(-16))/(5-(-4))](x - (-4))
y + 16 = 2(x + 4)
y = 2x - 8
y - 2 = 2(x - 5)
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The truss in the illustration is in the form of an isosceles triangle. Each of the two equal sides is 9 feet less than the third side. A truss has its outer border in the shape of an isosceles triangle. If the perimeter is 21 feet, find the length (in ft) of each side.
The side lengths of the truss which is an isosceles triangle are 13 feet , 4 feet and 4 feet.
In geometry, an isosceles triangle is a triangle with at least two sides of identical length. It can be defined as having exactly two equal-length sides or as having at least two equal-length sides, with the equilateral triangle being an exception to the second definition.
The base of the triangle is its third side, while its two equal sides are referred to as its legs. Simple formulas can be used to calculate the triangle's height, area, and perimeter from the lengths of the triangle's legs and base.
Let us consider the length of the longest side of the isosceles triangle be x.
Therefore length of smaller sides = x - 9
The perimeter of the triangle is given as 21 meters.
Therefore :
x + (x - 9) + (x - 9) = 21
or, 3x -18 = 21
or, 3x = 39
or , x = 13 feet
Therefore length of two smaller sides of the triangle = x - 9 = 4 feet
Therefore the length of the three sides of the triangle are
4 feet , 4 feet and 13 feet.
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Find the intercepts and graph each line.
a.) x – 4y =− 4
b.) 2x + 5y =-10
Answer:
(1,5) I think that's what it is
HELP ASAP 100 POINTS
A group of seniors decide to spend their spring break away from home, attending at least one major event. The following table summarizes their preferences.
Attend a concert Attend a festival
Stay in state 214 186
Travel out of state 391 59
What percent want to attend a festival, given that they want to travel out of state?
6.94%
13.11%
24.08%
31.72%
Answer:
I think 31.72 because it 39156+214186 is 253342 it would be to low
A percentage is a number or ratio that represents a fraction of 100.
The percentage who want to attend a festival, given that they want to travel out of state is 6.94%.
What is a percentage?A percentage is a number or ratio that represents a fraction of 100.
We have,
Attend a concert Attend a festival
Stay in state 214 186
Travel out of state 391 59
The number of groups who wants to attend a concert while staying in a state:
= 214
The number of groups who wants to attend a concert while traveling out of the state:
= 391
The number of groups who wants to attend a festival while staying in a state:
= 186
The number of groups who wants to attend a festival while traveling out of the state:
= 59
The percentage who want to attend a festival, given that they want to travel out of state:
= The number of groups who wants to attend a festival while staying out of the state/ Total number of groups from all four conditions
= 59 / ( 214 + 391 + 186 + 59 ) x 100
= 59 / 850 x 100
= 6.94%
Thus,
The percentage who want to attend a festival, given that they want to travel out of state is 6.94%.
Option A is the correct answer.
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7.
A coach travels from the station to the beach, a distance of 576km away in 6hrs.
The coach is only allowed to travel at a maximum speed of 90km/h. Did the coach
break the speed limit?
Answer:
Yes.
Step-by-step explanation:
distance = rate x time
Let r = rate
567 = r6 Divide both sides by 6
94.5 = r If he was driving at a constant rate the whole time then he was driving at 94.5 miles per hour which is faster than 90
write an algebraic expression that represent the purchase. Baseball= $7, Sweat socks = $6, Soccer ball = $30. Mr. Mishra bought x number of soccer balls and 5 baseballs. Write an expression for the cost of Mr. Mishra's purchase.
Answer:
[tex]\text{Total cost=30x+35}[/tex]Step by step explanation:
The total cost of a purchase is represented by:
[tex]\text{ cost of the unit}\cdot\text{ number purchased=total cost}[/tex]Then, since Mr. Mishra bought:
x number of soccer balls and each soccer ball is $30.
5 baseballs and each baseball is $7
Then, the algebraic expression of the purchase is given by:
[tex]\begin{gathered} \text{Total cost=30x+7}\cdot5 \\ \text{Total cost=30x+35} \end{gathered}[/tex]A person is standing 18 feet away from a street light that is 20.2 feet tall. How long is his shadow if he is 5.2 feet tall? Enter the exact value of the answer.
The shadow of the person is 6.24 feet long.
The height of the street light is 20.2 feet.The height of the person is 5.2 feet.The person is standing at a distance of 18 feet from the street light.Let the length of the shadow be "x".The above configuration of the street light, the person, and the shadow creates a situation of similar triangles.Using the properties of the similar triangles, we know that the ratio of the corresponding sides of the similar triangles is the same.The height of the street light divided by the sum of the length of the shadow of the person plus the distance of the person from the street light is equal to the height of the person divided by the length of the shadow.20.2/(x+18) = 5.2/x20.2x = 5.2x + 18*5.215x = 18*5.2x = (18*5.2)/15x = 6.24Thus, the length of the shadow is 6.24 feet.To learn more about similar triangles, visit :
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You are testing the null hypothesis that the population proportion equals 45, using data you collected from a sample of 100 adults. You sample proportion equals .30. What does Z equal? (Round to two decimal places)
When one is testing the null hypothesis that the population proportion equals 45, the. z score is -3.02.
How to illustrate the information?The null and alternative hypothesis is illustrated as the population proportion equals 45,
P0 = 0.45
1 - P0 = 1 - 0.45 = 0.55
Test statistic = z
This will be illustrated as:
= 0.30 - 0.45/ [✓(0.45 * 0.55) / 100]
Test statistic = z = -3.02
Therefore, the z score is -3.02.
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what is −2 1 4 + (−1 3 4 )
Using subtraction we get the value as -3 4 8 .
Subtraction is a mathematical operation that mimics the removal of elements from a collection.
Subtraction involves several important patterns. Because it is anticommutative, changing the sequence also changes the sign of the solution. Additionally, it is not associative, so the technique employed affects when removing more than two numbers. Since 0 is the additive identity, removing it from a number has no impact. Subtraction follows the same ideas as related operations like addition and multiplication. Each of these principles may be proved, starting with integer subtraction and progressing to real numbers and beyond.The study of these patterns of subtraction in common binary operations is done in abstract algebra.The given expression is −2 1 4 + (−1 3 4 )
Simplifying we get : -2 1 4 - 3 1 4
now using subtraction we get : - 3 4 8
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Peter has a tire with a diameter of 26 inches. How many inches is the radius of his tire?
Answer
The radius of Peter's tire = 13 inches
Explanation
The radius of a circular figure is related to its diameter through
Radius = ½ (Diameter)
Diameter = 26 inches
Radius = ½ (Diameter) = ½ (26) = 13 inches
Hope this Helps!!!
A journalist believes that, in reality, these components have a shorter lifespan than what the company is reporting.
He tests a random group of 35 of these components and finds a mean lifespan of 4.05 years.
a. What is the mean of the sampling distribution of sample means when 35 random components are tested?
The mean of the sampling distribution of sample means when 35 random components are tested will be 4.05 years.
How to illustrate the information?From the information, it should be noted that the journalist believes that, in reality, these components have a shorter lifespan than what the company is reporting.
He the tests a random group of 35 of these components and finds a mean lifespan of 4.05 years.
Therefore from the information, we acne see that the. mean lifespan has already been given.
Therefore, the mean lifespan is 4.05 years.
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Please help me with Calculus homework, Question 2 only***Answer to question 1: -3.938
As given by the question
There are given that the function:
[tex]f(x)=x^3-6x[/tex]Now,
According to the answer to question #1 and the graph of the function:
It is the area under the curve from x = 0 and x = 3;
So,
Some of it is below the x-axis means negative area and some of it is above the x-axis which means positive area.
Then,
Yes, they will cancel out and the integral gives the exact net result.
Explained!
4a + 8b/7 - a^2
if a = 3 and b = -1
Simplify the expression.
___________________
Solutions:Exact Form - [tex]\frac{13}{7}[/tex]Decimal Form - 1.8.....Mixed Number Form - [tex]1\frac{6}{7}[/tex]...^Here are all three solutions depending on what you are looking for. Hope this helps! If so, please lmk! Thanks and good luck!
A frog jumped 0.7 meters. Then jumped another 0.98 meters. How many meters did he jump
A restaurant has two types of tables, rectangular
ones that can each seat 4 people and circular
tables that can each seat 8 people. If 144 people
are enough to fill all 30 tables at the restaurant,
how many rectangular tables does the restaurant
have?
A) 12
B) 16
C) 20
D) 24
Answer:
D) 24
Step-by-step explanation:
c + r = 30 Eq. 1
4r + 8c = 144 Eq. 2
r = rectangular tables
c = circular tables
From Eq. 1:
c = 30 - r Eq. 3
Replacing Eq. 3 in Eq. 2:
4r + 8(30-r) = 144
4r + (8*30 + 8*-r) = 144
4r + 240 - 8r = 144
-4r = 144 - 240
-4r = -96
r = -96/-4
r = 24
from Eq. 3
c = 30 - r
c = 30 - 24
c = 6
Check:
from Eq. 2
4r + 8c = 144
4*24 + 8*6 = 144
96 + 48 = 144