Answer: As the two cylinders have the same height, the cylinder with the greater radius will have the greater volume. Therefore, the statement "The cylinder with radius has a greater volume than the cylinder with radius /2" is true.
Step-by-step explanation:
One number is 8 less than twice a second number. Find a pair of such numbers so that their product is as small as possible. These two numbers are ____. (Use a comma to separate your numbers.)
The smallest possible product is ____.
These two numbers are -4, 2. The smallest possible product is -8.
To find a pair of numbers that satisfy the given conditions, we can use algebra. Let x be the first number and y be the second number. According to the problem, one number is 8 less than twice a second number. This can be written as:
x = 2y - 8
We need to find the product of these two numbers, which is x*y. Substituting the value of x from the equation above, we get:
x*y = (2y - 8)*y
= 2y^2 - 8y
To find the smallest possible product, we need to minimize this expression. We can do this by finding the vertex of the parabola represented by this equation. The vertex of a parabola in the form ax^2 + bx + c is given by (-b/2a, f(-b/2a)). In this case, a = 2, b = -8, and c = 0. So, the vertex is:
(-b/2a, f(-b/2a)) = (-(-8)/(2*2), f(-(-8)/(2*2)))
= (2, f(2))
Substituting y = 2 into the equation for the product, we get:
x*y = 2(2)^2 - 8(2)
= 8 - 16
= -8
So, the smallest possible product is -8. To find the pair of numbers that give this product, we can substitute y = 2 into the equation for x:
x = 2y - 8
= 2(2) - 8
= -4
Therefore, the pair of numbers are -4 and 2.
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An electric utility company charges $24 per month maintenance fee to customers generating their own solar power, but refunds them $0.08{ per kilowatt-hour of net electricity returned to the grid.
How many kilowatt-hours per month does one have to return to the grid to break even (do not pay nor receive any money)? Round your answer to the nearest kilowatt-hour. Do not include units in your answer
300 kilowatt-hours per month does one have to return to the grid to break even.
To find out how many kilowatt-hours per month one has to return to the grid to break even, we need to set up an equation and solve for x, where x is the number of kilowatt-hours returned to the grid.
The equation would be:
24 = 0.08x
To solve for x, we need to isolate the variable on one side of the equation. We can do this by dividing both sides of the equation by 0.08:
24/0.08 = x
x = 300
Therefore, one would have to return 300 kilowatt-hours per month to the grid to break even.
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What’s the measure of the bolder arc is 4cm is in the boldes arc and 118 is outside
As a consequence, the bolded arc is approximately 8.24 centimetres long when the bolder arc is 4cm long.
what is angle ?The vertex of the angle is the common endpoint shared by two rays or line segments that make up an angle in geometry. The sides or limbs of the angle are the rays or line segments that make up the angle. Angles are used to describe the amount of spin or turn between two lines or objects. They are usually measured in degrees or radians. Numerous areas of mathematics, such as geometry, trigonometry, and calculus, depend on angles.
given
We must apply the following algorithm to determine the size of the bolded arc:
arc length is equal to (angle / 360) times 2r.
where r is the radius, angle is the central angle in degrees, and is a mathematical constant roughly equivalent to 3.14.
Inputting the numbers provided yields:
(118/360) * 2 = arc length (4)
8.24 cm arc length is equal to (0.3278) × (25.12) arc length.
As a consequence, the bolded arc is approximately 8.24 centimetres long when the bolder arc is 4cm long.
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Let P be the poset with vertices {A,B,C,D,E,F,G,H}, and comparisons A≤B≤C≤D≤E, F≤D, G≤H≤D, along with all reflexive comparisons V≤V for every vertex V, and all transitive comparisons formed from these (for example, A≤C, since A≤B and B≤C; or F≤E, since F≤D and D≤E). How many linear extensions does the poset P have?
The final answer is 16 - 1 = 15 linear extensions of the poset P.
The poset P has 8 linear extensions. We can determine the number of linear extensions by counting the number of topological sorts of the poset. A topological sort of a poset is a linear ordering of its vertices that respects the partial ordering. There are two ways to obtain a topological sort of the poset P:
1. Start with vertex A, then choose either B or F, then choose either C or G, then choose either D or H, and finally choose E. This gives us a total of 2 x 2 x 2 x 1 = 8 topological sorts.
2. Start with vertex F, then choose either A or G, then choose either B or H, then choose either C or D, and finally choose E. This also gives us a total of 2 x 2 x 2 x 1 = 8 topological sorts.
Therefore, the total number of linear extensions of the poset P is 8 + 8 = 16.
However, we need to subtract the number of topological sorts that are counted twice. These are the topological sorts that start with A, then choose F, then choose G, then choose H, and finally choose E. There are 1 x 1 x 1 x 1 x 1 = 1 of these topological sorts.
So the final answer is 16 - 1 = 15 linear extensions of the poset P.
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Directions
Math-Alg.F-IF.6 Standard Review (copy)
1
The function f(x) contains the set of points {(-2,4), (-1,8), (0, 12), (1, 15), (2, 18)} over the interval [-2,2].
What is the average rate of change of f(x) over the interval [0, 2]?
Check the picture below.
[tex]\begin{array}{llll} f(x)~from\\\\ x_1 ~~ to ~~ x_2 \end{array}~\hfill slope = m \implies \cfrac{ \stackrel{rise}{f(x_2) - f(x_1)}}{ \underset{run}{x_2 - x_1}}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array} \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} x_1=0\\ x_2=2 \end{cases}\implies \cfrac{f(2)-f(0)}{2 - 0}\implies \cfrac{18-12}{2}\implies \cfrac{6}{2}\implies \text{\LARGE 3}[/tex]
y is directly proportionalto the cube root of (x+3) when x=5 y(2)/(3 ) find y when x=24
when x=24, y = y(3)/(2).
Given that y is directly proportional to the cube root of (x+3), we can write this relationship as:
y = k * cube root (x+3)
Where k is the constant of proportionality. We can use the given values of x and y to find k:
y(2)/(3 ) = k * cube root (5+3)
y(2)/(3 ) = k * cube root (8)
y(2)/(3 ) = k * 2
k = y(2)/(3 ) / 2
Now we can use this value of k to find y when x=24:
y = k * cube root (24+3)
y = (y(2)/(3 ) / 2) * cube root (27)
y = (y(2)/(3 ) / 2) * 3
y = y(2)/(3 ) * (3/2)
y = y(3)/(2)
Therefore, when x=24, y = y(3)/(2).
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If h(w)=36w^(5)+36w^(4)+7w^(2)+12w+39, use synthetic division to find h(-1) Submit
Using synthetic division, we can find that h(-1) = -5.
To find h(-1) using synthetic division, we can use the following steps:
Write the coefficients of the polynomial in descending order of the exponents: 36, 36, 0, 7, 12, 39
Write the value of -1 to the left of the coefficients: -1 | 36 36 0 7 12 39
Bring down the first coefficient: -1 | 36 36 0 7 12 39
36
Multiply the first coefficient by -1 and write the result under the second coefficient: -1 | 36 36 0 7 12 39
36 -36
Add the second coefficient and the result: -1 | 36 36 0 7 12 39
36 0
Repeat steps 4 and 5 for the remaining coefficients: -1 | 36 36 0 7 12 39
36 0 0 -7 -5
The last number in the bottom row is the remainder, which is the value of h(-1): h(-1) = -5
Therefore, using synthetic division, we can find that h(-1) = -5.
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(c) In a test of a new package design, you drop a carton of a dozen eggs from a height of 1 foot and count the number of broken eggs
The number of broken eggs will give you an idea of how well the new package design protects the eggs from impacts. If there are no broken eggs, the package design may be effective. However, if there are several broken eggs, the package design may need to be improved to provide better protection.
When testing a new package design, it is important to simulate real-world conditions as much as possible. In this case, dropping a carton of a dozen eggs from a height of 1 foot is a good way to simulate the types of impacts that the package may experience during shipping and handling.
To conduct the test, you will need to follow these steps:
Obtain a carton of a dozen eggs and the new package design.Place the eggs inside the package according to the manufacturer's instructions.Find a suitable location to drop the package from a height of 1 foot. Make sure the area is clear and there is nothing that could interfere with the drop or damage the package.Drop the package from a height of 1 foot.Open the package and examine the eggs. Count the number of broken eggs.
The number of broken eggs will give you an idea of how well the new package design protects the eggs from impacts. If there are no broken eggs, the package design may be effective.
However, if there are several broken eggs, the package design may need to be improved to provide better protection.
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How to prove |z+w| |z| + |w|, z and w
are two complex numbers. (Use algebraic proof,
please don't use geometric proof! I want detailed explanation.
)
To prove that |z+w| ≤ |z| + |w| for two complex numbers z and w, we can use an algebraic proof.
First, let's rewrite z and w in terms of their real and imaginary parts:
z = a + bi
w = c + di
Now, we can use the definition of the absolute value of a complex number to write:
|z+w| = |(a+c) + (b+d)i|
= √((a+c)² + (b+d)²)
Similarly, we can write:
|z| = |a + bi| = √(a² + b²)
|w| = |c + di| = √(c² + d²)
Now, we can use the triangle inequality to prove that |z+w| ≤ |z| + |w|:
√((a+c)² + (b+d)²) ≤ √(a² + b²) + √(c² + d²)
Squaring both sides of the inequality gives us:
(a+c)² + (b+d)² ≤ (a² + b²) + (c² + d²) + 2√((a² + b²)(c² + d²))
Expanding the left-hand side of the inequality gives us:
a² + 2ac + c² + b² + 2bd + d² ≤ a² + b² + c² + d² + 2√((a² + b²)(c² + d²))
Simplifying and rearranging terms gives us:
2ac + 2bd ≤ 2√((a² + b²)(c² + d²))
Dividing both sides of the inequality by 2 gives us:
ac + bd ≤ √((a² + b²)(c² + d²))
Squaring both sides of the inequality again gives us:
(a² + b²)(c² + d²) - (ac + bd)² ≥ 0
Expanding and simplifying gives us:
a²c² + a²d² + b²c² + b²d² - a²c² - 2abcd - b²d² ≥ 0
a²d² + b²c² - 2abcd ≥ 0
(a²d² - 2abcd + b²c²) ≥ 0
(a² - 2ab + b²)(d² - 2cd + c²) ≥ 0
(a - b)²(d - c)² ≥ 0
Since the square of any real number is always greater than or equal to zero, this inequality is always true. Therefore, |z+w| ≤ |z| + |w| for any two complex numbers z and w. This completes the algebraic proof.
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In Exercises 22–26, give a set-theoretic description of the given points as a subset W of R^2. 22. The points on the line x – 2y = 1 23. The points on the x-axis 24. The points in the upper half-plane 25. The points on the line y = 2 26. The points on the parabola y = x2
The set W of points described in Exercises 22-26 can be written as a subset of R2 as follows: 22. The points on the line x – 2y = 1: W = {(x, y) | x - 2y = 1}, 23. The points on the x-axis: W = {(x, y) | y = 0}, 24. The points in the upper half-plane: W = {(x, y) | y > 0}, 25. The points on the line y = 2: W = {(x, y) | y = 2} and 26. The points on the parabola y = x2: W = {(x, y) | y = x2}
In other words, the set W contains all points (x, y) that satisfy the equations given in Exercises 22-26. As such, it is a subset of the two-dimensional Euclidean space R2. In Exercises 22–26, give a set-theoretic description of the given points as a subset W of R² is a problem where we need to find a set-theoretic description of the given points in each exercise.
Therefore, we can write the set W asW = { (x, y) ∈ R² | y ≥ 0 } The points on the line y = 2The equation of the line is y = 2Therefore, we can write the set W asW = { (x, y) ∈ R² | y = 2 } The points on the parabola y = x²The equation of the parabola is y = x²Therefore, we can describe the set W asW = { (x, y) ∈ R² | y = x² }
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Sides 10x and 6x+8 what is the perimeter of the square
Answer:
Step-by-step explanation:
SHORT ANSWER: perimeter = 4 * (6x+8) = 24x + 32
So, depending on which expression represents the length of the side of the square, the perimeter would be either 40x or 24x + 32.
LONG ANSWER WITH STEP-BY STEP:
If one side of the square has a length of 10x, then all sides have a length of 10x. Alternatively, if one side has a length of 6x+8, then all sides have a length of 6x+8. Therefore, we can write:
perimeter = 4 * side length
Substituting in the given expressions for the side length, we get:
perimeter = 4 * (10x) = 40x
or
perimeter = 4 * (6x+8) = 24x + 32
So, depending on which expression represents the length of the side of the square, the perimeter would be either 40x or 24x + 32.
LINEAR EQUATIONS AND INEQUALITES Solving a linear equation with several o Solve for v. 3(v-4)-6=-7(-4v+4)-7v Simplify. your answer as much as possible. v
To solve for v in the equation 3(v-4)-6=-7(-4v+4)-7v, we need to simplify the equation and isolate the variable v on one side of the equation. Here are the steps:
Step 1: Distribute the 3 and -7 on the left and right sides of the equation respectively:
3v - 12 - 6 = 28v - 28 - 7v
Step 2: Combine like terms on both sides of the equation: 3v - 18 = 21v - 28
Step 3: Move the variable terms to one side of the equation and the constant terms to the other side: 3v - 21v = -28 + 18
Step 4: Simplify both sides of the equation:
-18v = -10
Step 5: Divide both sides of the equation by -18 to solve for v: v = -10/-18
Step 6: Simplify the fraction: v = 5/9
Therefore, the solution for v is 5/9.
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A soccer player scores 3 goals in 2 games.How many goals are they expected to score in 9 games?
Answer: I could be wrong but its either going to be 13.5 or 13.
Step-by-step explanation:
Answer:
13.5 goals
Step-by-step explanation:
We can use a ratio to solve
3 goals x
--------------- = -------------
2 games 9 games
Using cross products
3 * 9 = 2x
27 = 2x
Divide each side by 2
27/2 = x
13.5 goals
You are holding a kite string in your hand. The angle of elevation from your hand to the kite is 40° and the distance to the kite is 289 feet. Your hand is 4 feet above the ground. How high is the kite? Round your answer to the nearest tenth of a foot.
Please help (will give brainiest)
Answer:
The height of the kite from the ground nearest to the tenth will be 443.6 feet.
Step-by-step explanation:
What does a dot plot show us that a box and whisker doesn't?
The dot plot allows us to know the exact data we are graphing.
What does a dot plot show us that a box and whisker doesn't?In a dot plot, we have the different possible values on the horizontal axis, and dots above each of these values that count how many times that value has appeared in an experiment.
Instead, in a whisker plot or a box plot we have a representation in a kinda of "rectangle with whiskers".
This type of plot is usefull to find the quartiles of the data set, but not to know exactly how many of each data points we have, like in the dot plot.
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if I get an annual income of 420 600,000 and get an increase of 8.2% calculate my new income
Answer:
Step-by-step explanation:
To calculate your new income after an increase of 8.2%, you can use the following formula:
New income = Old income + (Percentage increase * Old income)
Plugging in the values given in the problem, we get:
New income = 420,600,000 + (8.2% * 420,600,000)
New income = 420,600,000 + (0.082 * 420,600,000)
New income = 420,600,000 + 34,524,120
New income = 455,124,120
Therefore, your new income after an increase of 8.2% would be 455,124,120.
Determine if the vectors are linearly independent. Justify your answer. [[4],[0],[0]],[[9],[3],[-6]],[[6],[6],[-18]]
The vectors [[4],[0],[0]],[[9],[3],[-6]],[[6],[6],[-18]] are linearly independent.
To determine if the vectors are linearly independent, we can use the determinant of the matrix formed by the vectors. If the determinant is not equal to 0, then the vectors are linearly independent.
The matrix formed by the vectors is:
\begin{bmatrix}
4 & 9 & 6\\
0 & 3 & 6\\
0 & -6 & -18
\end{bmatrix}
The determinant of this matrix is:
\begin{vmatrix}
4 & 9 & 6\\
0 & 3 & 6\\
0 & -6 & -18
\end{vmatrix} = 4\begin{vmatrix}
3 & 6\\
-6 & -18
\end{vmatrix} - 9\begin{vmatrix}
0 & 6\\
0 & -18
\end{vmatrix} + 6\begin{vmatrix}
0 & 3\\
0 & -6
\end{vmatrix} = 4(-54-(-36)) - 9(0-0) + 6(0-0) = 4(-18) = -72
Since the determinant is not equal to 0, the vectors are linearly independent. Therefore, the vectors [[4],[0],[0]],[[9],[3],[-6]],[[6],[6],[-18]] are linearly independent.
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can any of you help me please
The two equivalent expressions are the second one and the last:
(56x + 24)/8 and 3 + 7x
How to identify the equivalent expression?We want to find an equivalent expression to 56x + 24 divided by 8, so we want to simplify the expression:
(56x + 24)/8 (that is the second expression)
We can distribute that division so we get:
(56x + 24)/8 = (56x)/8 + 24/8
Now we can simplify these two quotients so we get:
(56x)/8 + 24/8 = 7x + 3 (that is the last expression).
Then the two equivalent expressions are:
(56x + 24)/8 and 3 + 7x
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Determine whether or not the given vectors in R" form a basis for R". V1 =(0,0,2), V2 =(5,2,9), V3 =(7,2,11)
To determine whether the given vectors in $\mathbb{R}^3$ form a basis for $\mathbb{R}^3$, we need to check if they are linearly independent and span $\mathbb{R}^3$.
To check for linear independence, we set up the augmented matrix:
[
0
5
7
0
0
2
2
0
2
9
11
0
]
⎣
⎡
0
0
2
5
2
9
7
2
11
0
0
0
⎦
⎤
We reduce this to echelon form:
[
1
0
2
0
0
1
1
0
0
0
0
0
]
⎣
⎡
1
0
0
0
1
0
2
1
0
0
0
0
⎦
⎤
Since there is a row of zeros, the rank of the matrix is less than 3, which means the vectors are linearly dependent.
Therefore, the given vectors do not form a basis for $\mathbb{R}^3$.
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a line segment is drawn between (4,7) and (9,7). find it’s gradient.
Considering the expression of a line, the gradient or slope is zero and the line is horizontal.
Definition of linear equationA linear equation o line can be expressed in the form y = mx + b
where
x and y are coordinates of a point.m is the slope. The gradient or slope of a line tells how steep it is.b is the ordinate to the origin.Knowing two points (x₁, y₁) and (x₂, y₂) of a line, the slope of the line can be calculated using:
m= (y₂ - y₁)÷ (x₂ - x₁)
Gradient in this caseIn this case, being (x₁, y₁)= (4, 7) and (x₂, y₂)= (9, 7), the slope m can be calculated as:
m= (7 -7)÷ (9 -4)
m= 0÷ 5
m= 0
Finally, the gradient or slope is zero.
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From a population that is normally distributed with unknown mean µ and a standard deviation of 1.25, a sample of size 36 is taken.
(i) What is the standard error of the sample mean? (2 marks)
(ii) What is the probability that the sample mean is within 0.5 units of the population mean? (4 marks)
(iii) How large a sample size must we take to be 98% confident that the sample mean is within one-half of the population mean? (4 marks)
Thhe standard error of the sample mean is 0.208333. The probability that the sample mean is within 0.5 units of the population mean is 0.9918. We need to take a sample size of at least 7 to be 98% confident that the sample mean is within one-half of the population mean.
(i) The standard error of the sample mean is calculated by dividing the standard deviation of the population by the square root of the sample size. In this case, the standard error of the sample mean is 1.25 / √36 = 1.25 / 6 = 0.208333.
(ii) To find the probability that the sample mean is within 0.5 units of the population mean, we need to use the standard normal distribution. We can find the z-score for 0.5 units away from the mean by dividing 0.5 by the standard error of the sample mean, which is 0.5 / 0.208333 = 2.4.
Using a standard normal table, we can find the probability that the sample mean is within 2.4 standard deviations of the population mean, which is 0.9918.
(iii) To be 98% confident that the sample mean is within one-half of the population mean, we need to find the sample size that corresponds to a z-score of 2.33 (the z-score for a 98% confidence interval). We can use the formula for the standard error of the sample mean to solve for the sample size:
0.5 = 1.25 / √n
√n = 1.25 / 0.5
n = (1.25 / 0.5)²
n = 6.25
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In the figure, line m is parallel to line n. The measure of <3 is 58 degrees. What is the measure of <7?
In the parallel line measure of angle [tex]m\angle 7[/tex] is 32°.
What is parallel lines?In a plane, two lines are said to be parallel if they never cross at any point. A pair of lines that never cross paths and do not have a common junction point are said to be parallel. Parallel lines are represented by the symbol "||".
Here we know that If two lines which are parallel are intersected by a transversal then the pair of corresponding angles are equal.
Then,
=> [tex]m\angle3= m\angle10[/tex]
Here the given [tex]m\angle3=58\textdegree[/tex] the [tex]m\angle10=58\textdegree[/tex].
Now we know that sum of all angles in straight line is 180°.Then,
=> [tex]m\angle6+m\angle7+m\angle10=180\textdegree[/tex]
=> [tex]90\textdegree+m\angle7+58\textdegree=180\textdegree[/tex]
=> [tex]m\angle7=180\textdegree-90\textdegree-58\textdegree=32\textdegree[/tex]
Hence the measure of [tex]m\angle 7[/tex] is 32°.
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Which rational expression is equivalent to this expression? (4)/(x-3) A. (x-3)/(x+2)-:(4)/(x+2) B. (x+2)/(x-3)-:(4)/(x+2) C. (x+2)/(x-3)*(4)/(x+2) D. (x-3)/(x+2)*(x+2)/(x^(4))
Option C. (x+2)/(x-3)*(4)/(x+2) is the equivalent rational expression to (4)/(x-3).
The rational expression that is equivalent to this expression (4)/(x-3) is option C. (x+2)/(x-3)*(4)/(x+2).
We can simplify the rational expression (x+2)/(x-3)*(4)/(x+2) by canceling out the common factor (x+2) from the numerator and denominator. This will give us the equivalent rational expression:
(x+2)/(x-3)*(4)/(x+2) = (4)/(x-3)
Therefore, option C. (x+2)/(x-3)*(4)/(x+2) is the equivalent rational expression to (4)/(x-3).
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Find the missing side
Answer: 30
Step-by-step explanation:
Use Pythagorean's Theorem, a² + b² = c²
18² + 24² = c²
324 + 576 = c²
900 = c²
√900 = c
c = 30
The missing side is 30.
Hope this helps!
Find the missing variable and indicated
angle measure.
D
X =
7xº
G
I
62°
F
E
m²FDG =
Will mark brainiest if you answer fast please
Answer:
FDG is 28 degrees, and x = 4
Step-by-step explanation:
We can see for angle EDG that there is a little box at the angle. This means that it is a right angle, which means it is 90 degrees. This, in turn, means that the two angles formed there (angle EDF, and angle FDG) will add up to that number, 90 degrees because they are within the initial angle.
Let us create an equation using our angles. We know one is 62 degrees and the other is 7x degrees.
62 +7x = 90
7x = 28
x =4
This means that the unknown value is 4, and since the angle was 7x, it is really 7(4) = 28 degrees, and we know the other angle is 62 degrees.
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TOFAS A testing tofaseducation Please do the following calculation. need to simplify your answer. (+(13)/(18))+(-(1)/(3))
The expression is (+(13)/(18))+(-(1)/(3)). To simplify this expression, we need to combine like terms. First, we can combine the fractions with different denominators. The LCD (Least Common Denominator) for this expression is 18. Therefore, we need to convert each fraction to an equivalent fraction with a denominator of 18.
For the first fraction, (+(13)/(18)), 13/18 can be reduced to 1/2. To get this, multiply both the numerator and denominator by 2, resulting in 13/18 = 2/4 = 1/2.
For the second fraction, +(-(1)/(3)), 1/3 can be reduced to 6/18. To get this, multiply both the numerator and denominator by 6, resulting in 1/3 = 6/18.
Now that all the fractions have a common denominator, we can add the two fractions together. 1/2 + 6/18 = 8/18. Therefore, the simplified form of the expression is 8/18.
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12. Solve the following triangles using Law of Sines or Law of Cosines (round to nearest tenth when necessary and find all solutions) *Must show path/process/work for full credit": a. A-58 a. B- b=12
Answer:
sin(B)/b = sin(A)/a
sin(B)/12 = sin(58)/a
a = 12(sin(58)/sin(B))
Now we can use the Law of Cosines to find the remaining sides of the triangle:
a^2 = b^2 + c^2 - 2bc*cos(A)
a^2 = 12^2 + c^2 - 2(12)(c)*cos(58)
c^2 - 24c*cos(58) + 144 - a^2 = 0
Using the quadratic formula, we get:
c = (24*cos(58) ± sqrt((24*cos(58))^2 - 4(1)(144 - a^2)))/2(1)
c = 12*cos(58) ± sqrt(144*cos(58)^2 - 4(144 - a^2))
c = 12*cos(58) ± sqrt(576*cos(58)^2 - 4a^2)
c = 12*cos(58) ± sqrt(576*(1 - sin(58)^2) - 4a^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 4a^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 4(12(sin(58)/sin(B)))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/sin(B))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/sin(B))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/sin(B))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/sin(B))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/sin(B))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/sin(B))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/sin(B))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/sin(B))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(180 - A - B)))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(180 - 58 - B)))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - B)))^2)
Now we can substitute the value we found for a into the equation for c to get:
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - B)))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - arcsin(a/b))))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - arcsin(12/a))))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - arcsin(12/(12(sin(58)/sin(B)))))))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - arcsin(sin(58)/sin(B))))))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - arcsin(sin(58)/(12*sin(58)/a))))))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - arcsin(a/12))))))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - arcsin(1/12)*a))))))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - 4.98)*a))))))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(117.02)*a))))))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(0.97*a))))))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(1.03*a/sin(58))))))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(1.03*(12*sin(58)/sin(B))/sin(58))))))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(1.03*(12/sin(B)))))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(1.03*(12/sin(180 - A - B)))))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(1.03*(12/sin(180 - 58 - B)))))^2)
c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(1.03*(12/sin(122 - B)))))^2)
Now we can solve for c using the two possible values of B:
B = arcsin(b*sin(A)/a)
B = arcsin(12*sin(58)/a)
B = arcsin(12*sin(58)/(12*sin(58)/sin(B)))
B = arcsin(sin(B))
B = 58
or
B = 180 - arcsin(b*sin(A)/a)
B = 180 - arcsin(12*sin
Suppose a golf club company has designed a new club, which it claims will allow a professional golfer to make a hole in 120% of the time and an amateur golfer 10% of the time. Professional an amateur golfers sign up to play 5 games of 18 holes each
A professional golfer to make about 40.7 holes over 5 rounds of golf with the new club, while an amateur golfer would only make about 1.6.
First, let's define some variables to represent the probabilities of making a hole for a professional golfer and an amateur golfer:
Let p be the probability that a professional golfer makes a hole with the new club.
Let q be the probability that an amateur golfer makes a hole with the new club.
According to the company's claims, we know that:
p = 1.2q (since the professional golfer makes a hole 120% of the time, which is 1.2 times the probability of the amateur golfer making a hole)
Next, we need to determine the probability of each golfer making a hole during one round of golf, which consists of 18 holes. Let's assume that each hole is independent of the others, meaning that the outcome of one hole does not affect the outcome of another. In that case, the probability of making at least one hole in a round can be calculated using the complement rule:
The probability that a professional golfer makes at least one hole in a round is 1 minus the probability that the golfer misses every hole: [tex]1 - (1-p)^{18} .[/tex]
The probability that an amateur golfer makes at least one hole in a round is[tex]1 - (1-q)^{18} .[/tex]
Now, let's use these probabilities to calculate the expected number of holes each golfer will make in 5 rounds of golf:
The expected number of holes made by a professional golfer in 5 rounds is 5 times the expected number of holes made in one round, which is [tex](1 - (1-p)^{18} )\times18.[/tex]
The expected number of holes made by an amateur golfer in 5 rounds is 5 times the expected number of holes made in one round, which is [tex](1 - (1-q)^{18} )\times18.[/tex]
We can simplify these expressions using the relationship between p and q:
The expected number of holes made by a professional golfer in 5 rounds is [tex]518(1 - (1-1.2q)^{18} ).[/tex]
The expected number of holes made by an amateur golfer in 5 rounds is [tex]518(1 - (1-q)^{18} ).[/tex]
We can now evaluate these expressions using the values of p and q:
[tex]p = 1.2q, so q = p/1.2[/tex]
Substituting this into the expressions above, we get:
The expected number of holes made by a professional golfer in 5 rounds is[tex]518(1 - (1-1.2(p/1.2))^{18} ) = 518(1 - (1-p)^{18} ).[/tex]
The expected number of holes made by an amateur golfer in 5 rounds is [tex]518(1 - (1-p/1.2)^{18} ).[/tex]
Finally, we can evaluate these expressions using the given probabilities:
The expected number of holes made by a professional golfer in 5 rounds is[tex]518(1 - (1-1.2q)^{18} ) = 518(1 - (1-1.2(0.1))^{18} ) = 40.7.[/tex]
The expected number of holes made by an amateur golfer in 5 rounds is [tex]518(1 - (1-q)^{18} ) = 518(1 - (1-0.1/1.2)^{18} ) = 1.6.[/tex]
So according to these calculations, we would expect a professional golfer to make about 40.7 holes over 5 rounds of golf with the new club, while an amateur golfer would only make about 1.6
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[(0)/(1) Points ] DETAILS PREVIOL Perform the indicated division. (4x^(3)-5x^(2)+8x-8)/(x^(2)-3x)
The division of polynomials "(4x^(3)-5x^(2)+8x-8)/(x^(2)-3x)" gives the expression "4x+7+(29x-8)/(x^(2)-3x)".
To perform the indicated division, we will use polynomial long division.
First, we will divide the first term of the dividend, 4x^3, by the first term of the divisor, x^2, to get 4x. This will be the first term of our quotient.
Next, we will multiply 4x by the divisor, x^2-3x, to get 4x^3-12x^2. We will then subtract this from the dividend to get 7x^2+8x-8.
We will then repeat this process by dividing the first term of the new dividend, 7x^2, by the first term of the divisor, x^2, to get 7. This will be the second term of our quotient.
We will then multiply 7 by the divisor, x^2-3x, to get 7x^2-21x. We will subtract this from the new dividend to get 29x-8.
Since the degree of the new dividend, 29x-8, is lower than the degree of the divisor, x^2-3x, we are done with the division and 29x-8 will be our remainder.
Therefore, the final answer is:
(4x^(3)-5x^(2)+8x-8)/(x^(2)-3x) = 4x+7+(29x-8)/(x^(2)-3x)
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When QuickSort splits the list exactly in half we saw that T(n) = 2T(n/2) + O(n) and then T(n) = (n lgn) by the Master Theorem. Here we're going to examine the case where QuickSort splits the list into one sublist of size an and another sublist of size (1 –a)n for some 0
The running time of QuickSort in this case is O(n log(n)), which is the same as the running time when QuickSort splits the list exactly in half.
When QuickSort splits the list into one sublist of size an and another sublist of size (1-a)n, the recurrence relation for the running time of QuickSort becomes T(n) = T(an) + T((1-a)n) + O(n). This is because the two sublists have different sizes and therefore take different amounts of time to sort.
To solve this recurrence relation, we can use the recursion tree method. The recursion tree for this recurrence relation looks like this:
```
T(n)
/ \
T(an) T((1-a)n)
/ \ / \
T(a^2n) T(a(1-a)n) T(a(1-a)n) T((1-a)^2n)
...
```
At each level of the recursion tree, the size of the subproblems decreases by a factor of a or (1-a), and the number of subproblems doubles. The work done at each level is O(n), since the partitioning step takes O(n) time.
The recursion tree has log_{1/a}(n) levels, since the size of the subproblems decreases by a factor of a at each level. Therefore, the total work done by QuickSort is O(n log_{1/a}(n)) = O(n log(n)), since log_{1/a}(n) = log(n)/log(1/a) = log(n)/(-log(a)) = -log(n)/log(a) = O(log(n)).
So the running time of QuickSort in this case is O(n log(n)), which is the same as the running time when QuickSort splits the list exactly in half.
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