if csc(x) = 7 and x is in quadrant II, then cos(x/2) = -√((7 - 4√(3))/14).
How to solve the question?
We can start by using the reciprocal identity of csc to find the sin of x.
csc(x) = 7 can be rewritten as sin(x) = 1/7
Next, we can use the Pythagorean identity to find the cosine of x:
cos²(x) = 1 - sin²(x) = 1 - (1/7)²= 48/49
Since x is in quadrant II, cosine is negative, so cos(x) = -√(48/49) = -4√(3)/7
Now we can use the half-angle identity for cosine to find cos(x/2):
cos(x/2) = ±√((1 + cos(x))/2)
We know that cos(x) is negative and that x is in quadrant II, which means that x/2 is also in quadrant II.
Therefore, cos(x/2) = -√((1 - 4√(3)/7)/2)
We can simplify this expression by rationalizing the denominator:
cos(x/2) = -√((7 - 4√(3))/14)
In summary, if csc(x) = 7 and x is in quadrant II, then cos(x/2) = -√((7 - 4√(3))/14).
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The ratio of union members to nonunion members working for a company is 4 to 5. If there are 140 nonunion members working for the company,
what is the total number of employees?
The total number of employees is 112.
Explain numbers
Numbers are symbols or representations used to quantify or count objects, quantities, or measurements. They form the basis of mathematical operations, such as addition, subtraction, multiplication, and division, and are used in various fields such as science, finance, and engineering. Numbers can be positive, negative, whole, or fractional, and are essential for communication and calculation in our daily lives.
According to the given information
Let's use x to represent the total number of employees.
According to the problem, the ratio of union members to nonunion members is 4 to 5. This means that out of every 4 + 5 = 9 employee, 4 are union members and 5 are nonunion members.
So, we can set up the following proportion:
4/9 = x/(x - 140)
To solve for x, we can cross-multiply and simplify:
4(x - 140) = 9x
4x - 560 = 9x
560 = 5x
x = 112
Therefore, the total number of employees is 112.
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Solve by using matrices.
2x -y + 3z = 180
-4x + 2y + 3z = 225
3x - 4y = 270
X
= -66, y = [?], z =
Enter
Solving the system of equations using matrices is : = -66, y = 163, and z = 11.
Solving the system of equations using matrices ?To solve this system of equations using matrices, we can write it in the form AX = B, where:
A = coefficient matrix
X = variable matrix (containing x, y, and z)
B = constant matrix (containing the constants on the right-hand side of each equation)
So, we have:
| 2 -1 3 | | x | | 180 |
| -4 2 3 | x | y | = | 225 |
| 3 -4 0 | | z | | 270 |
We can solve for X by multiplying both sides of the equation by the inverse of A:
X = A^-1 * B
First, we need to find the inverse of A. We can do this by using the formula:
A^-1 = (1 / det(A)) * adj(A)
where det(A) is the determinant of A and adj(A) is the adjugate (transpose of the cofactor matrix) of A.
| 2 -1 3 |
| -4 2 3 |
| 3 -4 0 |
det(A) = 2(20 - 3(-4)) - (-1)(-40 - 33) + 3(-4*(-1) - 2*3) = 16
| 2 -1 3 |
| -4 2 3 |
| 3 -4 0 |
The cofactor matrix is:
| 2 9 6 |
| 12 0 -2 |
| 13 -9 8 |
Taking the transpose of the cofactor matrix gives us the adjugate of A:
| 2 12 13 |
| 9 0 -9 |
| 6 -2 8 |
So, we have:
A^-1 = (1 / det(A)) * adj(A) = (1 / 16) *
| 2 12 13 |
| 9 0 -9 |
| 6 -2 8 |
Multiplying A^-1 by B gives us:
| x | | -66 |
| y | = | 163 |
| z | | 11 |
Therefore, x = -66, y = 163, and z = 11.
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Al has a goal to read 2,000 pages over
summer break. He has read 1,248 pages.
How many more pages does Al need to
read to reach his goal?
Al needs to read 752 more pages to reach his goal of 2,000 pages.
What is subtraction?The act of deleting items from a collection is represented by subtraction. Subtraction is denoted by the minus sign.
To find out how many more pages Al needs to read to reach his goal of 2,000 pages, we can subtract the number of pages he has already read from his goal:
Number of pages Al needs to read = Goal - Pages already read
Number of pages Al needs to read = 2,000 - 1,248
Number of pages Al needs to read = 752
Therefore, Al needs to read 752 more pages to reach his goal of 2,000 pages.
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Tentor, Inc., purchases disposable coffee cups on which to print logos for sporting events, proms, birthdays, and other special occasions. The owner received a large shipment of 861 cups this afternoon, and to ensure the quality of the shipment, he selected a random sample of 410 cups and identified 353 as defective.
What is the estimated proportion of defectives in the population? (Round the final answer to 3 decimal places.)
Answer
What is the standard error of the sample proportion? (Round your answer to 3 decimal places.)
Answer
What are the upper and lower bounds for a 98% confidence level? (Round the final answers to 3 decimal places.)
Upper bound is Answer
Lower bound is Answer
It is estimated that 0.861 percent of the population is faulty. The sample proportion's standard error is 0.022. A 98% confidence level has an upper bound of 0.910 and a lower bound of 0.812.
What is a proportion?The comparative relationship between two or more things in terms of their size, amount, or number is referred to as a "proportion." Either a ratio or a fraction can be used to express it. The term "proportion" in statistics refers to the division of the total number of events by the frequency of each event.
The formula p = x/n, where p is the estimated proportion of defectives in the population, x is the number of defectives in the sample, and n is the sample size, can be used to determine the estimated proportion of defectives in the population.
When we substitute values, we obtain:
p = 353/410 = 0.861
As a result, the population's estimated defectiveness rate is 0.861.
The formula SE = √(p(1-p)/n), where SE is the standard error and n is the sample size, can be used to get the standard error of the sample percentage.
When we substitute values, we obtain:
SE is equal to√(0.861(1.0.861)/410) = 0.022.
As a result, the sample proportion's standard error is 0.022.
Using the following formula, the upper and lower bounds for a 98% confidence level can be determined:
Lower bound = z*SE - p
Upper bound = z*SE + p
where z is the z-score for a 98% degree of confidence.
We discover that the z-score corresponding to a 98% confidence level is roughly 2.33 using a z-table or calculator.
When we substitute values, we obtain:
Lower bound is equal to 0.861 - 2.33*0.022, or 0.812.
Upper bound is equal to 0.861 + 2.33 * 0.022 = 0.910.
Consequently, the range of a 98% confidence level is as follows:
Maximum: 0.910
Upper limit: 0.812
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It is estimated that 0.861 percent of the population is faulty. The sample proportion's standard error is 0.022. A 98% confidence level has an upper bound of 0.910 and a lower bound of 0.812.
What is a proportion?
The comparative relationship between two or more things in terms of their size, amount, or number is referred to as a "proportion." Either a ratio or a fraction can be used to express it. The term "proportion" in statistics refers to the division of the total number of events by the frequency of each event.
The formula p = x/n, where p is the estimated proportion of defectives in the population, x is the number of defectives in the sample, and n is the sample size, can be used to determine the estimated proportion of defectives in the population.
When we substitute values, we obtain:
p = 353/410 = 0.861
As a result, the population's estimated defectiveness rate is 0.861.
The formula SE = √(p(1-p)/n), where SE is the standard error and n is the sample size, can be used to get the standard error of the sample percentage.
When we substitute values, we obtain:
SE is equal to[tex]\sqrt{\frac{0.861(1.0.861)}{410)}[/tex]= 0.022.
As a result, the sample proportion's standard error is 0.022.
Using the following formula, the upper and lower bounds for a 98% confidence level can be determined:
Lower bound = z*SE - p
Upper bound = z*SE + p
where z is the z-score for a 98% degree of confidence.
We discover that the z-score corresponding to a 98% confidence level is roughly 2.33 using a z-table or calculator.
When we substitute values, we obtain:
Lower bound is equal to 0.861 - 2.33*0.022, or 0.812.
Upper bound is equal to 0.861 + 2.33 * 0.022 = 0.910.
Consequently, the range of a 98% confidence level is as follows:
Maximum: 0.910
Upper limit: 0.812
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Workout 461÷4 give your answer as a whole number and a reminder
Step-by-step explanation:
the answer is 115 remainder 1
Which is the largest of the followin fractions? 7/12, 2/3,5/6,3/4, 5/8, (a) 2/3 (b) 3/4 (c) 5/ 5
Answer:
Step-by-step explanation:it is 5/5 because it is a whole number and all the others aren’t
Is the graph increasing, decreasing, or constant?
-8 -6 4
A. Decreasing
B. Increasing
C. Constant
-2
4-
2
-2
AY
2
4
6
8
SUBMIT
Is the graph increasing, decreasing, or constant: C. Constant.
What is a graph?In Mathematics and Geometry, a graph can be defined as a type of chart that is typically used for the graphical representation of data points or ordered pairs on both the horizontal and vertical lines of a cartesian coordinate, which are the x-coordinate (x-axis) and y-coordinate (y-axis) respectively.
By critically observing the graph shown above, we can reasonably infer and logically deduce that the graph is constant because it assumes only one y-value, which is represented by this equation y = 3.
In conclusion, the above is constant, rather than increasing or decreasing.
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ZA =
Round your answer to the nearest hundredth.
Angle A equals 41.81°
c:
The number of potholes in any given 1 mile stretch of freeway pavement in Pennsylvania has a bell-shaped distribution. This distribution has a mean of 62 and a standard deviation of 11. Using the empirical rule (as presented in the book), what is the approximate percentage of 1-mile long roadways with potholes numbering between 51 and 84?
Determine the domain and range of the graph of this function.
x ≤ 4, -2≤ y ≤ 2 are the domain and range of the graph of this function.
What is the domain and domain of the function in the graph?
Another way to identify domains and feature sets is with graphs. Domain refers to the set of possible input values, so a chart's domain consists of all input values displayed on the x-axis.
A range is the set of possible output values plotted on the y-axis. To find the domain and domain of the equation y = f(x), find the value of the independent variable x for which the function is defined.
y = f(x)
according to the graph value of y
x ≤ 4,
-2≤ y ≤ 2
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x=10 3x+5y=20 in the system of equations, what is the value of x
A coordinate plane with 2 lines drawn. The first line is labeled f(x) and passes through the points (0, negative 2) and (1, 1). The second line is labeled g(x) and passes through the points (negative 4, 0) and (0, 2). The lines intersect at about (2.5, 3.2)
How does the slope of g(x) compare to the slope of f(x)?
The slope of g(x) is the opposite of the slope of f(x).
The slope of g(x) is less than the slope of f(x).
The slope of g(x) is greater than the slope of f(x).
The slope of g(x) is equal to the slope of f(x)
Therefore, the correct answer is: The slope of g(x) is less than the slope of f(x).
Where do the X and Y axes intersect on the coordinate plane, at position 0 0?The origin is the location where the two axes meet. On both the x- and y-axes, the origin is at 0. The coordinate plane is divided into four portions by the intersection of the x- and y-axes. The term "quadrant" refers to these four divisions.
We can use the slope formula to get the slopes of the lines f(x) and g(x):
slope of f(x) = (change in y)/(change in x) = (1 - (-2))/(1 - 0) = 3/1 = 3
slope of g(x) = (change in y)/(change in x) = (2 - 0)/(0 - (-4)) = 2/4 = 1/2
The slope of g(x) is 1/2, which is less than the slope of f(x), which is 3.
Therefore, the correct answer is: The slope of g(x) is less than the slope of f(x).
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Answer:
B
Step-by-step explanation:
pls help :)
Angie is really fit and wants to test her endurance by competing in a swimming competition over four days. On the first day, she covers 25 per cent of the distance. On the second day, she covers one third of the remaining distance, on the third day, 25 per cent of the remaining distance and, on the final day, she swims half of the remaining distance. She still has 15km to swim to complete the competition. Form a linear equation and find the total distance of the swimming competition. (Use x as the total distance of the swimming competition).
Answer: Total distance of the swimming competition is 55.56 km.
Step-by-step explanation:
Let x be the total distance of the swimming competition.
On the first day, Angie covered 25% of the total distance, which means she swam 0.25x km.
On the second day, she covered one-third of the remaining distance, which is (1 - 0.25)x = 0.75x km.
So, she swam (1/3) * 0.75x = 0.25x km on the second day.
On the third day, she covered 25% of the remaining distance, which is (1 - 0.25 - 1/3)x = 0.42x km.
So, she swam 0.25 * 0.42x = 0.105x km on the third day.
On the final day, she swam half of the remaining distance,
which is (1 - 0.25 - 1/3 - 0.5)x = 0.125x km.
The total distance she swam is the sum of the distances swum on each day:
0.25x + 0.25x + 0.105x + 0.125x = 0.73x
We know that the total distance of the competition is x - 15 km (since she still has 15 km left to swim).
So we can set up the equation:
0.73x = x - 15
Solving for x, we get:
x-0.73x = 15
0.27x = 15
x = 55.56
Therefore, the total distance of the swimming competition is 55.56 km.
The angle measures for 2 supplementary angles are 3x and 2x +40.
What is the value of x?
Answer:
28°
Step-by-step explanation:
If the angles are supplementary, they add up to 180°.
(3x) + (2x + 40) = 180
5x + 40 = 180
- 40 - 40
5x = 140
x = 28°
Find the gradients of lines A and B
I'm afraid you forgot to add a picture?
What is the equation of the circle with centre
(1/2, 0)and radius 2?
Responses (attached)
The equation of the circle is (x - 1/2)^2 + y^2 = 15/4.
How to calculate the equationThe equation of a circle with center (a,b) and radius r is given by the equation:
(x - a)^2 + (y - b)^2 = r^2
Using the given values, the equation of the circle with center (1/2, 0) and radius 2 is:
(x - 1/2)^2 + (y - 0)^2 = 2^2
Expanding and simplifying, we get:
(x - 1/2)^2 + y^2 = 4 - 1/4
Therefore, the equation of the circle is:
(x - 1/2)^2 + y^2 = 15/4
So, the equation of the circle is (x - 1/2)^2 + y^2 = 15/4.
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please answer in detail
Answer:
y = 2x + 4
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = 2x + 3 ← is in slope- intercept form
with slope m = 2
• Parallel lines have equal slopes , then
slope of line AB is m = 2
line AB crosses the y- axis at (0, 4 ) ⇒ c = 4
y = 2x + 4 ← equation of line AB
please help meeeee. What is the value of k?
Answer:
k = 10
Step-by-step explanation:
the exterior angle of a triangle is equal to the sum of the 2 opposite interior angles.
∠ MYZ is an exterior angle of the triangle , then
4k + 5 + 6k + 10 = 115
10k + 15 = 115 ( subtract 15 from both sides )
10k = 100 ( divide both sides by 10 )
k = 10
Answer:
k = 10----------------------
Exterior angle of a triangle is equal to the sum of remote interior angles.
In the given picture, the exterior angle is 115°, and remote interior angles are (4k + 5)° and (6k + 10)°.
Set up equation and solve for k:
4k + 5 + 6k + 10 = 11510k + 15 = 11510k = 100k = 10Therefore the value of k is 10.
if the pink lines are parallel, solve for n
The option that is true about the equations for these two lines iis that they represent the same lines.
How to explain the informationThe trick to questions like this is to get both equations into the slope-intercept form. That is done for our first equation (y = 3x + 5). However, for the second, some rearranging must be done:
5y – 25 = 15x; 5y = 15x + 25; y = 3x + 5
Note: Not only do these equations have the same slope (3), they are totally the same; therefore, they represent the same equation.
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Two lines are described by the equations:
y = 3x + 5 and 5y – 25 = 15x
Which of the following is true about the equations for these two lines?They represent perpendicular lines.
None of the other answers
They represent non-perpendicular, intersecting lines.
They represent the same lines.
They represent parallel lines.
The value of a certain collectible card in dollars from January 2020 to May 2021 can be modeled by the function V(m)=250m² -3,500m + 18,000, where mis
the approximate number of months since the start of 2020.
Over what period was the value of the card declining?
To respond to the question, we must identify the m-values for which V(m) equation decreases. The value of the card decreased during the course of the time from January 2020 to July 2020 (about 7 months),
What is equation?A mathematical equation is a formula that connects two claims and uses the equals symbol (=) to denote equivalence. An equation in algebra is a mathematical statement that establishes the equivalence of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign places a space between the variables 3x + 5 and 14. The relationship between the two sentences that are written on each side of a letter may be understood using a mathematical formula. The symbol and the single variable are frequently the same. as in, 2x - 4 equals 2, for instance.
We must identify the values of m for which the function V(m) is dropping in order to estimate the time span during which the value of the card was declining.
The function V(m)'s derivative is provided by:
V'(m) = 500m - 3,500
By setting V'(m) = 0 and solving for m, we may determine the crucial point(s):
500m - 3,500 = 0
m = 7
V''(m) = 500
To respond to the question, we must identify the m-values for which V(m) decreases. The value of the card decreased during the course of the time from January 2020 to July 2020 (about 7 months), as we know that V(m) is falling for m 7.
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Home values in a town have declined 26% per year for each of the past
four years. What was the total percentage decrease in home values
during the four-year period?
Answer: 104%
Step-by-step explanation: 26% times 4 years
A principal of $1500 is invested at 8.5% interest, compounded annually. How much will the investment be worth after 14 years?
Use the calculator provided and round your answer to the nearest dollar.
Find the inradius of triangle ABC.
Find the circumradius of triangle ABC.
The sides of the triangle are 5, 29, and 42.
Identify the correct equation of the graph.
-10
O f(b) = (6+4)² +8
O f(b) = (b+8)² +4
Of(b)=(6-8)²-4
O
-5
10
5
-5
-10
V
5
O f(b) = (b-8)² +4
Of(b) = (6-4)²-8
Of(b) (6-4)² +8
10
Check
Thus, the correct equation for the given parabolic graph is found as: f(b) = (b – 8)² + 4.
Explain about the quadratic function in vertex form:A parabola has a lowest point if it opens upward. A parabola has a highest point if it opens downward.
The vertex of the parabola is located at this lowest or highest point.
Vertex form of a quadratic function:
f(x) = a(x – h)² + k, where a, h, and k are constants.
The vertex of the parabola is at because it is translated h horizontal units and k vertical units from the origin (h, k).
(h,k) are the vertex of parabola.
From the given graph:
f(b) is the given function:
Vertex (h,k) = (8, 4)
Thus, h= 8 and k = a = 1, x = b.
Put the values in quadratic function:
f(b) = 1(b – 8)² + 4
f(b) = (b – 8)² + 4
Thus, the correct equation for the given parabolic graph is found as: f(b) = (b – 8)² + 4.
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On January 1, Year 2, Kincaid Company's Accounts Receivable and the Allowance for Doubtful Accounts carried balances of $74,600 and $3,700, respectively. During Year 2, Kincaid reported $208,000 of credit sales, wrote off $2,000 of receivables as uncollectible, and collected cash from receivables amounting to $258,900. Kincaid estimates that it will be unable to collect one percent (1%) of credit sales.
What effect will recognizing the uncollectible accounts expense for Year 2 have on the elements of the financial statements?
Multiple Choice
Decrease total assets and net income
Decrease total assets and increase retained earnings
Increase total assets and decrease net income
Increase total assets and retained earnings
The correct answer is: Decrease total assets and net income.
What is statistics?Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of numerical data. It involves the use of methods and techniques to gather, summarize, and draw conclusions from data.
The recognition of uncollectible accounts expense for Year 2 will decrease total assets and net income.
When uncollectible accounts expense is recognized, it is recorded as a debit to the allowance for doubtful accounts and a credit to the uncollectible accounts expense account. This reduces the balance of accounts receivable, which is a current asset, and thus decreases total assets. Additionally, recognizing this expense reduces net income by the same amount.
Therefore, the correct answer is: Decrease total assets and net income.
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Draw place value disk to represent the value of the following expressions
5 times 2=
When the two sections are combined, the total number of circles would be ten, representing the value of 5 times 2, which is 10.
What is number?Number is a mathematical concept used to quantify or measure a quantity. It is a symbol or group of symbols that stands for a quantity, such as 2, 4, 5, 10, and so on. Numbers are used to express amounts, to count, and to measure. They are also used to represent relationships between quantities and to compare quantities.
A place value disk can be used to represent the value of 5 times 2, which is equal to 10. The disk would have two sections, one representing the number 5 and the other representing the number 2. The number 5 would be represented by five single circles in the “ones” section, and the number 2 would be represented by two single circles in the “ones” section. When the two sections are combined, the total number of circles would be ten, representing the value of 5 times 2, which is 10.
The place value disk is a useful visual tool to help students understand the concept of multiplication. It allows them to clearly see how the number of circles in each section are multiplied to get the final answer. This can help them better understand the concept of multiplication and how it works.
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Complete questions as follows-
A retail manager constructs a 95% confidence interval to estimate the mean amount of money each customer spends per visit to the retail store. Assume that all conditions have been met. The one-sample t-interval is ($10.53, $31.89). The owner of the retail store will issue the manager a bonus if the customers spend $35, on average, per visit. Is it reasonable to believe this manager will receive a bonus? No. Because a different sample might give different results, no conclusion can be drawn. No. Because the interval does not contain $35, the manager would not be issued the bonus. Yes. Because the interval representing the mean amount customers spend per visit is entirely below $35, it is reasonable to believe the manager will be issued a bonus. Yes. Because the interval representing the mean amount customers spend per visit does not contain $35, it is reasonable to believe the manager will be issued a bonus.
The correct option is: No. Because the interval does not contain $35, the manager would not be issued the bonus.
What is an interval?An interval refers to a range of values between two points on a numerical scale. It can be represented by two numbers, called endpoints, and includes all the values that lie between these endpoints.
The two common types of intervals are closed intervals and open intervals.
According to the given information:
In a confidence interval, the range of values within which the true population parameter (in this case, the mean amount of money each customer spends per visit) is likely to fall is estimated. A 95% confidence interval means that there is a 95% probability that the true population parameter falls within the interval.
In this case, the confidence interval is ($10.53, $31.89), which means that we are 95% confident that the true mean amount customers spend per visit falls between $10.53 and $31.89. Since the interval does not contain $35, it is not reasonable to believe that the manager will receive a bonus, as the estimated mean amount spent per visit is below $35 according to the confidence interval.
The correct answer is: No. Because the interval does not contain $35, the manager would not be issued the bonus.
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Approximately of the Earth's surface is made up of the oceans. What fraction of the surface is not made up of oceans?
The fraction of Earth that is not made up of ocean = 1/4.
Explain about the fraction:The numbers we are familiar with are whole numbers, such as 1, 2, and so on.
Numbers expressed as fractions have a numerator and a denominator, separated by a line known as a vinculum.
In essence, a fraction explains how a portion of a group interacts with the entire group.
Given that-
fraction of Earth made up of water = 3/4The fraction of Earth that is not made up of ocean = 1 - fraction of Earth made up of water
The fraction of Earth that is not made up of ocean = 1 - 3/4
The fraction of Earth that is not made up of ocean = (4 - 3)/4
The fraction of Earth that is not made up of ocean = 1/4
Thus, the fraction of Earth that is not made up of ocean = 1/4.
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Complete question:
Approximately 3/4 of the Earth's surface is made up of the oceans. What fraction of the surface is not made up of oceans?
50 Points! Write the expression x^4+5x^2-8 in quadratic form, if possible. Photo attached. Thank you!
The expression x^4 + 5x^2 - 8 in quadratic form is: (x^2 + 8)(x^2 - 1)
How to solve the expressionIt should be noted that to express the given expression x^4+5x^2 - 8 in quadratic form, we need to identify a suitable substitution that will allow us to rewrite the expression as a quadratic in a new variable.
One possible substitution is to let u = x^2, so that we can write:
x^4 + 5x^2 - 8 = u^2 + 5u - 8
We can then factor this quadratic expression as:
u^2 + 5u - 8 = (u + 8)(u - 1)
Substituting back u = x^2, we get:
x^4 + 5x^2 - 8 = (x^2 + 8)(x^2 - 1)
Therefore, the expression x^4 + 5x^2 - 8 in quadratic form is:
(x^2 + 8)(x^2 - 1)
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What are the answers to these questions?
A=?
B=?
f''(A)=?
f''(B)=?
Thus f(x) has a local max or min at A and a local max or min at B.
We can say that f(x) has a local maximum at A and a local minimum at B.
What is Function?A function is a mathematical rule that assigns a unique output value to each input value. It describes the relationship between the input and output variables.
Critical numbers of a function are the values of the independent variable where the derivative is zero or undefined, indicating possible extrema or inflection points.
According to the given information:
To find the critical numbers of the function f(x), we need to find the values of x where f'(x) = 0 or f'(x) is undefined.
f(x) = cos(x) + √2/2
f'(x) = -sin(x)
Setting f'(x) = -sin(x) = 0, we get sin(x) = 0, which is true for x = nπ, where n is an integer. However, we only need to consider the values of x in the interval [0,2].
For n = 0, x = 0 is a critical number.
For n = 1, x = π is a critical number.
For n = 2, x = 2π is not in the interval [0,2], so we don't need to consider it.
Therefore, the critical numbers of f(x) in the interval [0,2] are A = 0 and B = π.
To find the nature of the critical points, we need to find the second derivative of f(x).
f''(x) = -cos(x)
Evaluating f''(x) at A and B, we get:
f''(A) = -cos(0) = -1, which means that f(x) has a local maximum at A.
f''(B) = -cos(π) = 1, which means that f(x) has a local minimum at B.
Therefore, we can say that f(x) has a local maximum at A and a local minimum at B.
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