Let f(x,y) = x⁴ + y⁴ – 4xy +1. Find all critical points. For each critical point, determine whether it is a local maximum, a local minimum, or a saddle point. (At least with my approach, for this problem you'll need to factor x⁹ - x. This factors as x(x² - 1)(x² + 1)(x⁴ + 1)

Answer:

Combining like terms,

(4x - 6) + (5x + 1) = 9x - 5

Answer:

√429.

Step-by-step explanation:

Triangles BCD and ABC are similar, so corresponding sides are in the same ratio, so:

x/ AC = DC/x

x/39 = 11/x

x^2 = 11*39

x^2 = 429

x = √429

Pair C of adjacent angles is supplementary because both the angles make the sum of 180°.

Adjacent angles are those angles which have a common vertex and supplementary angles are those which on adding make sum of 180°. In the given question, only the adjacent angles of Pair C make the sum of 180°.

Supplementary angles are those that total 180 degrees. Angles 130° and 50°, for example, are supplementary angles since the sum of 130° and 50° equals 180°.

Complementary angles, on the other hand, add up to 90 degrees. When the two additional angles are brought together, they form a straight line and an angle.

It should be emphasized, however, that the two supplementary angles do not have to be adjacent to each other. As a result, any two angles can be supplementary if their sum is equal to 180°.

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Correct question:

Which pair of adjacent angles is complementary?

A. Pair A

B. Pair B

C. Pair C

D. Pair D

Image is attached below.

The central angle is approximately 51.6 degrees.

To solve this problem, we can use the formula for arc length of a circle:

arc length = θ × r

where θ is the central angle in radians, and r is the radius of the circle.

We know that the arc length is 4.5 feet and the radius is 5 feet. So we can rearrange the formula to solve for θ:

θ = arc length / r

θ = 4.5 / 5

θ = 0.9 radians

To find the central angle in degrees, we can convert radians to degrees by multiplying by 180/π:

θ = 0.9 × (180/π)

θ ≈ 51.6 degrees

Therefore, the central angle is approximately 51.6 degrees.

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Answer:

$2,240

Step-by-step explanation:

1400×3=4,200

4200×20=84,000

84,000÷100=840

840+1400=2,240

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The y-intercept of the graph is: C. "the temperature in degrees Fahrenheit when the temperature is zero degrees Celsius."

In mathematical terms, the y-intercept is the point where a line or curve intersects the y-axis on a coordinate plane.

In practical terms, the y-intercept of a situation often represents the starting point or initial value of a variable or function.

In the graph given above that shows the relationship between temperature in degrees Fahrenheit and degrees Celsius, the y-intercept is 32 degrees Fahrenheit, which means: C. "the temperature in degrees Fahrenheit when the temperature is zero degrees Celsius."

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The function M(2) from the equation of the line of best fit M(t)=-0.15t+3.70 is M(2) = 3.4

The equation of the line of best fit is given as

M(t) = -0.15t + 3.70

The function M(2) means the value of the variable t is 2

i.e. t = 2

To calculate M(2), we substitute the known values of t = 2 in the equation M(t) = -0.15t + 3.70

So, we have

M(2) = -0.15 * 2 + 3.70

Evaluate the product in the expression

This gives

M(2) = -0.30 + 3.70

Evaluate the sum/difference in the expression

This gives

M(2) = 3.4

The above is the value of M(2)

Hence, the calculated value of M(2) is 3.4

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The area of a rectangle can be used to find triangle, rhombus and parallelogram since the shapes are related to one another

The area of a rectangle can be used to determine the area of a triangle, rhombus and parallelogram because these figures are somewhat similar to a rectangle.

When we join two triangles together at opposite end or we cut transversally between a rectangle from the diagonal, we would have two triangle.

When a rhombus is split across it's diagonal, it will produce two congruent triangles.

When a parallelogram is divided across it's diagonals, it produces two congruent triangles.

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The volume of prism is 75.6 [tex]in^3[/tex]. The correct answer is option d,

We are given that the rectangle represents the base of a right rectangular prism and the height of the prism is 6 inches. To find the volume of the prism, we need to multiply the area of the rectangle by the height.

The area of the rectangle can be found by multiplying its length and width. However, we are not given any specific values for the length and width of the rectangle. Therefore, we cannot directly calculate its area.

Since we are given answer choices, we can use them to check which option gives the correct volume of the prism. We can start by assuming that the length, width, and height of the prism are integers, and then calculate the volume for each option until we find the one that matches.

Option a: 87.9 in^3 - This is not an integer, so we can eliminate this option.

Option b: 105.8 in^3 - This is not an integer, so we can eliminate this option.

Option c: 12.6 in^3 - This is too small, so we can eliminate this option.

Option d: 75.6 in^3 - To check if this is the correct answer, we can calculate the area of the rectangle by dividing the volume by the height.

Volume of the prism = area of rectangle x height

75.6 in^3 = (length x width) x 6 in

Length x width = 12.6 in^2

Now, we need to find two integers whose product is 12.6 in^2 and whose sum is 16 in (since the rectangle represents the base of a right rectangular prism). After some trial and error, we find that 3.15 in and 4 in satisfy these conditions.

Therefore, the length of the rectangle is 4 inches and the width is 3.15 inches.

Now, we can calculate the area of the rectangle by multiplying its length and width:

Area of rectangle = length x width = 4 in x 3.15 in = 12.6 in^2

Finally, we can calculate the volume of the prism:

Volume of prism = area of rectangle x height = 12.6 in^2 x 6 in = 75.6 in^3

Therefore, the correct answer is option d, 75.6 [tex]in^3[/tex].

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Answer:The answer is B

Step-by-step explanation:

A group of 8 friends went to lunch and spent a total of $76, which included the food bill and a tip of $16. They decided to split the bill and tip evenly among themselves. Which equations and solutions describe the situation? Select two options. The equation StartFraction 1 over 8 EndFraction (x + 16) = StartFraction 76 over 8 EndFraction represents the situation, where x is the food bill. The equation StartFraction 1 over 8 EndFraction (x + 16) = 76 represents the situation, where x is the food bill. The solution x = 60 represents the total food bill. The solution x = 60 represents each friend’s share of the food bill and tip. The equation 8 (x + 16) = 76 represents the situation, where x is the food bill.

Answer:

Vertical Compression by factor of 1/4

Step-by-step explanation:

Two methods:

Method 1. Transformations

Method 2. Algebraic input-output tables

Method 1. Transformations

The Main concept of this question is about Transformations of functions -- specifically, multiplying on the outside by a positive number less than 1.

The transformation that occurs when multiplying a function by a positive number on the outside of the function is a vertical stretch or compression.

Positive numbers larger than 1 will stretch it vertically, whereas positive numbers smaller than 1 will compress it vertically.

Therefore, multiplying by 1/4 on the outside, a positive number less than 1, will vertically compress the function down to one-fourth the size.

This means that for g(x), all points on the original function f will have their heights reduced to 1/4 their original height (or depth) -- making all points on g(x) 1/4 their previous distance from the x-axis on the "f" function.

Method 2. Algebraic input-output tables

Observe on the graph three points on the function f:

Using the equation relating f and g, [tex]g(x)=\frac{1}{4}f(x)[/tex], we can find how those points would look like on the new function g(x).

For [tex]f(0)=0[/tex]

[tex]g(0)=\frac{1}{4}[f(0)]\\g(0)=\frac{1}{4}[0]\\g(0)=0[/tex]

For [tex]f(1)=4[/tex]

[tex]g(1)=\frac{1}{4}[f(1)]\\g(1)=\frac{1}{4}[4]\\g(1)=1[/tex]

For [tex]f(3)=0[/tex]

[tex]g(3)=\frac{1}{4}[f(3)]\\g(3)=\frac{1}{4}[0]\\g(3)=0[/tex]

These known points should correctly identify the graph from the possible choices.

Average speed going to work is 50 mph.

Let's assume that the average speed going to work is "x" miles per hour.

Since you can travel to work in the same time it takes for you to make it 28 miles on the trip back home, we can set up an equation using the formula:

time = distance / speed

The time it takes to travel 35 miles to work is:

time to work = 35 / x

The time it takes to travel 28 miles on the return trip is:

time to return = 28 / (x - 10)

We know that both times are equal, so we can set them equal to each other and solve for "x":

[tex]35 / x = 28 / (x - 10)[/tex]

Multiplying both sides by x(x - 10), we get:

[tex]35(x - 10) = 28x[/tex]

Expanding the left side and simplifying, we get:

[tex]35x - 350 = 28x[/tex]

[tex]7x = 35[/tex]

[tex]x = 50[/tex]

Therefore, your average speed going to work is 50 miles per hour.

To solve this problem, we used the formula for speed, distance, and time. We t up an equation using the fact that the time it takes to travel to work is equal to the time it takes to travel 28 miles on the return trip. We then solved for the average speed going to work by setting the two times equal to each other and solving for "x". Finally, we found that the average speed going to work is 50 miles per hour.

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The value of m∠Q as shown in the right-angled triangle below is 62.59°.

A right-angled triangle is a triangle, that has one of its interior angles equal to 90 degrees or any one angle is a right angle.

To calculate the value of m∠Q, we use the formula below

Formula:

Where:

Substitute these values into equation 1

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It will take approximately 15.27 years for your investment to reach $5,000 at a growth rate of 4.55% per year.

Using the compound interest calculation, we can determine how long it will take for your investment to grow to $5,000 at a growth rate of 4.55% per year:

[tex]A = P(\frac{1 + r}{n} )^(^n^t^)[/tex]

Where:

A = the final amount ($5,000)

P = the initial deposit ($2,500)

r = the interest rate (4.55%)

n = the number of times interest is compounded per year (assuming yearly compounding, n=1)

t = the number of years

Plugging in the values, we get:

[tex]\$5,000 = \$2,500(\frac{1 + 0.0455}{1} )^(^1^t^)[/tex]

Simplifying, we get:

[tex]2 = (1.0455)^t[/tex]

Taking the natural logarithm of both sides, we get:

ln(2) = t ln(1.0455)

Solving for t, we get:

t = ln(2) / ln(1.0455) = 15.27 years

Therefore, it will take approximately 15.27 years for your investment to reach $5,000 at a growth rate of 4.55% per year.

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Answer:

3! = 6

Step-by-step explanation:

Once she teaches the puppy one trick there are 3 possible tricks left. After teaching the second trick there are 2 and after the third there is 1. Therefore, we multiply these numbers together to get 3(2)(1)=6 which is 3!.

Answer:

D.5

Step-by-step explanation:

8x – 6 > 12 + 2x

= 8x−2x>12+6= 6x>18

= x>38x−2x>12+6

= 6x>18

= x>3

From the given options the only value which is greater than 3 is Option (D) 5

The angular velocity of the point on the rim of the wheel is 25 radians per second.

The linear velocity of a point on a wheel's rim is determined by:

v = rω

v = linear velocity

r = radius of the wheel

ω = angular velocity.

In this case, the diameter of the wheel is 8 feet, so the radius is 4 feet. The linear velocity is given as 100 feet per second, so we have:

100 = 4ω

Solving for ω, we get:

ω = 25 radians per second

Therefore, the angular velocity of the point on the rim of the wheel is 25 radians per second.

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Answer:

h = 30 cm

Step-by-step explanation:

Given:

V (volume) = 2560π cm^3

d (diameter) = 32 cm (r (radius) = 0,5 × 32 = 16 cm)

Find: h (height) - ?

[tex]v = \frac{1}{3} \times\pi {r}^{2} \times h[/tex]

[tex]h = \frac{v}{ \frac{1}{3} \times \pi {r}^{2} } = \frac{2560\pi}{ \frac{1}{3} \times \pi \times {16}^{2} } = 30[/tex]

[tex]h = \frac{2560\pi}{ \frac{1}{3} \times \pi \times {16}^{2} } [/tex]

The volume of the solid generated when the right triangle below is rotated about side IK is: 37.7 units²

The three-dimensional figure that is formed by rotating a triangle about it's height is called a Cone.

Where:

The triangle base length will be seen to become the radius of the cone

The triangle height will be seen to become the height of the cone

The formula for the volume of a cone is expressed as:

V = ¹/₃πr²h

Where:

r refers to the radius

h refers to the height

Therefore, we can say that the volume will be expressed as:

V = ¹/₃ * π * 2² * 9

V = 37.7 units²

Thus, that is the  volume of the solid generated when the right triangle below is rotated about side IK.

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The mode of the number of push-ups is 40.

A stem-and-leaf plot is a way of organizing data where the stems (the tens digit) and leaves (the ones digit) are separated. Each row represents a stem and the leaves represent the values that belong to that stem.

Here's the stem-and-leaf plot for the number of push-ups:

3 | 5 6 8

4 | 0 0 1 2 2 3 5 6 8 9

5 | 0 1 3 4 5 5 7 8 9

6 | 0 1 2 2 3 4 5 7 8 9

7 | 0 2 5 8

8 | 1 2 4

9 | 0

To find the mode, we look for the value that appears most frequently. In this case, the number 40 appears three times, which is more than any other value. Therefore, the mode of the number of push-ups is 40.

Note that the stem-and-leaf plot makes it easy to see the distribution of the data. For example, we can see that there are a lot of values between 40 and 49, and relatively few values above 60.

We can also see that there are no values between 90 and 99.

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The experimental probability that the next customer buys a red notebook with 150 pages is 53/386 or 13.73%.

To find the experimental probability of the next customer buying a red notebook with 150 pages, we need to first identify the total number of red 150-page notebooks sold and then divide that by the total number of notebooks sold.

From the table, we can see that 53 red 150-page notebooks were sold. The total number of notebooks sold is 386.

The experimental probability is therefore the ratio of red 150-page notebooks sold to the total number of notebooks sold:

Probability = (Number of red 150-page notebooks) / (Total number of notebooks)
Probability = 53 / 386

The simplified fraction for the experimental probability is 53/386 or 13.73%.

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There are 616 freshmen in the school according to fraction, percentage and number of different types of student.

Let us represent the total number of students in school as x. So, the equation that will form is -

Number of freshmen + sophomores + juniors + seniors = total number of students

35x/100 + x/4 + 9x/40 + 308 = x

Adding the values and converting into decimal.

0.825x + 308 = x

Rewrite the equation

308 = (1 - 0.825)x

308 = 0.175x

x = 308/0.175

Divide the values

x = 1760

So, number of freshmen = 0.35 × 1760

Multiply the numbers

Number of freshmen = 616

Thus, there are 616 freshmen.

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Note that where the above function is given, the equation in function notation is f(x) = (-2/3)x + 490. This function represents the profit Sal makes from lunch specials based on the number of sandwich and wrap lunch specials sold. See the graph attached.

To change the given equation to slope-intercept form, we need to solve for y.

2x + 3y = 1470

3y = -2x + 1470

y = (-2/3)x + 490

Therefore, the slope of the line is -2/3 and the y-intercept is 490.

To graph this line using the slope-intercept method, we can plot the y-intercept first, which is (0, 490). Then, using the slope of -2/3, we can find another point by moving 2 units to the right and 3 units down from the first point. We can continue this pattern to plot additional points and then draw a straight line through them.

The equation in function notation is f(x) = (-2/3)x + 490. This function represents the profit Sal makes from lunch specials based on the number of sandwich and wrap lunch specials sold.

To graph the function, we can plot the intercepts (0, 490) and (735, 0), where 735 is the x-intercept. Then, using the slope of -2/3, we can find other points and draw a straight line through them.

If Sal's total profit on lunch specials for the next month is $1,593, then the equation would be 2x + 3y = 1593. The graphs of the functions for both months would have the same slope of -2/3, indicating that the profit per lunch special sold remains constant.

However, the y-intercept would be different, indicating a different starting profit for the month. The graphs would have different intercepts and intersect the y-axis at different points, reflecting the difference in starting profits.

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The markup on retail is 47%. The correct option is c.

he markup on retail price is calculated to determine the percentage increase from the wholesale price to the retail price. In this case, the wholesale price is $17 and the retail price is $25. By subtracting the wholesale price from the retail price ($25 - $17),

we find that the markup is $8. Dividing this markup by the wholesale price ($8 / $17) gives us a ratio. Multiplying this ratio by 100 converts it to a percentage, which is approximately 47.06%.

This means that the retail price is approximately 47% higher than the wholesale price. Option c, 47%, correctly represents the calculated markup on the retail price.

Therefore, the markup on retail is 47%, so the answer is (c).

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The equation of the line passing through the points (0,-2) and (2,0) is y = x - 2.

An equation is a mathematical statement that shows that two expressions are equal. It typically consists of variables, constants, and mathematical operations, such as addition, subtraction, multiplication, and division, and can be represented using symbols and/or words. Equations are used to solve problems in mathematics, science, engineering, and other fields.

In the given question,

We can use the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept.

First, we need to find the slope of the line passing through the points (0,-2) and (2,0):

slope = (change in y)/(change in x)

slope = (0 - (-2))/(2 - 0)

slope = 2/2

slope = 1

Now that we have the slope, we can use one of the given equations and substitute the coordinates of one of the points to find the y-intercept:

y = mx + b

-2 = 1(0) + b

b = -2

So the equation of the line passing through the points (0,-2) and (2,0) is y = x - 2.

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The cost of a Powerade is $0.75 if 2 bags of chips and 2 Powerades cost $3.50 and one bag of chips and 3 Powerades cost $3.75.

Let the cost of one Powerade be x

The cost of one bag of chips be y

Cost of 2 bags of chips and 2 Powerades = $3.50

Cost of 2 bags of chips = 2y

Cost of 2 Powerades = 2x

2x + 2y = 3.50 ----(i)

Cost of one bag of chips and 3 Powerades= $3.75

Cost of 1 bag of chips = y

Cost of 3 Powerades = 3x

x + 3y = 3.75

Multiply the above equation by 2

2x + 6y = 7.50 -----(ii)

Subtract (ii) and (i)

4y = 4

y = $1

The cost of one bag of chips is $1

x + 3 = 3.75

x = $0.75

The cost of one Powerade is $0.75

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The value of the calculated percent markup of the shirt is 11.1%

From the question, we have the following parameters that can be used in our computation:

A department store buys 200 shirts at a cost of ​$ 3600 and sells them at a selling price of ​$ 20 each.

This means that

Cost price = 3600/200

Evaluate

Cost price = 18

The percent markup of the shirt is then calculated as

Percentage = (20 - 18)/18

Evaluate

Percentage = 11.1%

Hence, the percent markup of the shirt is11.1%

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The maximum number of students allowed to attend the field trip is 59.

To determine the maximum number of students allowed to attend the field trip, given the ticket cost and money raised, we'll need to follow these steps:
1. Subtract the adult ticket cost from the total money raised.
2. Divide the remaining amount by the cost of a student ticket.
Subtract the adult ticket cost from the total money raised.
$580 (money raised) - $75 (adult ticket cost) = $505 (remaining amount)
Divide the remaining amount by the cost of a student ticket.
$505 (remaining amount) / $8.50 (student ticket cost) = 59.41
Since we can't have a fraction of a student, we round down to the nearest whole number.
The maximum number of students allowed to attend the field trip is 59.

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