Question

Select the correct answer from each drop-down menu. Nick bought apples at a farmers market where 5 apples cost $4. 45. If Nick bought 7 apples, he paid $. If Nick paid $9. 79 for some apples, he bought

apples.

p and q both are equal and p=q=20.

Given are an equation show equivalent fractions 2p/5q = 40/100

We need to find the p and q,

So

2p/5q = 40/100

2p/5q = 2×20/2×20

p/q = 20/20

p/q = 1

p = q

Therefore p and q both are equal and p=q=20.

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The coordinates of the points in the table indicates that the coordinates of the image following the reflections, can be presented on the coordinate plane as shown in the attached graph created with MS Excel.

A reflection transformation is one in which the mirror image of the preimage is created across a line of reflection.

The coordinate points of the image after the specified reflections can be presented as follows;

Original   [tex]{}[/tex]Reflection across the x-axis   Reflection across the y-axis   y = -x

(0, 15)      [tex]{}[/tex] (0, -15)                                      (0, 15)                                   (-15, 0)

(1,  15)      [tex]{}[/tex] (1, -15)                                       (-1, 15)                                   (-15, -1)

(1, 13)      [tex]{}[/tex]  (1, -13)                                       (-1, 13)                                    (-13, -1)

(3, 15)      [tex]{}[/tex] (3, -15)                                      (-3, 15)                                   (-15, -3)

(3, 12)      [tex]{}[/tex] (3, -12)                                      (-3, 12)                                   (-12, -3)

(1, 10)      [tex]{}[/tex]  (1, -10)                                      (-1, 10)                                     (-10, -1)

(1, 8)      [tex]{}[/tex]   (1, -8)                                        (-1, 8)                                      (-8, -1)

(3, 10)      [tex]{}[/tex](3, -10)                                      (-3, 10)                                    (-10, -3)

(3, 7)      [tex]{}[/tex]  (3, -7)                                       (-3, 7)                                      (-7, -3)

(1, 5)      [tex]{}[/tex]  (1, -5)                                        (-1, 5)                                      (-5, -1)

(1, 2)      [tex]{}[/tex]  (1, -2)                                        (-1, 2)                                      (-2, -1)

(4, 4)      [tex]{}[/tex] (4, -4)                                       (-4, 4)                                      (-4, -4)

(5, 7)      [tex]{}[/tex] (-5, 7)                                       (5, -7)                                      (-7, -5)

(7, 8)      [tex]{}[/tex] (7, -8)                                       (-7, 8)                                      (-8, -7)

(6, 5)     [tex]{}[/tex]  (6, -5)                                      (-6, 5)                                      (-5, -6)

(8, 6)     [tex]{}[/tex]  (8, -6)                                      (-8, 6)                                      (-6, -8)

(9, 9)     [tex]{}[/tex]  (9, -9)                                      (-9, 9)                                      (-9, -9)

(11, 10)     [tex]{}[/tex] (11, -10)                                   (-11, 10)                                    (-10, -11)

(10, 7)     [tex]{}[/tex]  (10, -7)                                   (-10, 7)                                     (-7, -10)

(12, 8)     [tex]{}[/tex]  (12, -8)                                   (-12, 8)                                    (-8, -12)

(13, 6)     [tex]{}[/tex]  (13, -6)                                   (-13, 6)                                    (-6, -13)

(11, 5)      [tex]{}[/tex]  (11, -5)                                    (-11, 5)                                     (-5, -11)

(14, 4)      [tex]{}[/tex] (14, -4)            [tex]{}[/tex]                       (-14, 4)                                     (-4, -14)

(12, 3)     [tex]{}[/tex]  (12, -3)                                   (-12, 3)                                     (-3, -12)

(9, 4)      [tex]{}[/tex]  (9, -4)                                    (-9, 4)                                       (-4, -9)

(7, 3)      [tex]{}[/tex]  (7, -3)                                     (-7, 3)                                       (-3, -7)

(10, 2)    [tex]{}[/tex]  (10, -2)                                   (-10, 2)                                     (-2, -10)

(8, 1)      [tex]{}[/tex]  (8, -1)                                      (-8, 1)                                       (-1, -8)

(5, 2)     [tex]{}[/tex]  (5, -2)                                     (-5, 2)                                      (-2, -5)

(2, 0)     [tex]{}[/tex]  (2, 0)                                      (-2, 0)                                      (0, -2)

The above coordinate points can be used to plot the graphs showing the image of the points following the specified reflections across the x-, y-, and y = -x, axis.

Please find attached the required graph of the coordinate points following the reflection transformations, created with MS Excel.

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We can use the point-slope form of the equation of a line to find the equation of the tangent line.

To find the equation of the line that is tangent to circle A from point C (2, 8), we need to first find the point of tangency, which is the point where the line intersects the circle.

Point C (2, 8) is outside the circle, so the tangent line will be perpendicular to the line connecting the center of the circle to point C and will pass through point C.

Step 1: Find the center of the circle

The center of the circle A is at (6, 5).

Step 2: Find the slope of the line connecting the center of the circle to point C

The slope of the line connecting the center of the circle (6, 5) and point C (2, 8) is:

m = (8 - 5) / (2 - 6) = -3/4

Step 3: Find the equation of the line perpendicular to the line from Step 2 passing through point C

The slope of the line perpendicular to the line from step 2 is the negative reciprocal of the slope:

m_perp = -1 / (-3/4) = 4/3

Now we can use the point-slope form of the equation of a line to find the equation of the tangent line:

y - 8 = (4/3)(x - 2)

Simplifying, we get:

y = (4/3)x + 4.67

So the equation of the line that is tangent to circle A from point C (2, 8) is y = (4/3)x + 4.67.

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The volume of the funnel is approximately 42.39 in³. The correct answer is option 2.

To calculate the volume of the funnel, which is shaped like a cone, we need to use the formula for the volume of a cone: V = (1/3)πr²h.

Given:
Height (h) = 4.5 inches
Diameter = 6 inches
Radius (r) = Diameter / 2 = 6 / 2 = 3 inches
Pi (π) ≈ 3.14

Now, plug the values into the formula:

V = (1/3) × 3.14 × 3² × 4.5
V ≈ (1/3) × 3.14 × 9 × 4.5
V ≈ 3.14 × 3 × 4.5
V ≈ 42.39 in³

So, the volume of the funnel is approximately 42.39 in³. The correct answer is option 2.

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Doug is filing singly, and his net taxable income is $80,575. The tax amount for this income range is $16,539. Every week, $304 is withheld from his earnings for income tax. Doug can expect he will owe an additional $731. So option d is the correct answer.

Since the result is a positive number, Doug will owe an additional $731 when his taxes are due. Therefore, the correct answer is d. Doug will owe an additional $731.

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a) The partial derivative of W with respect to T is always positive, which means that increasing T always increases W.

b) Increasing V decreases W if V is greater than

[tex]((0.8T - 0.6215) / 5.71)^{(1/0.16)} .[/tex]

c) The largest domain in which the inequality derived in (b) holds true is:

T > 0.7769. This means that the wind chill formula can be used only for

air temperatures above 0.7769 degrees Fahrenheit.

(a) To show that increasing T always increases W, we need to calculate the partial derivative of W with respect to T and show that it is always positive.

∂W/∂T = [tex]0.6215/0.4275V^{0.16} - (35.75V^{0.16})/0.4275TV^{0.16}^{2}[/tex]

Simplifying this expression, we get:

∂W/∂T = [tex]1.44(0.6215 - 0.0275V^{0.16T}) / V^{0.16}T^{2}[/tex]

Since 1.44 and[tex]V^{0}.16T^{2}[/tex] are always positive, the sign of the partial derivative depends on the sign of[tex](0.6215 - 0.0275V^{0.16T} ).[/tex]

Since 0.0275 is always positive and [tex]V^{0.16T}[/tex] is also always positive, we see that [tex](0.6215 - 0.0275V^{0.16T} )[/tex] is always positive.

(b) To find the conditions under which increasing V decreases W, we need to calculate the partial derivative of W with respect to V and show that it is always negative.

∂W/∂V = [tex](-35.750.16V^{(-0.84)} (35.74+0.6215T-35.75V^{0.16} )-0.6215V^{(-0.16} ))/0.4275TV^{(0.16)}[/tex]

Simplifying this expression, we get:

∂W/∂V = [tex]-0.16(0.6215+5.71V^{0.16-0.8T} ) / TV^{0.84}[/tex]

The sign of the partial derivative depends on the sign of [tex](0.6215+5.71V^{0.16-0.8T} ).[/tex]

If [tex]0.6215+5.71V^{0.16-0.8T} < 0[/tex], then the partial derivative is negative and increasing V decreases W.

Solving this inequality for V, we get:

[tex]V > ((0.8T - 0.6215) / 5.71)^{(1/0.16)}[/tex]

(c) Assuming that W should always decrease when V is increased, we need to find the largest domain in which the inequality derived in (b) holds true.

Since the expression inside the parentheses must be positive for a real solution, we have:

0.8T - 0.6215 > 0

T > 0.7769

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The quadratic function in standard form that passes through the given points would be f (x ) = 5x ² - 70x + 225

A quadratic function in standard form is given by the equation:

f ( x ) = ax ²  + bx + c

The system of equations would be:

25 a + 5b + c = 0

81 a + 9b + c = 0

49 a + 7b + c = -20

With a series of calculations, we can find the value of b and c :

b = - 14 a = - 14 ( 5 ) = -70

25 ( 5 ) - 70 (5) + c = 0

125 - 350 + c = 0

c = 225

This gives us the quadratic function of :

f ( x ) = 5x ² - 70x + 225

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The number of flu virus particles in a person's body after 1 day is 28.46 billion, based on the given polynomial function.

The polynomial given to us is, y = −0.74x⁴ + 2.8x³ + 26.4 which describes the billions of flu virus particles in a person’s body. To find the number of virus particles in a person's body after 1 day, we need to substitute x = 1 into the given polynomial and evaluate it.

So, we have,

y = -0.74(1)⁴ + 2.8(1)³ + 26.4

= -0.74 + 2.8 + 26.4

= 28.46

Therefore, the number of virus particles in a person's body after 1 day is 28.46 billion.

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Complete question - The polynomial y=−0.74x⁴ + 2.8x³ + 26.4 describes the billions of flu virus particles in a person’s body x days after being infected. find the number of virus particles, in billions, after 1 day.

The inches of farming the picture require is 45 inches

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

Base = 5 in

High = 4 in

Scale factor = 2.5

The inches of farming the picture require is calculated as

Perimeter = 2 * (Base + High) * Scale factor

Substitute the known values in the above equation, so, we have the following representation

Perimeter = 2 * (5 + 4) * 2.5

Evaluate

Perimeter = 45

Hence, the inches of farming the picture require 45 inches

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The distance between the lions and the monkeys is 64 feet.

We can set up a proportion to find the distance between the lions and monkeys. Here's the step-by-step explanation:

1. Proportion: Since the path from zebras to monkeys is parallel to the path from tigers to elephants, we can set up a proportion using the given distances: 52/78 = x/96.

2. Cross-multiply: To solve for x, we can cross-multiply: 52 * 96 = 78 * x, which simplifies to 4992 = 78x.

3. Solve: Now we just need to solve for x. Divide both sides of the equation by 78: x = 4992 / 78. This gives x ≈ 64.

So, the distance between the lions and the monkeys is approximately 64 feet.

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The margin error for the given poll having 95% confidence level with sample size of 410 is equal to 3.15%.

Sample size n = 410

Confidence level = 95%

Margin of error for this poll, use the formula,

ME = Z× (√(p₁(1-p₁) / n))

where Z is the z-score corresponding to the desired level of confidence.

p₁ is the sample proportion = 0.12

Using attached z-score table,

For a 95% confidence level, the corresponding z-score is 1.96.

Substituting the given values, we get,

ME = 1.96 × (√(0.12× (1-0.12) / 410))

Simplifying the expression inside the parentheses, we get,

⇒ME = 1.96 ×  0.0160

⇒ME = 0.0315

Margin of error for this poll at the 95% confidence level is approximately 0.0315.

Therefore,  95% confidence level represents that the true proportion of people who like dogs is within 3.15% of the observed proportion of 12%.

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The ordered pairs that satisfy the function is B)(0,20) and D)(10,-20).

What is function?

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.

Here the given function is y=-4x+20.

Now put x=6 and y=4 then,

=> 4=-4(6)+20

=> 4 = -24+20

=> 4 ≠ -4.

Then the coordinate (6,4) dost not satisfy the function.

Put x=0 and y=20 then,

=> 20 = -4(0)+20

=> 20= 0+20

=> 20=20

Hence the coordinate (0,20) satisfy the function.

Now put x=-4 and y=20 then,

=> 20 = -4(-4)+20

=> 20 = 16+20

=> 20 ≠ 36

Hence the coordinate (-4,20) does not satisfy the function.

Now put x=10 and y=-20 then,

=> -20 = -4(10)+20

=> -20 = -40+20

=> -20=-20

Then the coordinate (10,-20) satisfy the function.

Hence the ordered pairs that satisfy the function is B)(0,20) and D)(10,-20).

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The only critical point of f(x) is x = -2.

a) To find the derivative of f(x), we can use the chain rule and the power rule of differentiation.

f(x) = 3sqrt(x + 2)^2

f'(x) = 3 * 2 * sqrt(x + 2) * (x + 2)^1/2-1 * (1)

Applying the power rule, we simplify the expression as:

f'(x) = 6(x + 2)^1/2

Therefore, the derivative of f(x) is f'(x) = 6(x + 2)^1/2.

b) To solve f'(x) = 0, we set f'(x) equal to zero and solve for x:

f'(x) = 6(x + 2)^1/2 = 0

Dividing both sides by 6, we get:

(x + 2)^1/2 = 0

Squaring both sides, we get:

x + 2 = 0

Subtracting 2 from both sides, we get:

x = -2

Therefore, the only critical point of f(x) is x = -2.

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Howler monkeys are heavier on average than spider monkeys (17.75 lbs vs. 11.25 lbs).

Calculate the mean:

Some of the weights divided by the number of monkeys

Calculate the MAD:

Find the absolute deviation (difference) of each weight from the mean

Calculate the average of these deviations

Mean Howler Monkey = (16+17+18+18+18+20+22+23)/8 = 142/8 = 17.75 lbs

Mean Spider Monkey = (8+10+10+11+11+12+14+14)/8 = 90/8 = 11.25 lbs

Now, we calculate the MAD for each type of monkey:

MAD Howler Monkey=  1.875 lbs

MAD Spider Monkey=  1.5625 lbs

we calculate the means-to-MAD ratio for both types of monkeys:

Howler Monkey: Mean/MAD = 17.75/1.875 = 9.466

Spider Monkey: Mean/MAD = 11.25/1.5625 = 7.2

Inference:

Howler monkeys are heavier on average than spider monkeys (17.75 lbs vs. 11.25 lbs).

The means-to-MAD ratio shows that howler monkeys have more consistent weights (9.466) compared to spider monkeys (7.2), as a higher ratio indicates less variability in weights.

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Answer:

it's ither 1 2 or 4 I'm still thinking but I'm pretty sure it's one of those answers

The completed table of equivalent ratios would be:
13.5 square meters : 3 liters of paint
4.5 square meters : 1 liter of paint
45 square meters : 10 liters of paint.

To find the missing ratios, we need to set up proportions using the given information.

13.5 square meters is covered by 3 liters of paint, so we can write:

13.5/3 = ?/1

To solve for the missing ratio, we can cross-multiply:

13.5 x 1 = 3 x ?

? = (13.5 x 1) / 3

? = 4.5

So, 4.5 square meters can be covered by 1 liter of paint.

To find the last missing ratio, we can use the same method:

13.5/3 = ?/10

10 x 13.5 = 3 x ?

? = (10 x 13.5) / 3

? = 45

So, 45 square meters can be covered by 10 liters of paint.

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The value that will complete Pythagorean triple would be = 24,143, 145 )

To calculate the missing value of a triangle that completes a Pythagorean triple that formula that should be used is given as follows.

That is;

C ² = a² + b²

C = Missing value of the Pythagorean triple

a = 24

b.= 143

C² = 24²+143²

= 576+20,449

C =√21,025

= 125

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If he drives at an average speed of 70 km/h for 45 min, the distance of the journey from Mr. Linden's home to his office is 52.5 km.

To find the distance of Mr. Linden's journey from his home to his office, we can use the formula:

Distance = Speed x Time

Since Mr. Linden drives at an average speed of 70 km/h for 45 minutes, we first need to convert the time to hours:

45 minutes = 0.75 hours

Now, we can plug in the values we have into the formula:

Distance = 70 km/h x 0.75 hours
Distance = 52.5 km

Therefore, the distance of the journey from Mr. Linden's home to his office is 52.5 km.

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Answer: C

Step-by-step explanation:

since slope is rise/run its 2 and since the line is a negative slope the slope of the line is -2. and the y-intercept of the line is -2.

y = mx+b

m = -2

b= -2

Answer: C. y= -2x -2

The simplest formula for working out probability is the following:

Probability (P) = Number of favorable outcomes (F) / Total number of possible outcomes (T)

In this formula, the probability of an event occurring is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Probability is a measure of the likelihood or chance of an event occurring. It is represented as a value between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event.

To calculate the probability, you need to determine the number of favorable outcomes, which are the desired outcomes or the outcomes you are interested in. Then, you divide that by the total number of possible outcomes, which is the number of equally likely outcomes in the given situation.

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Without this information, I cannot evaluate the repeated integral or determine the correct answer choice. Please provide additional information so I can assist you better.

 To evaluate the repeated integral, we must first understand the given question. It appears that some information is missing, making it difficult to provide a complete answer.

Please provide the complete problem statement, including the limits of integration for each variable (x, y, and z). This will allow me to accurately evaluate the repeated integral and provide you with the correct answer.

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Question 1: The probability of picking 1 almond cookie followed by another almond cookie is 1/5.

Question 2: The probability of drawing 2 blue marbles in a row is 1/4.

Question 3: The probability of Chang wearing the white shirt and tan pants is 1/4.

In the problem, the chef draws two cookies without replacement from a box containing six cookies of three different types. The probability of picking one almond cookie followed by another almond cookie can be found by using the multiplication rule of probability.

The probability of picking the first almond cookie is 3/6, and since the first cookie is not replaced, there are now only 2 almond cookies left in the box out of a total of 5 cookies. Therefore, the probability of picking another almond cookie is 2/5. Using the multiplication rule, we multiply these probabilities together to get:

P(almond, then almond) = (3/6) x (2/5) = 1/5

In the problem, Peter draws two marbles randomly from a bag containing three different colors of marbles. The probability of drawing two blue marbles in a row can be found by using the multiplication rule of probability again.

The probability of drawing a blue marble on the first draw is 4/8, and since the marble is replaced, there are still 4 blue marbles left out of a total of 8 marbles. Therefore, the probability of drawing another blue marble on the second draw is also 4/8. Using the multiplication rule, we multiply these probabilities together to get:

P(blue, then blue) = (4/8) x (4/8) = 1/4

In the problem, Chang has two shirts and two pairs of pants, and he chooses one of each at random to wear.

The probability of him choosing the white shirt is 1/2, and the probability of him choosing the tan pants is also 1/2. Using the multiplication rule, we multiply these probabilities together to get:

P(white shirt and tan pants) = (1/2) x (1/2) = 1/4

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By using the method of undetermined coefficients, The general solution is y = ae^(4x)cos(3x) + be^(4x)sin(3x) + (7/2cos(3t) + 5/2sin(3t))e^(4t). The solution to the initial value problem is y = 3e^(2x) + 14e^(4x) - 3e^(3x).

By using the method of undetermined coefficients, the associated homogeneous equation is y''-8y'+297=0, which has the characteristic equation r^2-8r+297=0. The roots of this equation are r=4+3i and r=4-3i, so the homogeneous solution is yh=a*e^(4x)cos(3x)+be^(4x)*sin(3x).

To find the particular solution, we make the ansatz yp = (Acos(3t) + Bsin(3t))e^(4t), where A and B are constants to be determined. Substituting this into the differential equation, we get

y" - 8y' + 297 = (16A - 18B)e^(4t)cos(3t) + (16B + 18A)e^(4t)sin(3t)

On the right-hand side, we have 48e^4tcos(3t) + 80e^4tsin(3t), which suggests setting

16A - 18B = 48, and

16B + 18A = 80

Solving these equations simultaneously, we get A = 7/2 and B = 5/2. Therefore, the particular solution is

yp = (7/2cos(3t) + 5/2sin(3t))e^(4t)

And the general solution is

y = yh + yp = ae^(4x)cos(3x) + be^(4x)sin(3x) + (7/2cos(3t) + 5/2sin(3t))e^(4t)

For the second problem, the associated homogeneous equation is y''-6y'+8y=0, which has the characteristic equation r^2-6r+8=0. The roots of this equation are r=2 and r=4, so the homogeneous solution is yh=ae^(2x)+be^(4x).

To find the particular solution, we make the ansatz yp = Ce^3x, where C is a constant to be determined. Substituting this into the differential equation, we get

y" - 6y' + 8y = 9Ce^3x - 18Ce^3x + 8Ce^3x = (8C - 9C)e^3x = -C*e^3x

On the right-hand side, we have 3e^x, which suggests setting -C = 3. Therefore, the particular solution is

yp = -3e^(3x)

And the general solution is

y = yh + yp = ae^(2x) + be^(4x) - 3e^(3x)

To find the values of a and b, we use the initial conditions

y(0) = a + b - 3 = 14

y'(0) = 2a + 4b - 9 = 29

y''(0) = 2a + 8b = 25

Solving these equations simultaneously, we get a = 3 and b = 14. Therefore, the solution to the initial value problem is

y = 3e^(2x) + 14e^(4x) - 3e^(3x)

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--The given question is incomplete, the complete question is given

"  (1 point) Use the method of undetermined coefficients to find a solution of a y" – 8y' + 297 = 48e4t cos(3t) + 80e4t sin(3t) + 3 - Use a and b for the constants of integration associated with the homogeneous solution. Use a as the constant in front of the cosine term. y = yh + yp = - = (1 point) Find y as a function of x if ' y" – 6y" + 8y' = 3e", - - = y(0) = 14, y'(0) = 29, y"(0) = 25."--

Answer:

Step-by-step explanation:

Answer:

The maximum is when both X and y = 1.The maximum value of the function is 3.When both X and YY are equal to 0, the minimum value is 0

The absolute maximum value is 3 and the absolute minimum value is 0.

To find the absolute maximum and minimum values of f(x,y) = x^2 + 2y^2 on the square [0,1] x [0,1], we need to consider both the interior and the boundary of the square.

First, check for critical points in the interior by finding the partial derivatives and setting them equal to zero:

fx = 2x and fy = 4y

Setting them equal to zero, we have:
2x = 0 => x = 0
4y = 0 => y = 0

The only critical point in the interior is (0,0).

Next, evaluate f(x,y) on the boundary of the square [0,1] x [0,1]. The boundary consists of four segments: x=0, x=1, y=0, and y=1.

1. x=0: f(0,y) = 2y^2 (for y in [0,1])
2. x=1: f(1,y) = 1 + 2y^2 (for y in [0,1])
3. y=0: f(x,0) = x^2 (for x in [0,1])
4. y=1: f(x,1) = x^2 + 2 (for x in [0,1])

Now, compare the values of f at the critical point and boundary points to find the absolute maximum and minimum:

Absolute minimum: f(0,0) = 0
Absolute maximum: f(1,1) = 3

So the absolute maximum value is 3 and the absolute minimum value is 0.

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The home repair crew worked for 9 days.

To determine the number of days the home repair crew worked for, we can use algebra. Let's assume that the number of days they worked for is represented by "d". We know that they charge $75 per day plus a $250 service fee, so we can set up the following equation:

75d + 250 = 925

Simplifying the equation, we get:

75d = 675

Dividing both sides by 75, we get:

d = 9

However, we need to keep in mind that the $250 service fee is a one-time charge, not a daily charge. So we need to subtract that from the total amount to get the actual amount charged for the days worked:

925 - 250 = 675

Dividing 675 by the daily rate of $75, we get:

675 / 75 = 9

Therefore, the home repair crew worked for 9 days.

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The coordinates of the translated triangle A'B'C' are: A'(-1, -1), B'(4, 0), and C'(0, -2).

To find the coordinates of the vertices of the translated figure, we simply apply the given translation rule to each vertex of the original triangle.

For vertex A(-3, 3):
(x, y) --> (x+2, y-4)
(-3, 3) --> (-3+2, 3-4)
(-1, -1)

So, the translated coordinates of vertex A are (-1, -1).

For vertex B(2, 4):
(x, y) --> (x+2, y-4)
(2, 4) --> (2+2, 4-4)
(4, 0)

So, the translated coordinates of vertex B are (4, 0).

For vertex C(-2, 2):
(x, y) --> (x+2, y-4)
(-2, 2) --> (-2+2, 2-4)
(0, -2)

So, the translated coordinates of vertex C are (0, -2).

Therefore, the vertices of the translated triangle are A'(-1, -1), B'(4, 0), and C'(0, -2).

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If a teacher has an annual salary of 98,500, the teacher makes  $947.12 per biweekly period.

To calculate the teacher's biweekly salary, we need to divide their annual salary by the number of weeks in a year, and then divide that result by 2 (since there are 2 weeks in a biweekly period).

There are a few different ways to approach this calculation, but one common method is to use the following formula:

Biweekly Salary = (Annual Salary / Number of Weeks in a Year) / 2

Using this formula, we can calculate the teacher's biweekly salary as follows:

Biweekly Salary = (98,500 / 52) / 2

Biweekly Salary = (1,894.23) / 2

Biweekly Salary = 947.12

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The profit of the company after 8 years is approximately $105,085.11.

The value of the truck after 6 months is approximately $3,677.49.

The population of the city in 1999 is approximately 1,457.66 people.

Let P(t) be the profit in year t, where t is the number of years after 1998. The initial profit in 1998 is $35,000.

The profit increases by 18% per year, which means the profit at time t is 1.18 times the profit at time t-1. Therefore, the equation to model the situation is: [tex]P(t) = 35000 * 1.18^t[/tex]

To find the profit of the company after 8 years:

[tex]P(8) = 35000 * 1.18^8[/tex] = $105,085.11

Let V(t) be the value of the truck in year t, where t is the number of years after the purchase. The initial value of the truck is $4,000.

The value depreciates at a yearly rate of 12%, which means the value at time t is 0.88 times the value at time t-1. Therefore, the equation to model the situation is: [tex]V(t) = 4000 * 0.88^t[/tex]

To find the value of the truck after 6 months (0.5 years):

[tex]V(0.5) = 4000 * 0.88^0^.^5[/tex] = $3,677.49

Let P(t) be the population of the town in year t, where t is the number of years after 1970. The initial population in 1970 is 600.

The population increases by 2.5% per year, which means the population at time t is 1.025 times the population at time t-1. Therefore, the equation to model the situation is: [tex]P(t) = 600 * 1.025^t[/tex]

To find the population of the city in 1999 (29 years after 1970):

[tex]P(29) = 600 * 1.025^2^9 = 1,457.66[/tex]

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The critical points of f(x) are x = -3 and x = 4.

Derivative of 4y = sin(cos(8x)):
We can find the derivative of the given function using the chain rule of differentiation. Let u = cos(8x), then we have:
y = sin(u)
dy/du = cos(u) (derivative of sin(u))
du/dx = -8sin(8x) (derivative of cos(8x) using chain rule)
dy/dx = dy/du * du/dx = cos(cos(8x)) * (-8sin(8x))
Therefore, the derivative of 4y = sin(cos(8x)) is:
dy/dx = -32sin(8x)cos(cos(8x))
Critical points of f(x) = x√(x+6):
To find the critical points of f(x), we need to find where the derivative of the function is equal to zero or undefined.
f(x) = x√(x+6)
f'(x) = (√(x+6) + x/2√(x+6))
To find the critical points, we need to set f'(x) equal to zero and solve for x:
(√(x+6) + x/2√(x+6)) = 0
Multiplying both sides by 2√(x+6), we get:
2x + 6 = -x√(x+6)
Squaring both sides, we get:
4[tex]x^2[/tex] + 24x + 36 = [tex]x^3[/tex] + 6[tex]x^2[/tex]
Rearranging, we get:
[tex]x^3[/tex] + [tex]2x^2[/tex] - 24x - 36 = 0
We can solve this cubic equation using numerical methods or by factoring.

By testing values, we can see that x = -3 is a root of the equation. Dividing the equation by (x+3), we get:
[tex]x^2[/tex]- x - 12 = 0
This quadratic equation can be factored as (x-4)(x+3) = 0.
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