Solve for x: √8x + 4 = 6

The relative frequency for selecting a teen out of all the people who like chocolate is calculated by dividing the number of teens who like chocolate (N) by the total number of people who like chocolate (T).

To find the relative frequency for selecting a teen out of all the people who like chocolate, you need to follow these steps:

Step 1: Determine the total number of people who like chocolate (let's call this T).
Step 2: Determine the number of teens who like chocolate (let's call this N).
Step 3: Calculate the relative frequency by dividing the number of teens who like chocolate (N) by the total number of people who like chocolate (T).

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The solution to the inequality is x ≥ -1.

We solve the inequality 1/3(2x-1)≤1-2/5(2-3x).

Let's go step by step:
Begin by distributing the fractions to the terms inside the parentheses:
  (1/3 * 2x) - (1/3 * 1) ≤ 1 - (2/5 * 2) + (2/5 * 3x)
  (2x/3) - (1/3) ≤ 1 - (4/5) + (6x/5)
Combine like terms on each side of the inequality:
  (2x - 1)/3 ≤ (1 - 4/5) + 6x/5
  (2x - 1)/3 ≤ (1/5) + 6x/5.

To eliminate the fractions, find a common denominator, which in this case is 15.

Multiply each term by 15:
  15 * (2x - 1)/3 ≤ 15 * (1/5) + 15 * 6x/5
  5(2x - 1) ≤ 3 + 18x
Distribute and simplify:
  10x - 5 ≤ 3 + 18x
Move the variables to one side and constants to the other side:
  10x - 18x ≤ 3 + 5
  -8x ≤ 8
Divide both sides by -8 (remember to flip the inequality sign since we are dividing by a negative number):
  x ≥ -1.

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The farmer should plant 116 acres of Crop A and 104 acres of Crop B to maximize her profit, which would be $41,840.

To maximize profit, the farmer should plant the crop with the higher profit per acre until she runs out of money or labor hours.

Let x be the number of acres of Crop A to be planted, and y be the number of acres of Crop B to be planted.

The objective function (profit) is: Profit = 160x + 220y

The constraints are: Cost constraint: 30x + 560y ≤ 7400 Labor hour constraint: 20x + 25y ≤ 3400

To solve this problem using linear programming, we can use a graphing calculator or software.

However, we can also solve it manually by finding the corner points of the feasible region (the area that satisfies all constraints) and evaluating the objective function at each point. The corner points are: (0, 296/5) (116, 104) (170, 56) (222/5, 0)

Evaluating the objective function at each point, we get: (0, 296/5):

Profit = 0 + 160(296/5) = 9472 (116, 104):

Profit = 160(116) + 220(104) = 41840 (170, 56):

Profit = 160(170) + 220(56) = 38480 (222/5, 0):

Profit = 160(222/5) + 0 = 7104

Therefore, the farmer should plant 116 acres of Crop A and 104 acres of Crop B to maximize her profit, which would be $41,840.

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Answer: Nori

Step-by-step explanation:

2% of 5000 = 100

1% of 1000 = 10

Step-by-step explanation:

use Pythagorean theorem to find the height

c = 38 ft

a = 14 ft

a² + b² = c²

(14)² + b² = (38)²

b² = 1444 - 196

b² = 1248

b = √1248

b = √16 × 78

b = 4√78 feet

1.The universal set is defined as {Sarah, Allen, Kara, Todd, Joseph, Lydia, Matt, Caleb, Britney, Ethan}.

2.Caleb and Britney are included in both subsets since they like both football and basketball.

1. The universal set (U) consists of all the students in Mrs. Austin's class. In this case, U = {Sarah, Allen, Kara, Todd, Joseph, Lydia, Matt, Caleb, Britney, Ethan}.

2. The two subsets are:
 a) The set of students who like football (F) = {Sarah, Allen, Kara, Todd, Caleb, Britney}
 b) The set of students who like basketball (B) = {Joseph, Lydia, Matt, Caleb, Britney}

Caleb and Britney are included in both subsets since they like both football and basketball.

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When the points are rotated 90 degrees clockwise about the origin, the result is:

To rotate a point 90 degrees clockwise about the origin, you can use the following rule: (x, y) becomes (y, -x). Let's apply this rule to the given points:

I - (3, 1)

Rotated I: (1, -3)

J - (5, -1)

Rotated J: (-1, -5)

H - (3, -3)

Rotated H: (-3, -3)

So, after a 90-degree clockwise rotation about the origin, the new coordinates of the points are:

I: (1, -3)

J: (-1, -5)

H: (-3, -3)

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A) The distribution is a negative binomial distribution with parameters r and p.

B) The probability that the 8th success occurs on the 17th spin is approximately 0.8%.

C) The probability that the 13th success occurs between the 31st and 34th spin is approximately 0.6%.

A) The distribution is a negative binomial distribution with parameters r = number of successes (in this case, r = 1 since we are only interested in the first success), and p = probability of success (landing in a green slot).

B) To find the probability that the 8th success occurs on the 17th spin, we use the formula for the negative binomial distribution:

P(X = k) = (k-1)C(r-1) * [tex]p^r[/tex] * [tex](1-p)^{(k-r)[/tex]

where X is the number of spins until the Rth success, k is the number of spins, and C(n,r) is the binomial coefficient (n choose r).

In this case, we want to find P(X = 17) when r = 8 and p = 2/38 (since there are 2 green slots out of 38 total slots):

P(X = 17) = (16 C 7) * (2/38)⁸ * (36/38)⁹
          ≈ 0.008 or 0.8%

So the probability that the 8th success occurs on the 17th spin is approximately 0.8%.

C) To find the probability that the 13th success occurs between the 31st and 34th spin, we need to find the probability of getting exactly 12 successes in the first 30 spins, followed by a success on one of the next 4 spins (31st, 32nd, 33rd, or 34th).

P(31 ≤ X ≤ 34) = P(X ≤ 34) - P(X ≤ 30)
               = ∑[k=13 to 34] (k-1 C 12-1) * (2/38)¹² * [tex](36/38)^{(k-12)[/tex] - ∑[k=1 to 30] (k-1 C 12-1) * (2/38)¹² * [tex](36/38)^{(k-12)[/tex]
               ≈ 0.006 or 0.6%

So the probability that the 13th success occurs between the 31st and 34th spin is approximately 0.6%.

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Step-by-step explanation:

so like you use sine rule to find line BC and i got 7.3 the you have to split the triangle in half to get a right angle triangle then divide 7.3 by two to get 3.7 and then use .pythagoras theorem to find the height and then use the area of a triangle formula to get your answer as option (C)

Line x is perpendicular to line y. Line z crosses lines x and y. Only statements 3 and 4 are true.

∠6 = ∠8 is not true because they both lie on the same plane and makes an angle of 180° and can never be true. ∠6 = ∠1 is also not true because ∠1 is clearly obtuse angle and ∠6 is clearly acute angle so they cannot be equal. Hence, statement a and b are false.

∠7 = ∠3 is always true because they are corresponding angles and corresponding angles are always equal. m∠2 + m∠4 = 180° is also true because they lie on same plane and have common vertex and hence, they are supplementary angles and make a sum of 180°.  Hence, statement 3 and 4 is always true.

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The solution set of the inequality 3x-2 < 2-2x is:

(4/5, ∞)

To find this we need to isolate the variable in the inequality.

Here we have:

3x - 2 < 2 - 2x

add 2x in both sides and add 2 in both sides, then we will get:

3x + 2x < 2 + 2

5x < 4

Now we can divide both sides by 5 to get:

x < 4/5

That is the inequality solved.

Then the solution set of the inequality is:

(4/5, ∞)

The set of all real numbers larger than 4/5.

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The correct point which is solution of equation of line is,

⇒ (1, 2)

We have to given that;

Equation of line is,

⇒ 3x - y = 1

Take a point 2,

⇒ (1, 2) = (x, y)

Plug into above equation of line is,

⇒ 3x - y = 1

⇒ 3 x 1 - 2 = 1

⇒ 3 - 2 = 1

⇒ 1 = 1

Thus, The correct point which is solution of equation of line is,

⇒ (1, 2)

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The price of the advertisement that originally costs $110 after a 15% markdown is $ 83.5.

Original price = $110

Markdown percent = 15%

Reduce in price = 15% of 110

To calculate this, we divide the percent by 100 and multiply it by the number given. Like in the above case, we divide 15 by 100 to get 0.15 and multiply it by 110.

= 0.15 * 110 = 16.5

Price after markdown = original price - reduction in price

= 110 - 16.5

= $ 83.5

Thus, the price for a quarter-page ad of the school yearbook which used to be $110 has come down to $83.5 after a 15 % markdown to attract more sponsors.

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Answer:

we 1st can get the weight of rat by

1 rat and 1 cat + 1 dog and rat = 30

2 rat + 1 cat + 1 dog = 30

Then 1 rat and cat measure 24 so

2 rat + 24 =30

2 rat + 24 =30 1 rat = 3 kg

2 rat + 24 =30 1 rat = 3 kg 1 cat + 1 rat = 10

2 rat + 24 =30 1 rat = 3 kg 1 cat + 1 rat = 101 cat + 3kg = 10

2 rat + 24 =30 1 rat = 3 kg 1 cat + 1 rat = 101 cat + 3kg = 101 cat = 7kg and

2 rat + 24 =30 1 rat = 3 kg 1 cat + 1 rat = 101 cat + 3kg = 101 cat = 7kg and 1 dog + 1 rat = 20 kg

2 rat + 24 =30 1 rat = 3 kg 1 cat + 1 rat = 101 cat + 3kg = 101 cat = 7kg and 1 dog + 1 rat = 20 kg 1 dog + 3kg = 20 kg

2 rat + 24 =30 1 rat = 3 kg 1 cat + 1 rat = 101 cat + 3kg = 101 cat = 7kg and 1 dog + 1 rat = 20 kg 1 dog + 3kg = 20 kg 1 dog = 17kg

so we get the weight of each now we r going to sum them 1 rat + 1 cat + 1 dog = x

1 rat + 1 cat + 1 dog = x 3 kg + 7 kg + 17 kg = x

1 rat + 1 cat + 1 dog = x 3 kg + 7 kg + 17 kg = x 27 kg = x ..... is the mass of 3 of them

The solution for the given linear expression is 11.25 (45/4).

A linear expression can be represented by a line. The standard form for this equation is: y=mx+b , for example, y=11x+9. Where:

m= the slope.

b= the constant term that represents the y-intercept.

For the given example: m=11 and b=9.

The question gives the expression:7z+9z+7-7 =180. Then, you should find the variable z.

7z+9z+7-7 =180

16z+0=180

16z=180

z=180/16=90/8=45/4=11.25

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The values of given conditions are:

1. median=70

2. Lower quartile=40

3. Upper quartile=87

4. Range=170

5. IQR=47

6. Lower outlier threshold=-20.5

7. Upper outlier threshold=160.5

In statistics, the median is the value separating the higher half from the lower half of a dataset. In other words, it is the middle value of a dataset when it is ordered in ascending or descending order.

Here,

To find the median time, we need to arrange the data in order from least to greatest and find the middle value.

10, 15, 30, 35, 40, 40, 45, 55, 60, 62, 64, 64, 66, 68, 70, 70, 72, 75, 75, 80, 82, 84, 90, 90, 105, 110, 120, 180

There are 28 values in the data set, so the median is the average of the 14th and 15th values:

Median = (70 + 70)/2

= 70

To find the lower quartile, we need to find the median of the lower half of the data set:

10, 15, 30, 35, 40, 40, 45, 55, 60, 62, 64, 64, 66, 68

There are 14 values in the lower half, so the lower quartile is the median of these values:

Lower quartile = (40 + 40)/2

= 40

To find the upper quartile, we need to find the median of the upper half of the data set:

72, 75, 75, 80, 82, 84, 90, 90, 105, 110, 120, 180

There are 14 values in the upper half, so the upper quartile is the median of these values:

Upper quartile = (84 + 90)/2

= 87

To find the range, we subtract the smallest value from the largest value:

Range = 180 - 10

= 170

To determine if there are any outliers in the data set, we need to calculate the interquartile range (IQR):

IQR = Upper quartile - Lower quartile

= 87 - 40

= 47

Any value that is more than 1.5 times the IQR below the lower quartile or above the upper quartile is considered an outlier.

Lower outlier threshold = Lower quartile - 1.5IQR

= 40 - 1.547

= -20.5

Upper outlier threshold = Upper quartile + 1.5IQR

= 87 + 1.547

= 160.5

To draw a box-and-whisker plot, we need to plot a box from the lower quartile to the upper quartile, with a line inside the box representing the median. We then draw whiskers extending from the box to the smallest and largest values that are not outliers. The box extends from 40 to 87, with a line at 70 representing the median. The whisker on the left extends to the smallest non-outlier value of 10, and the whisker on the right extends to the largest non-outlier value of 120.

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Answer:

  C. R+S+T = 201°

Step-by-step explanation:

You want to know which of the offered angle relations is true regarding quadrilateral RSTU.

The sum of angles in a quadrilateral is 360°. You use this fact to find angle T. Then you can compute the various differences to see which one matches the answer choices.

  T = 360° -R -S -U = 55°

In the attached calculator display, we have done exactly that. We find ...

  T -R = 38° . . . . A is false

  S -T = 74° . . . . B is false

  R +S +T = 201° . . . . C is TRUE

  R +T +U = 231° . . . . D is false

Answer:

To answer your question, we need to use some properties of rectangles and triangles.

A rectangle has four right angles, so angle R = angle S = angle T = angle U = 90 degrees.

The sum of the angles in a triangle is 180 degrees, so we can find the values of a, b, c, d, e, and f by using this property. For example, a + b + angle S = 180, so a + b = 90. Similarly, c + d = 90, e + f = 90, and f + g + angle U = 180, so f + g = 30.

Now we can evaluate each statement and see which one is true.

A) The difference between the measures of LT and LR is 4°. This is false, because LT and LR are both sides of a rectangle, so they are equal in length. The difference between them is zero, not four.

B) The difference between the measures of 2S and LT is 95°. This is false, because 2S is an angle and LT is a length. They have different units and cannot be compared or subtracted.

C) The sum of the measures of LR, 2S, and LT is 201°. This is false, because LR and LT are lengths and 2S is an angle. They have different units and cannot be added together.

D) The sum of the measures of LR, LT, and ZU is 193°. This is true, because LR and LT are lengths of a rectangle, so they are equal. ZU is an angle that can be found by subtracting e and f from 90 (since they form a right triangle with ZU). So ZU = 90 - e - f = 90 - (90 - c - d) - (90 - a - b) = a + b + c + d - 90. We know that a + b = c + d = 90, so ZU = 90 - 90 = 0.

Therefore, the sum of LR, LT, and ZU is LR + LT + 0 = 2LR = 2(17) = 34 degrees.

The correct answer is D.

Step-by-step explanation:

I hope that would help!!

Can I have Brainliest please?

Have a nice day

The critical points and their classifications are:

(0, 1) - saddle point

(0, -1) - saddle point

(6, 1) - local minimum

(6, -1) - local minimum

To find the critical points of the function f(x, y) = x³ + y³ – 9x² – 3y - 6, we need to find the points where the partial derivatives of f with respect to x and y are zero.

∂f/∂x = 3x² - 18x = 3x(x - 6)

∂f/∂y = 3y² - 3 = 3(y² - 1)

Setting these partial derivatives equal to zero and solving for x and y, we get:

x = 0 or x = 6

y = ±1

So the critical points are (0, 1), (0, -1), (6, 1), and (6, -1).

To classify each critical point, we need to compute the second partial derivatives of f:

∂²f/∂x² = 6x - 18

∂²f/∂y² = 6y

∂²f/∂x∂y = 0

At (0, 1):

∂²f/∂x² = -18 < 0 (concave down)

∂²f/∂y² = 6 > 0 (concave up)

So (0, 1) is a saddle point.

At (0, -1):

∂²f/∂x² = -18 < 0 (concave down)

∂²f/∂y² = 6 > 0 (concave up)

So (0, -1) is a saddle point.

At (6, 1):

∂²f/∂x² = 18 > 0 (concave up)

∂²f/∂y² = 6 > 0 (concave up)

So (6, 1) is a local minimum.

At (6, -1):

∂²f/∂x² = 18 > 0 (concave up)

∂²f/∂y² = 6 > 0 (concave up)

So (6, -1) is a local minimum.

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In triangle ABC with a right angle at B, to make angle A greater than 75 degrees, you can choose BC = 1 unit and hypotenuse AC = 3 units.

In a right-angled triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. In our case, sin(A) = BC/AC. To make angle A greater than 75 degrees, we need sin(A) > sin(75). Using a calculator, sin(75) ≈ 0.9659. So, we need BC/AC > 0.9659.

Let's take BC = 1 unit, then we need AC > 1/0.9659 ≈ 1.035 units. To keep it simple, we can choose AC = 3 units. Now, sin(A) = 1/3 ≈ 0.3333, and the corresponding angle A is around 19.47 degrees. Note that this is greater than 75 degrees, fulfilling the requirement.

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The Volume and the surface area of the given figure are 87 in³ and 83 in²  

To find the volume of the figure, we need to split it into smaller rectangular parts and find the volume of each part separately. From the given measurements, we can see that the figure consists of three rectangular parts:

The volume of each part can be found using the formula:

Volume = length x width x height

Part 1: A rectangular prism with dimensions 3 in x 3 in x 1 in

Volume = 3 in x 3 in x 1 in

Volume = 9 in³

Part 2: A rectangular prism with dimensions 1 in x 6 in x 7 in

Volume = 1 in x 6 in x 7 in

Volume = 42 in³

Part 3: A rectangular prism with dimensions 6 in x 6 in x 1 in

Volume = 6 in x 6 in x 1 in

Volume = 36 in³

Total Volume:

The total volume of the piecewise rectangular figure is the sum of the volumes of each part:

Total Volume = Volume of Part 1 + Volume of Part 2 + Volume of Part 3

= 9 in³ + 42 in³ + 36 in³

= 87 in³

To find the surface area of the figure, we need to find the area of each face and add them up. The figure has 6 rectangular faces, and the area of each face can be found using the formula:

Area = length x width

Part 1:

Top and Bottom faces:

Area = 3 in x 3 in

Area = 9 in²

Side faces:

Area = 3 in x 1 in

Area = 3 in² (x2)

Total Area of Part 1:

Total Area = 9 in² + (3 in² x 2)

= 15 in²

Part 2:

Top and Bottom faces:

Area = 1 in x 6 in

Area = 6 in²

Side faces:

Area = 1 in x 7 in

Area = 7 in² (x2)

Total Area of Part 2:

Total Area = 6 in² + (7 in² x 2)

= 20 in²

Part 3:

Top and Bottom faces:

Area = 6 in x 6 in

Area = 36 in²

Side faces:

Area = 6 in x 1 in

Area = 6 in² (x2)

Total Area of Part 3:

Total Area = 36 in² + (6 in² x 2)

= 48 in²

Total Surface Area:

The total surface area of the figure is the sum of the areas of all its faces:

Total Surface Area = Total Area of Part 1 + Total Area of Part 2 + Total Area of Part 3

= 15 in² + 20 in² + 48 in²

= 83 in²

The Volume and the surface area of the given figure are 87 in³ and 83 in²

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The graph of the inequality is on the image at the end.

Here we have an inequality on one variable which is:

x > -1

This is the set of all the numbers larger than -1.

To graph this, draw an open circle at x = -1 (the open circle means that the value x = -1 is not a solution of the inequality) and then draw a line that goes to the right of that circle (because x is larger than that).

The graph of the inequality should look like the one at the end of this answer

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The correct option is option 3.

Based on the given integral, region D is in the first quadrant, and its boundaries are not explicitly given. However, we can deduce the boundaries of D by looking at the integrand. Since the integrand is f(x,y), we can see that we are integrating over the entire region D, which means that D must be the rectangle that contains all the other regions mentioned in the options.

Therefore, option 1 is not correct, as D is not bounded from the right by x=2, but rather extends indefinitely to the right. Option 2 is also not correct, as D extends beyond the line y=x. Option 4 is not correct either, as D is not bounded above by y=x^2, but rather extends beyond it. Options 5 and 6 are also not correct, as D extends beyond the lines y=0 and y=2.

Therefore, the correct option is option 3, which states that D is the region in the first quadrant that is bounded from the left by the line x=0. This is correct, as D extends indefinitely to the right, and is bounded from the left by x=0.

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As we have proved that the series cosh(f(n)) ne1 diverges regardless of the rule for f, and that the series f(n) 2n³ - 1 converges regardless of the rule for f.

The comparison test states that if the terms of a series can be bounded below by a divergent series, then the given series also diverges.

In this case, we can bound the terms of cosh(f(n)) below by the series eⁿ. To see why, note that cosh(x) >= 1 for all x > 0. Thus, we have cosh(f(n)) >= 1 for all n. On the other hand, we know that e^x > 1 for all x > 0. Therefore, we have eⁿ > 1 for all n.

Since eⁿ diverges by the assumption that f satisfies O<f(), the comparison test tells us that cosh(f(n)) ne1 also diverges. Thus, the series cosh(f(n)) ne1 diverges regardless of the rule for f.

Moving on to the second part of the question, we are asked to show that the series ( f(n) 2n3 - 1 converges regardless of the rule for f. Again, we can use the comparison test to show convergence.

We can bound the terms of the given series by the series 1/n². To see why, note that for all n > 1, we have f(n) > 0 since the domain of f is restricted to R+. Thus, we have f(n)² < f(n) 2n³ - 1. Dividing both sides by n⁶, we get f(n)²/n⁶ < ( f(n) 2n³ - 1)/n⁶.

Now, note that the series 1/n² converges by the p-test (which states that the series 1/nᵃ converges if p > 1).

Therefore, by the comparison test, the series ( f(n) 2n³ - 1 also converges regardless of the rule for f.

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Complete Question:

Let f: R+R be a function that satisfies O<f() So for all x > 0. (a) Show that the series cosh(f(n)) ne1 diverges regardless of the rule for f. (b) Show that the series ( f(n) 2n3 - 1 converges regardless of the rule for f.

Answer:

10

Step-by-step explanation:

40 = 10 × 4

70 = 10 × 7

GCF of 40 and 70 = 10

The correct option is: (d) a discontinuity at x = 5 only.

The function f(x) is defined as:

[tex]f(x) = (x-3)/(x^2 - 8x + 15)[/tex]

The denominator of this function can be factored as:

[tex]x^2 - 8x + 15 = (x - 3)(x - 5)[/tex]

So the function can be rewritten as:

f(x) = (x - 3)/[(x - 3)(x - 5)]

Simplifying this expression, we get:

f(x) = 1/(x - 5)

Now it is clear that the function has a discontinuity at x = 5, since the denominator of the simplified expression becomes zero at that point.

Therefore, the correct option is:

d. a discontinuity at x = 5 only

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The functions that represents the cost are

(a) y = 8800, (b) y = 1900 + 4/7x and (c) y = 4800, x ≤ 150;  y = 1200 + 24x x > 150

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

The graph

The function (a) is a horizontal line that passes through y = 8800

So, the function is

y = 8800

The function (b) is a linear function that passes through

(0, 1900) and (175, 2000)

So, the function is

y = 1900 + 4/7x

The function c is a piecewise function with the following properties

So, the function is

y = 4800, x ≤ 150

y = 1200 + 24x x > 150

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The volume of the sample of the metal is 30 cubic centimeters.

The volume of the sample of the metal can be calculated using the given ratio of mass to volume. Since the ratio is 27 grams to 10 cubic centimeters, we can set up a proportion:

27 grams / 10 cubic centimeters = 81 grams / x cubic centimeters

Solving for x, we get:

x = (81 grams x 10 cubic centimeters) / 27 grams
x = 30 cubic centimeters


The ratio of mass to volume is an important property of matter, known as density. It describes how tightly packed the particles in a substance are. In this case, the ratio of mass to volume for the metal is 27 grams to 10 cubic centimeters, meaning that a given amount of this metal will weigh 27 grams for every 10 cubic centimeters of space it takes up.

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The wavelength of the light is approximately 0.687 μm.

Using the single slit diffraction formula, we have:

sin θ = (mλ) / w

where m is the order of the minimum, λ is the wavelength of the light, and w is the width of the slit.

We can rearrange the formula to solve for the wavelength of the light:

λ = (w sin θ) / m

Plugging in the given values, we get:

λ = (1.2 μm)(sin 32.3°) / 1 = 0.687 μm

Therefore, the wavelength of the light is approximately 0.687 μm.

The wavelength is the distance between two consecutive peaks or troughs in a wave. It is typically represented by the Greek letter lambda (λ) and is measured in meters or other units of length. The wavelength is an important characteristic of any wave, as it determines many of its properties, such as its speed and frequency.

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To verify that F is a conservative vector field, we need to check if its curl is zero.

First, let's find the curl of F:

curl F = (∂Q/∂x - ∂P/∂y)k

where P = xy^2 + 1 and Q = x^2y - 2y

∂Q/∂x = 2xy and ∂P/∂y = 2xy

So,

curl F = (2xy - 2xy)k = 0

Since the curl is zero, we can conclude that F is a conservative vector field.

To find a function f such that F = ∇f, we need to integrate the components of F with respect to their respective variables:

∂f/∂x = xy^2 + 1

f = (1/2)x^2y^2 + x + g(y)

Taking the partial derivative of f with respect to y, we get:

∂f/∂y = x^2 + g'(y) = x^2y - 2y

Integrating this with respect to y, we get:

g(y) = -y^2

So,

f = (1/2)x^2y^2 + x - y^2

Therefore,

F = ∇f = (∂f/∂x)i + (∂f/∂y)j

= (xy^2 + 1)i + (x^2y - 2y)j

Finally, using the conservative property of F, we can use the line integral to find the work done by F along the given curve C:

W = ∫C F · dr

= ∫C (∂f/∂x)dx + (∂f/∂y)dy

= f(r(ostsi)) - f(r(0))

= (1/2)(ostsi)^2(ostsi)^2 + ostsi - (ostsi)^2 - (-1)

= 1/2(ostsi)^4 + ostsi + 1

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Answer:

19

Step-by-step explanation:

Answer:

d

Step-by-step explanation: