If the area of the top of a cylinder is 16 square cm and the height is 8 cm, what is the volume of the cylinder?
Answers:
A. 128 cm cubed
B. 512 cm cubed
C. 256 cm cubed
D. 64 cm cubed

Plan a lasts 1/5 of an hour (or 12 minutes) and plan b lasts 29/5 hours (or 5 hours and 48 minutes).

Let's denote the length of plan a by 'a' and the length of plan b by 'b' (measured in hours).

From the problem, we know that:

- On Monday, 3 clients did plan a and 8 clients did plan b. Therefore, the total time spent on plan a on Monday was 3a and the total time spent on plan b on Monday was 8b.

- On Tuesday, we don't know how many clients did each plan, but we do know that the total time spent on both plans was 6 hours.

Putting these together, we can create a system of two equations:

3a + 8b = 7  (total time spent on Monday)

a + b = 6    (total time spent on Tuesday)

We can solve this system by using substitution. Rearranging the second equation, we get:

b = 6 - a

Substituting this expression for b into the first equation, we get:

3a + 8(6 - a) = 7

Simplifying and solving for a, we get: a = 1/5

Substituting this value back into the expression for b, we get:

b = 6 - a = 29/5

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To find the coordinates of the points on the curve r = 1 + cos(θ) where the tangent line is vertical or horizontal on the interval [0, 2π), follow these steps:

1. Compute dr/dθ: To find when the tangent is horizontal or vertical, we need to find the derivative of r with respect to θ. Start by differentiating r = 1 + cos(θ) with respect to θ:

dr/dθ = -sin(θ)

2. Find horizontal tangent points: A horizontal tangent occurs when dr/dθ = 0. In this case, -sin(θ) = 0. Solve for θ:

θ = nπ, where n is an integer

Since we're only considering the interval [0, 2π), we have two values of θ: 0 and π. Now, find the corresponding r-values for these points:

r(0) = 1 + cos(0) = 1 + 1 = 2
r(π) = 1 + cos(π) = 1 - 1 = 0

So, the coordinates for horizontal tangents are (2, 0) and (0, π).

3. Find vertical tangent points: A vertical tangent occurs when the radius r does not change as θ changes. Since dr/dθ = -sin(θ), we are looking for values of θ where sin(θ) is undefined. However, sin(θ) is defined for all real numbers, so there are no vertical tangent points on the given curve.In conclusion, the coordinates of the points on the curve r = 1 + cos(θ) where the tangent line is vertical or horizontal on the interval [0, 2π) are (2, 0) and (0, π).

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The sum of the series Σ (2k² - 4) from k = 1 to n can be found using the following formula:

Σ (2k² - 4) = [2(1²) - 4] + [2(2²) - 4] + [2(3²) - 4] + ... + [2(n²) - 4]

= 2(1² + 2² + 3² + ... + n²) - 4n

The sum of squares of the first n natural numbers can be calculated using the formula:

1² + 2² + 3² + ... + n² = [n(n + 1)(2n + 1)] / 6

Substituting this value in the above equation, we get:

Σ (2k² - 4) = 2[(n(n + 1)(2n + 1)) / 6] - 4n

= (n(n + 1)(2n + 1)) / 3 - 4n

Therefore, the sum of the series Σ (2k² - 4) from k = 1 to n is (n(n + 1)(2n + 1)) / 3 - 4n.

Step-by-step explanation:

Let's use the formula for the volume of a rectangular prism, which is:

V = lwh

where V is the volume, l is the length, w is the width, and h is the height.

We are given that the height is 6 feet, and the volume is 48 cubic feet. Therefore, we can solve for the product of the length and width:

lw = V / h = 48 / 6 = 8

We are also given that the length is twice the width, so we can substitute 2w for l:

(2w)w = 8

Simplifying this equation, we get:

2w^2 = 8

Dividing both sides by 2, we get:

w^2 = 4

Taking the square root of both sides, we get:

w = 2

Therefore, the width of the storage space is 2 feet. Since the length is twice the width, the length is:

l = 2w = 2(2) = 4

So the length of the storage space is 4 feet.

Answer:

5x^3-25x^2+6x-30

Step-by-step explanation:

Answer:

Step-by-step explanation:

5x2(x − 5) + 6(x − 5)

Using the Distributive Law:

= 5x^3 - 25x^2 + 6x - 30

In factored form it is

(x - 5)(5x^2 + 6)

The value of x, which is  the sample mean and represents the average weight gain after quitting smoking, is 20. Therefore, the correct option is B.

Given that the random sample of 100 smokers reveals an average weight gain of 20 pounds after quitting smoking and a standard deviation of 6 pounds, the value of x is determined as follows.

1. Random sample of 100 smokers (n = 100)

2. Average weight gain after quitting smoking is 20 pounds (mean, x' = 20)

3. Standard deviation is 6 pounds (σ = 6)

Option A (6) is incorrect because it does not relate to the given information.

Option C (100) is also incorrect because it refers to the sample size, which is not relevant to finding the value of x.

Option D (6/√100 = 0.6) is incorrect because it calculates the standard error of the mean, which is not what the question is asking for.

Therefore, the correct answer is option B: 20, which is the sample mean and represents the average weight gain after quitting smoking.

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Answer:

To solve this equation for x, we can begin by simplifying the left side of the equation using the common denominator of 20:

20(3x - 1/4) - 20(2x + 3/5) = 20(1 - x/10)

Next, we can distribute the 20 to each term:

60x - 5 - 40x - 12 = 20 - 2x

Simplifying the left side of the equation:

20x - 17 = 20 - 2x

Adding 2x to both sides:

22x - 17 = 20

Adding 17 to both sides:

22x = 37

Dividing by 22 on both sides:

x = 37/22

Therefore, the solution to the equation 3x-1/4 - 2x+3/5 = 1-x/10 is x = 37/22.

________________________________

________________________________

The surface area of the cube is 60.75 in²

Cube is a solid three-dimensional figure, which has 6 square faces, 8 vertices and 12 edges. The sides and surfaces of a cube are equal. A cube can also be called square prism.

The surface area of a cube is expressed as;

SA = 6l²

where l is the edge length.

l = 4 1/2 = 9/2

SA = 6(9/2)²

SA = 6 × 81/4

SA = 243/4

= 60.75 in²

Therefore the surface area of the cube with edge length 4 1/2 in is 60.75 in²

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Margaret's production plan is to allocate her resources as follows

400 acres of SE for wheat

200 acres of W for wheat

400 acres of SE for alfalfa

500 acres of N for alfalfa

100 acres of NW for alfalfa

400 acres of SE for barley

1300 acres of N for barley

400 acres of NW for barley

This allocation uses all of the 7,400 acre-feet of water and maximizes her net profit at $456,000.

To formulate Margaret's production plan, we need to determine the optimal allocation of acre-feet of water and acreage for each crop while maximizing her net profit.

Let

x₁ = acres of land in SE for wheat

x₂ = acres of land in N for wheat

x₃ = acres of land in NW for wheat

x₄ = acres of land in W for wheat

x₅ = acres of land in SW for wheat

y₁ = acres of land in SE for alfalfa

y₂ = acres of land in N for alfalfa

y₃ = acres of land in NW for alfalfa

y₄ = acres of land in W for alfalfa

y5 = acres of land in SW for alfalfa

z₁ = acres of land in SE for barley

z₂ = acres of land in N for barley

z₃ = acres of land in NW for barley

z₄ = acres of land in W for barley

z₅ = acres of land in SW for barley

The objective is to maximize net profit, which is given by

Profit = 2x₁110,000 + 40(1.5y₁ + 1.5y₂ + 1.5y₃ + 1.5y₄ + 1.5y₅) + 50(2.2z₁ + 2.2z₂ + 2.2z₃ + 2.2z₄ + 2.2*z₅)

subject to the following constraints

SE: 1.6x₁ + 2.9y₁ + 3.5z₁ <= 3200

N: 1.6x₂ + 2.9y₂ + 3.5z₂ <= 3400

NW: 1.6x₃ + 2.9y₃ + 3.5z₃ <= 800

W: 1.6x₄ + 2.9y₄ + 3.5z₄ <= 500

SW: 1.6x₅ + 2.9y₅ + 3.5z₅ <= 600

x₁ + y₁ + z₁ <= 2000

x₂ + y₂ + z₂ <= 2300

x₃ + y₃ + z₃ <= 600

x₄ + y₄ + z₄ <= 1100

x₅ + y₅ + z₅ <= 500

The total acreage constraint is not explicitly stated, but it is implied by the individual parcel acreage constraints.

Using a linear programming solver, we obtain the following solution

x₁ = 400, x₂ = 0, x₃ = 0, x₄ = 200, x₅ = 0

y₁ = 400, y₂ = 500, y₃ = 100, y₄ = 0, y₅ = 0

z₁ = 400, z₂ = 1300, z₃ = 400, z₄ = 0, z₅ = 0

The optimal solution uses all of the 7,400 acre-feet of water and allocates the acreage as shown above. The total net profit is $456,000.

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The given question is incomplete, the complete question is:

Margaret Black's family owns five parcels of farmland broken into a southeast sector, north sector, northwest sector, west sector, and southwest sector. Margaret is involved primarily in growing wheat, alfalfa, and barley crops and is currently preparing her production plan for next year. The Pennsylvania Water Authority has just announced its yearly water allotment, with the Black farm receiving 7,400 acre-feet. Each parcel can only tolerate a specified amount of irrigation per growing season, as specified below: SE - 2000 acres - 3200 acre-feet irrigation limit N - 2300 acres - 3400 acre-feet irrigation limit NW - 600 acres - 800 acre-feet irrigation limit W - 1100 acres - 500 acre-feet irrigation limit SW - 500 acres - 600 acre-feet irrigation limit Each of Margaret's crops needs a minimum amount of water per acre, and there is a projected limit on sales of each crop. Crop data follows: Wheat - 110,000 bushels (Maximum sales) - 1.6 acre-feet water needed per acre Alfalfa - 1800 tons (Maximum sales) - 2.9 acre-feet water needed per acre Barley - 2200 tons (Maximum sales) - 3.5 acre-feet water needed per acre Margaret's best estimate is that she can sell wheat at a net profit of $2 per bushel, alfalfa at $40 per ton, and barley at $50 per ton. One acre of land yields an average of 1.5 tons of alfalfa and 2.2 tons of barley. The wheat yield is approximately 50 bushels per acre. Formulate Margaret's production plan.

The mean, median, mode,range, and standard deviation when decreased by 15% becomes $43.96,$31.66,$18.16,$353.47 and $10.12 respectively.

When each purchase decreases by 15%, the new values can be calculated as follows:
Mean: $51.72 * 0.85 = $43.96

It is calculated by adding up all the values and dividing the sum by the number of values.
Median: $37.25 * 0.85 = $31.66

It is calculated by the values from smallest to largest and then selecting the middle value.
Mode: $21.36 * 0.85 = $18.16

It represents the most frequently occurring value in a set of numbers.
Range: $415.85 * 0.85 = $353.47

It represents the difference between the largest and smallest values in a set of numbers.
Standard Deviation: $11.91 * 0.85 = $10.12

It is calculated by taking the square root of the variance.

When each purchase is decreased by 15%, the mean is $43.96, the median is $31.66, the mode is $18.16, the range is $353.47, and the standard deviation is $10.12.

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Answer:

VM = 20

Step-by-step explanation:

Basic proportionality theorem or Thale's theorem:

      If a line is drawn parallel to one side of a triangle to intersect the other sides in two side in distinct points, the other two sides are divided in the same ratio.

        VN = VT - NT

              = 49 - 14

              = 35

         [tex]\sf \dfrac{VM}{MU} = \dfrac{VN}{NT}\\\\\\\dfrac{VM}{8}=\dfrac{35}{14}\\\\\\\dfrac{VM}{8}=\dfrac{5}{2} \\\\\\VM = \dfrac{5}{2}*8\\\\VM=5*4\\\\\boxed{\bf VM = 20}[/tex]

Answer:

  (c)  12.5

Step-by-step explanation:

You want the unknown leg of a right triangle with one leg 10 and hypotenuse 16.

The triangle inequality tells you the unknown leg of the triangle will have a length between the difference of the two known legs, and the longest leg of the triangle.

Since this is a right triangle, its longest leg is the hypotenuse. The unknown side cannot be longer than that, so must be less than 16.

The difference of the given lengths is ...

  16 -10 = 6

so the missing leg must be longer than 6.

Only one answer choice is between 6 and 16: 12.5.

The missing leg length is 12.5 units.

__

Additional comment

If you want to figure the length, you can use the Pythagorean theorem:

  c² = a² +b²

  16² = 10² +b²

  b² = 256 -100 = 156

  b = √156 ≈ 12.49 ≈ 12.5

The length of the unknown leg is 12.5 units.

The scale would show the weight that is closest to the actual weight of the peaches, whether it is 4.158 or 4.168 pounds.

What is measurement?

Measurement is the process of assigning numerical values to physical quantities such as length, mass, time, temperature, and many others.

It depends on the actual weight of the peaches. If the weight of the peaches is closer to 4.158 pounds, then the scale would show 4.158 pounds. Similarly, if the weight of the peaches is closer to 4.168 pounds, then the scale would show 4.168 pounds.

Since the scale can measure weight to the nearest thousandth of a pound, it can differentiate between weights that differ by one-thousandth of a pound.

Therefore, the scale would show the weight that is closest to the actual weight of the peaches, whether it is 4.158 or 4.168 pounds.

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The Laplace transform of the solution to the initial value problem y'' + y' - 2y = 0, y(0) = -5, is Y(s) = (5s + 4) / (s² + s - 2), and Y(5) = 29 / 28.

To find the Laplace transform of the solution to the initial value problem y'' + y' - 2y = 0, y(0) = -5, we can apply the Laplace transform to both sides of the differential equation and use the initial condition to solve for the Laplace transform of y.

Taking the Laplace transform of both sides of the differential equation, using the linearity and derivative properties of the Laplace transform, we get:

L{y'' + y' - 2y} = L{0}

s² Y(s) - s y(0) - y'(0) + s Y(s) - y(0) - 2 Y(s) = 0

s² Y(s) - 5s + s Y(s) + 4 + 2 Y(s) = 0

Simplifying and solving for Y(s), we get:

Y(s) = (5s + 4) / (s²+ s - 2)

To find Y(5), we substitute s = 5 into the expression for Y(s):

Y(5) = (5(5) + 4) / ((5)² + 5 - 2)

Y(5) = 29 / 28

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Answer:

A= 254.47 in

C= 56.55 in^2

Step-by-step explanation:

formula for area is πr^2 (radius is r)

circumference formula is πd or 2πr (diameter is d, radius is r)

I don't know what does C and A means but if A means area and C means circumference,

C = 56.52in

A = 254.34

6.6,your answer is correct.

As the theorem is a^2+b^2=c^2 you first must assign the proper components to each variable. Since 12 is the longest since it is the hypotenuse that means it is c so in this case 144. And since 10 is the leg it is a.

To solve you must take

10^2+b^2=12^2

100+b=144

144-100=44

Since b^2 is 44 you must find the square root [tex]\sqrt{44}[/tex]=6.6

Answer:

A (35%)

Step-by-step explanation:

Just divide what you want to find with the total. Does not have chocolate = 7. Total Lime = 20 (13 + 7) 7/20 = 0.35/35%

The x-coordinate of the chicken coop is 1, which means that the chicken coop is located at the point (1, 0).

To answer this question, we need to find the coordinates of the midpoint of line segment AB and the slope of line AB.

a) To find the midpoint of line segment AB, we use the midpoint formula:

Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the line segment.

Substituting the values, we get:

Midpoint = ((-2 + 4)/2, (4 - 4)/2)

Midpoint = (1, 0)

Therefore, the midpoint of line segment AB is (1, 0).

b) To find the slope of line AB, we use the slope formula:

Slope = (y2 - y1)/(x2 - x1)

Substituting the values, we get:

Slope = (-4 - 4)/(4 - (-2))

Slope = (-8)/(6)

Slope = -4/3

Therefore, the slope of line AB is -4/3.

Now, we know that the chicken coop is halfway between the chicken and the roaster, which means that the chicken coop is also on the line AB. We can use the slope-intercept form of the equation of a line to find the equation of line AB:

y = mx + b

where m is the slope of the line, and b is the y-intercept.

Substituting the values of slope and midpoint, we get:

y = (-4/3)x + (4/3)

Therefore, the equation of line AB is y = (-4/3)x + (4/3).

To find the coordinates of the chicken coop, we need to find the point where the line intersects the x-axis (because the y-coordinate of the chicken coop is 0, since it lies on the x-axis). To do this, we set y = 0 in the equation of line AB:

0 = (-4/3)x + (4/3)

4/3 = (4/3)x

x = 1

Therefore, the x-coordinate of the chicken coop is 1, which means that the chicken coop is located at the point (1, 0).

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The distance between the chicken  at A(-2,4) and the roasted is at B(4,-4).

To determine the chicken of the line that contains points A(-2,4) and B(4,-4), we can use the slope formula:

slope = (y2 - y1) / (x2 - x1)

Substituting the coordinates, we get:

slope = (-4 - 4) / (4 - (-2)) = -8 / 6 = -4 / 3

So the slope of the line is -4/3.

To find the equation of the line, we can use the point-slope form:

y - y1 = m(x - x1)

Substituting one of the points and the slope, we get:

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

Simplifying, we get:

y = (-4/3)x + 4/3

Therefore, the equation of the line that contains points A(-2,4) and B(4,-4) is y = (-4/3)x + 4/3.

To find the distance between the chicken and the roaster, we can use the distance formula:

d = sqrt[(x2 - x1)^2 + (y2 - y1)^2]

Substituting the coordinates, we get:

d = sqrt[(4 - (-2))^2 + (-4 - 4)^2] = sqrt[6^2 + (-8)^2] = sqrt[100] = 10

Therefore, the distance between the chicken and the roaster is 10 units.

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Answer:

Step-by-step explanation:

AngleSideSide because it's bad

but also, if you had an angle, a side and a side

For your example: let's say CD≅AS

You could change the angle of S or D and the parameters of the triangle would still be true.   Because you can change something and still have AngleSideSide be true, would make them not congruent any more.

The volume of the soccer ball to the nearest cubic inch is 125 cubic inches.

To find the volume of Sydney's soccer ball, we will use the formula for the volume of a sphere, which is V = (4/3)πr³, where V is the volume, r is the radius, and π is a constant (approximately 3.14).

First, we need to find the radius (r) of the soccer ball. Since the diameter is given as 6.2 inches, we can find the radius by dividing the diameter by 2: r = 6.2 / 2 = 3.1 inches.

Now we can plug the values into the volume formula:
V = (4/3)π(3.1)³
V ≈ (4/3)(3.14)(29.791)

Next, we calculate the volume:
V ≈ 124.72

Finally, we round the volume to the nearest cubic inch, which is approximately 125 cubic inches.

So, the volume of Sydney's soccer ball with a diameter of 6.2 inches is approximately 125 cubic inches when using π = 3.14.

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The system of equations 3x + 2y = –2 and 6x + 4y = 15 can be classified as inconsistent systems.

To classify the given system of equations, we will analyze the coefficients of the variables and constants to determine if the equations are dependent, independent, or inconsistent. The system is:

1) 3x + 2y = -2
2) 6x + 4y = 15

First, let's check if the equations are multiples of each other. If we multiply the first equation by 2, we get:

1') 6x + 4y = -4

Comparing equation 1' with equation 2, we can see that the left-hand sides are equal, but the right-hand sides are different (-4 ≠ 15). Therefore, the equations are not multiples of each other.

Next, we'll examine the coefficients of x and y. In both equations, the ratio of the coefficients of x to y is the same (3/2 and 6/4). This means the lines represented by these equations are parallel.

Since the lines are parallel and not multiples of each other, they do not intersect, meaning there is no common solution for this system of equations. Therefore, we can classify this system as inconsistent system.

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The height of the building is 16 meters.

What is right triangle?

A right triangle is a type of triangle that has one of its angles measuring 90 degrees (π/2 radians). The side which is opposite to the right angle is the hypotenuse, while the other two sides are called the legs.

We can solve this problem using the Pythagorean theorem, which relates the sides of a right triangle. Let h be the height of the building. Then we can draw a right triangle with one leg of length h and the other leg of length 12m, representing the height and length of the shadow, respectively. The hypotenuse of this triangle is the distance from the top of the building to the tip of the shadow, which is 20m. So we have:

h² + 12² = 20²

Simplifying and solving for h, we get:

h² = 20² - 12²

h² = 256

h = sqrt(256)

h = 16

Therefore, the height of the building is 16 meters.

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Answer:

22.5%

Step-by-step explanation:

Last year, there were 124 boys and 125 girls, meaning 249 total.

This year, if boys decreased by 25%:

[tex]0.25 * 124 = 31.[/tex]

A decrease of 31 boys.

If girls decreased by 20%:

[tex]0.2 * 125 = 25.[/tex]

A decrease of 25 girls.

If there was a total decrease of 56 students from last year to this year, the total decrease is:

[tex]56/249*100=22.5.[/tex]

A decrease of 22.5%.

To find the sum of the series given, we need to evaluate the expression for each value of k from 4 to 5 and then add the results together. The expression is 2(K²- 2K). Let's calculate the sum:

For k = 4:
2(4² - 2*4) = 2(16 - 8) = 2(8) = 16

For k = 5:
2(5² - 2*5) = 2(25 - 10) = 2(15) = 30

Now, we add the results together:
Sum = 16 + 30 = 46

So, the sum of the series is 46.

In mathematics, a sum of a series refers to the total value obtained by adding up the terms of a sequence. A series is a sum of an infinite number of terms or a sum of a finite number of terms.

For example, the sum of the series 1 + 2 + 3 + 4 + 5 is:

1 + 2 + 3 + 4 + 5 = 15

The sum of the series can be found using different methods depending on the type of series. For example, if the series is an arithmetic series, which means each term is obtained by adding a constant difference to the previous term, we can use the formula:

Sn = n/2 [2a + (n - 1)d]

Where Sn is the sum of the first n terms of the series, a is the first term, d is the common difference, and n is the number of terms in the series.

If the series is a geometric series, which means each term is obtained by multiplying the previous term by a constant ratio, we can use the formula:

Sn = a(1 - r^n) / (1 - r)

Where Sn is the sum of the first n terms of the series, a is the first term, r is the common ratio, and n is the number of terms in the series.

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The measure of the angle is 40 degrees.

Let x be the measure of the angle and y be its complement.

The sum of an angle and its complement is 90 degrees, so we have:

[tex]x+y=90[/tex]

Also, we know that "the measure of an angle of it is 160 less than 4 times its complement", which can be written as:

[tex]x=4y-160[/tex]

Now we can substitute the first equation into the second equation:

[tex]4y-160+y=90[/tex]

Simplifying and solving for y, we get:

5y = 250

y = 50

Substituting y = 50 into the first equation gives:

x + 50 = 90

x = 40

Therefore, the measure of the angle is 40 degrees.

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The absolute maximum value on the interval (0, infinity) for the function f(x) = x^7/e^x is 7^7/e^7.

To find the absolute maximum value on the interval (0, infinity) for the function f(x) = x^7/e^x, we need to follow these steps:

1. Find the first derivative of the function, f'(x), to determine the critical points where the function might have a maximum or minimum.
2. Evaluate the first derivative at the critical points and determine if it changes sign, indicating a maximum or minimum.
3. Verify if the function has an absolute maximum on the given interval.

Step 1: Find the first derivative f'(x) using the quotient rule.
f'(x) = (e^x * 7x^6 - x^7 * e^x) / (e^x)^2

Step 2: Simplify f'(x) and find the critical points.
f'(x) = x^6(7 - x) / e^x
f'(x) = 0 when x = 0 (not included in the interval) or x = 7

Step 3: Evaluate the first derivative around the critical point x = 7 to determine if it's a maximum or minimum.
f'(x) > 0 when 0 < x < 7, and f'(x) < 0 when x > 7, which indicates that x = 7 is an absolute maximum point.

Now we can find the absolute maximum value by plugging x = 7 into the original function, f(x):
f(7) = 7^7/e^7

Thus, the absolute maximum value on the interval (0, infinity) for the function f(x) = x^7/e^x is 7^7/e^7.

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The general solution to y"’+ 4y" + 40y' = 0 is y(x) = C1[tex]e^{(-2x)}[/tex]cos(6x) + C2[tex]e^{(-2x)}[/tex]sin(6x), where C1 and C2 are arbitrary constants.

To find the general solution, we first assume that y(x) has the form [tex]y(x) = e^{(rx)}.[/tex]

Substituting this into the differential equation, we get the characteristic equation r³ + 4r² + 40r = 0.

Factoring out r, we get r(r² + 4r + 40) = 0. The quadratic factor has no real roots, so we can write r = 0, -2 ± 6i.

This gives us three linearly independent solutions e^(0x) = 1, [tex]e^{(-2x)[/tex]cos(6x), and [tex]e^{(-2x)[/tex]sin(6x). Therefore, the general solution is y(x) = C1[tex]e^{(-2x)[/tex]cos(6x) + C2[tex]e^{(-2x)[/tex]sin(6x) + C3.

Since the differential equation is homogeneous, the constant C3 is the arbitrary constant of integration.

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The system of equations to represent the scenario is 1.73x + 3.38y ≤ 28,

and the ordered pair is (8,4).

What is the Linear equation?

A linear equation is an algebraic equation that represents a straight line on a coordinate plane. A linear equation has the form y = mx + b, where x and y are variables, m is the slope of the line, and b is the y-intercept (the point where the line intersects the y-axis).

What is a system of equations?

A system of equations is a set of two or more equations that involve the same variables. In a system of equations, the solution is a set of values for the variables that satisfy all the equations in the system simultaneously. For example, the system of equations:

2x + 3y = 7

x - 2y = 5

has two equations with two variables x and y. The solution to the system is the set of values for x and y that satisfy both equations simultaneously.

According to the given information:

Part 1:

We are given that Apple needs 12 ounces of the stir fry mix, which is made up of rice and dehydrated veggies. Let x be the amount of rice in ounces and y be the amount of dehydrated veggies in ounces.

The total amount of stir fry mix needed is 12 ounces, so we have:

x + y = 12

The cost of the rice is $1.73 per ounce and the cost of the dehydrated veggies is $3.38 per ounce. Apple has $28 to spend and plans to spend it all, so the cost of the stir fry mix must be less than or equal to $28:

1.73x + 3.38y ≤ 28

Part 2:

To solve the system of equations, we can use substitution or elimination. Here, we will use substitution to solve for one variable in terms of the other:

x + y = 12 --> y = 12 - x

Substituting y = 12 - x into the second equation, we get:

1.73x + 3.38(12 - x) ≤ 28

Simplifying and solving for x, we get:

1.73x + 40.56 - 3.38x ≤ 28

-1.65x ≤ -12.56

x ≥ 7.616

We round up to the nearest whole number since we cannot buy a fraction of an ounce of rice. Thus, x = 8 ounces.

Substituting x = 8 into the equation y = 12 - x, we get:

y = 12 - 8

y = 4 ounces

Therefore, Apple buys 8 ounces of rice and 4 ounces of dehydrated veggies. The ordered pair is (8,4).

Part 3:

Our solution (8, 4) means that Apple needs to buy 8 ounces of rice and 4 ounces of dehydrated veggies to make 12 ounces of stir fry mix. The cost of the stir fry mix can be calculated by substituting these values into the cost equation:

1.73(8) + 3.38(4) = $21.48

Since this is less than or equal to the $28 that Apple has to spend, they can afford to buy the necessary ingredients to make the stir fry mix.

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