The Battery Park Stable feeds and houses horses used to pull tourist-filled carriages through the streets of Charleston’s historic waterfront area. The stable owner, an ex-racehorse trainer, recognizes the need to set a nutritional diet for the horses in his care. At the same time, he would like to keep the overall daily cost of feed to a minimum. The feed mixes available for the horses’ diet are an oat product, a highly enriched grain, and a mineral product. Each of these mixes contains a certain amount of five ingredients (labeled as A, B, C, D, and E) needed daily to keep the average horse healthy. The table below shows these minimum requirements, units of each ingredient per pound of feed mix, and costs for the three mixes. In addition, the stable owner is aware that an overfed horse is a sluggish worker. Consequently, he determines that 6 pounds of feed per day are the most that any horse needs to function properly.
a) (4 pts) Is this a maximization or a minimization problem? What is the objective?
b) (8 pts) Formulate this problem mathematically. Use X1, X2, and X3 to represent the amount of
oat product, enriched grain, and mineral product, respectively. Write down or type the objective
function and all constraints clearly. (Don’t forget the nonnegativity constraints when applicable.)
c) (8 pts) Set up this problem in Excel and use Solver to find the optimal daily mix of the three
feeds, i.e., the optimal value for X1, X2, and X3.

Answer:

The single transformation that maps triangle P onto triangle R is a reflection about the origin followed by a translation of 5 units to the right and 2 units down.

Answer:

Slope-Intercept Form: y=2x+1

Standard Form: 2x-y=6

1. [tex]y=\frac{1}{3}x-1[/tex]

2. [tex]y=-\frac{2}{5}x[/tex]

3. [tex]y=x+8[/tex]

4. [tex]y=-\frac{1}{2}x+3[/tex]

5. [tex]y=2x+1[/tex]

6. [tex]y=-4x-5[/tex]

7. [tex]y=-x+2[/tex]

8. [tex]y=\frac{2}{3}x-4[/tex]

9. [tex]y=x\\[/tex]

The equation of the plane passing through the lines of intersection of the planes 2x - 7y + 4z = 3 and 3x - 5y + 4z + 11 = 0 and the point (-2, 1, 3) is

-13x - 8y + z + 90 = 0

We have,

To find the equation of the plane passing through the intersection of two planes, we need to first find the direction vector of the line of intersection of the two planes.

Let's call the two planes P1 and P2, which are given by the equations:

P1: 2x - 7y + 4z = 3

P2: 3x - 5y + 4z + 11 = 0

To find the direction vector of the line of intersection of P1 and P2, we can take the cross product of the normal vectors of P1 and P2. The normal vectors are:

n1 = <2, -7, 4>

n2 = <3, -5, 4>

Taking the cross-product of n1 and n2.

n1 x n2 = <-13, -8, 1>

This vector is parallel to the line of intersection of P1 and P2. We can use this vector as the direction vector for the line.

Now, we need to find a point on the line.

We can use the point of intersection of P1 and P2 as a point on the line.

To find this point, we can solve the system of equations:

2x - 7y + 4z = 3

3x - 5y + 4z + 11 = 0

Subtracting the first equation from the second, we get:

x + 2y + 7 = 0

Solving for x in terms of y,.

x = -2y - 7

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

-4y - 14 - 7y + 4z = 3

Simplifying.

-11y + 4z = 17

Choosing y = 1.

-11 + 4z = 17

Solving for z.

z = 7

Substituting y = 1 and z = 7 into the equation for x.

x = -2(1) - 7 = -9

So a point on the line is (-9, 1, 7).

Now, we can use the point (-9, 1, 7) and the direction vector <-13, -8, 1> to write the equation of the plane passing through the line of intersection of P1 and P2.

Let's call this plane P3.

Using the point-normal form of the equation of a plane.

-13(x + 9) - 8(y - 1) + (z - 7) = 0

Expanding and simplifying, we get:

-13x - 8y + z + 90 = 0

Thus,

The equation of the plane passing through the lines of intersection of the planes 2x - 7y + 4z = 3 and 3x - 5y + 4z + 11 = 0 and the point (-2, 1, 3) is:

-13x - 8y + z + 90 = 0

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The vertices of image N′M′O′ if the preimage is reflected over x = −2 include the following: D. N′(9, 2), M′(6, 1), O′(7, 3)

In Mathematics and Geometry, a reflection over or across the y-axis or line x = 0 is represented and modeled by this transformation rule (x, y) → (-x, y).

Similarly, the reflection of a point (x, y) over the line x = a is modeled by this transformation rule:

(x, y) → (-2a - x, y)

(x, y) → (-2(-2) - x, y)

(x, y) → (4a - x, y)

By applying a reflection over x = -2 to the coordinate of the given triangle NMO, we have the following coordinates:

(x, y)                             →                 (4 - x, y).

N (−5, 2)                        →                (9, 2)

M(−2, 1)                        →                 (6, 1)

O(−3 , 3)                       →                 (7, 3)

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Answer:

N′(−5, −6), M′(−2, −5), O′(−3, −7)

Step-by-step explanation:

I am in the middle of taking the test and even made a graph to check my work! This should be the correct answer

If you want to transform rectangular coordinates (x, y) into its polar counterpart (r, θ), do the following:

Firstly, obtain r by computing the distance from the origin to point (x,y), making use of the Pythagorean theorem or simply put, by evaluating r = sqrt(x^2 + y^2).

Secondly, acquire θ which denotes the angle formed between the positive x-axis and a line correlating the origin and the coordinate (x,y). Calculate this via inverse tangent function: θ = atan2(y, x).

Thirdly, in case the measurement for θ is indicated in radians while you require degrees instead, multiply by 180/π to convert it.

Usually expressed as (r, θ), where r indicates proximity disentangled between the origin and the point (x,y), whilst θ identifies the aforementioned angle.

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1. Distribute. Everything. 2x - 2 + 3x + 3 = 4x + 16.
2. Add your like terms. (must be on the same side) 5x + 1 = 4x + 16
3. Single out x by subtracting 1. 5x = 4x + 16 - 1.
4. Simplify. 5x = 4x + 15
5. Subtract 4x from both sides to combine like terms. 5x - 4x = 15.
6. Simplify. 1x = 15.

That means your final answer would be x = 15. If you were to input 15 into every x, it would simplify to 76 = 76, which means x is definitely 15.

1. Distribute. Everything. 2x - 2 + 3x + 3 = 4x + 16.
2. Add your like terms. (must be on the same side) 5x + 1 = 4x + 16
3. Single out x by subtracting 1. 5x = 4x + 16 - 1.
4. Simplify. 5x = 4x + 15
5. Subtract 4x from both sides to combine like terms. 5x - 4x = 15.
6. Simplify. 1x = 15.

That means your final answer would be x = 15. If you were to input 15 into every x, it would simplify to 76 = 76, which means x is definitely 15.

Answer:

x = 15

Step-by-step explanation:

Simplifying the expression:

Add the numbers:

Combine like terms:

Subtract 1 from both sides:

Subtract 4x from both sides:

Answer:

x = 15

Step-by-step explanation:

Simplifying the expression:

Add the numbers:

Combine like terms:

Subtract 1 from both sides:

Subtract 4x from both sides:

The domain is all real numbers, and the range is (y ≤ -4.855). (option b)

To sketch the graph of a quadratic equation, you can use a graphing calculator, which is a handy tool that can produce a visual representation of the equation. For instance, let's take the quadratic equation y = -0.5x² + 0.7x - 5.1 and sketch its graph using a graphing calculator.

When we enter this equation into the graphing calculator, we can see a U-shaped curve that is symmetric about the vertical line passing through the vertex. The vertex is the point where the parabola changes direction and is given by the formula x = -b/2a, y = f(x), where f(x) is the value of y at the vertex.

Now let's examine the given options and determine their domain and range based on the graph of the quadratic equation.

all real numbers, R: (y ≤ -4.855)

For this option, we can observe that the parabola opens upward, and the vertex is at (0.7, -5.35).

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The graph of the absolute value function y = -g(x) + 1 is shown in the image attached below.

The graph of the absolute value function y = 2h(x + 4) is shown in the image attached below.

In Mathematics, the translation a geometric figure or graph upward simply means adding a digit to the value on the y-coordinate of the pre-image or function.

In Mathematics, a vertical translation to the positive y-direction (upward) is modeled by this mathematical expression g(x) = f(x) + N.

Where:

In this scenario, we can logically deduce that the equation of the first parent absolute value function is g(x) = |2x| and it would be reflected over the y-axis, and then translated by 1 units upward;

g(x) = |2x|

y = -g(x) + 1 = |2x| + 1

h(x) = |2x + 4| - 4

y = 2h(x + 4)

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Answer:

this is the Pythagorean theorem this is the formula a^2 + b^2 = c^2

Step-by-step explanation:

please mark as brainliest have a great day :D

Polling companies survey 1060 individuals to estimate a population proportion with a margin of error of 3% with 95% confidence because it strikes a balance between cost and accuracy, and it is based on statistical calculations that aim to achieve a reasonable level of precision while minimizing the margin of error.

When a polling company selects a sample size of 1060 individuals, it is based on statistical calculations that aim to achieve a 3% margin of error with 95% confidence. The sample size needs to be large enough to reduce the margin of error and increase the level of confidence in the results.

The margin of error decreases as the sample size increases, and it is inversely proportional to the square root of the sample size. Therefore, a larger sample size leads to a smaller margin of error. However, increasing the sample size also increases the cost and time required to conduct the survey.

Polling companies aim to strike a balance between the cost and the desired level of accuracy, which is typically a margin of error of 3-5% with 95% confidence. A sample size of 1060 individuals is often considered a reasonable compromise between cost and accuracy, as it provides a small enough margin of error to be useful while being feasible to conduct within a reasonable time and budget.

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The number line is shown below:

The given inequality:

x -5 < 11

Add 5 on both sides

x -5 + 5 < 11 + 5

Add x < 16

Thus, this is represented in the number line below:

A graphical illustration of numbers in a line is known as a number line. It is utilized for indicating the relation between values, and also to do calculations such as subtraction, multiplication, division, etc. The figures get bigger from left to right and diminishes from right to left.

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Okay, let's analyze the r^2 values for the 4 options:

a. y=16.5839•1.0185^x

r^2 is undefined, as this is an exponential model. We cannot determine how well it fits the data based only on r^2.

b. y=0.985515x-2.81333

This is a linear model, so r^2 would be between 0 and 1. Without knowing the actual r^2 value, we cannot determine how good the fit is.

c. y=-0.0000758936x^3+0.0143444x^2+0.207549x+7.70667

This is a 3rd order polynomial model. Again, without the r^2 value, we cannot determine the quality of fit. It could be a good or poor fit.

d. y=0.00182197x^2+0.785098x+1.195

If this model has the highest r^2 value (closest to 1), it would indicate the best fit to the data of the options.

In summary, without seeing the actual r^2 values calculated from the data for each model, we cannot definitively say which option is the "best fit". The model with the r^2 closest to 1 would likely be the best, but r^2 only tells part of the story. Other factors like complexity, interpretability, and residual analysis also matter. But based solely on r^2, the quadratic model d seems the most likely to be the best fit, as long as its r^2 value is high.

Does this help explain the approach? Let me know if you have any other questions!

Answer: $1.182

Step-by-step explanation: 3.546 divided by 3

Answer:

$1,182

Step-by-step explanation:

I just took the total of $3546 and divided the total by the three grade levels so 3 and I got the answer 3,546. But honestly i'm not sure if it's right lol, so if it's incorrect let me know and I'll work into it more.

Answer:

a line graph

Step-by-step explanation:

We are given the equation of a parabola, [tex]y=-x^2+2[/tex]. The question asks us to find both values of a point given the the x-coordinate which is -3. To find the y-coordinate, simply plug in -3 for x into the equation of the parabola.

[tex]y=-x^2+2 \Longrightarrow y=-(-3)^2+2 \Longrightarrow y=-(9)+2 \Longrightarrow y=-9+2[/tex]

[tex]\Longrightarrow y=-7[/tex]

Thus, the point is (-3,-7)

Hence the correct option is D. 38inches long, to the nearest inch, should the strip of foil be.

The circumference of a circle  in mathematics is its perimeter. That is, if the circle were opened up and straightened out to a line segment, the circumference would be the length of the arc. In according to broader  the perimeter is the length of any closed figure of  curve.

A  straight line and cuts through the middle of a body or figure. particularly  the length of a diameter is a line segment passing through the center of a circle with its ends on the circumference.

We determine the circumference of the pie pan in order to determine the length of the aluminium foil strip required.

Circumference =2[tex]\pi[/tex]r.

where, r = circle of radius and [tex]\pi[/tex] = mathematical constant =3.14.

The pie pan of  radius = half of its diameter=12 inches/2= 6 inches.

Now, the pie pan of circumference is:

Circumference: 2 [tex]\pi[/tex]r=2[tex]\pi[/tex] *6 inches= 37.7 inches

Therefore , the length of the aluminium foil strip = 38 inches,

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Answer:

4

Step-by-step explanation:

1p:80°-140。＝40°，40°

Thus, decimal word problem that involves subtraction and division-

The two empty water container weighs 0.64 kg and 1.728 kg when filled with water. Find the water weigh?

The decimal point, which distinguishes between a decimal number's whole number and fractional component, is a little dot. Decimal numbers include 4.14, 3.786, 5.453, and 123.5687 as examples.

decimal word problem : (subtraction and division)

The two empty water container weighs 0.64 kg and 1.728 kg when filled with water. Find the water weigh?

Division: Weight of one water container = 0.64 / 2 = 0.34 kg.

Subtraction: Weight of water : 0.728 - 0.34 = 0.385 kg.

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Answer:

Step-by-step explanation:

Set up a simple system.

Price = 4x where x = price of donut

For the sale, the price = 3.78, but the x value is 0.5x as it is half off.

$3.78 = 4(0.5x)

Solve for x =$1.89

Therefore, the original price of the donuts were $1.89.

Check by doing 4 x 50%(.5) of 1.89 which equals 3.78!

Answer:

B represents a function.

The number line where we plotted 1 1/4 and 5/2 is added as an attachment

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

-1 1/4 and 5/2

To start with, we convert both numbers to the same form

i.e. decimal or fraction

When converted to fractions, we have

-5/4 and 5/2

Rewrite the denominators of the fractions

So, we have

-5/4 and 10/4

This means that we can plot -5/4 at -5 and 10/4 at point 10 where the difference in each interval is 1/4

Using the above as a guide, we have the following:

The number line is attached

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Part A:

To calculate the mean, we add up all the values and divide by the total number of values:

12 + 12 + 19 + 20 + 22 + 26 + 31 + 31 + 40 + 68 = 281

Mean = 281 / 10 = 28.1

To find the median, we need to find the middle value. Since there is an even number of values, we take the average of the two middle values:

Median = (22 + 26) / 2 = 24

The mode is the value that appears most frequently. In this case, there is no mode as no value appears more than once.

The range is the difference between the highest and lowest values:

Range = 68 - 12 = 56

Part B:

The median may be a better measure of central tendency in this scenario as there is an outlier value of 68 that can skew the mean. Since outliers can have a significant impact on the mean, reporting the median may give a better representation of the typical amount of time spent practicing for the tournament. Therefore, the tennis players should report the median value.

the coordinates of the point that represents the number of snowballs needed to make 2 snowmen should be option C. which is (2,6)

The identity (x² + y²)² = (x² - y²)² + (2xy)²

An identity is a mathematical equation which shows that one side of the equation is equivalent and equal to the other side of the equation.

To verify the following Pythagorean identity for all values of x and y:

(x² + y²)² = (x² - y²)² + (2xy)², we need to shows that

left-hand side L.H.S = right hand side

So, we proceed as follows

L.H.S = (x² + y²)²

Expanding the brackets, we have that

(x² + y²)² = (x² + y²)(x² + y²)

= (x²)² + 2x²y² + (y²)²

=  (x²)² + (y²)² + 2x²y²  

Now, we know that (a - b)² = a² + b² - 2ab. So,

a² + b² = (a - b)² + 2ab

Let a = x² and b = y²

So, we have that

(x²)² + (y²)² =  (x² - y²)² + 2x²y²

So, substituting this into the equation, we have that

(x²)² + (y²)² + 2x²y² = (x² - y²)² + 2x²y²  + 2x²y²

= (x² - y²)² + 4x²y²

= (x² - y²)² + 2²x²y²

= (x² - y²)² + (2xy)²

So, Since L.H.S = R.H.S

(x² + y²)² = (x² - y²)² + (2xy)²

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Answer:

a. The probability of getting an O on the first draw can be calculated by dividing the number of O tiles by the total number of tiles in the bag:

```

P(O on first draw) = 3/15 = 0.2

```

Therefore, the probability of getting an O on the first draw is 0.2 or 20%.

b. After removing one O tile, there are 14 tiles left in the bag, of which 2 are U tiles. Therefore, the probability of getting a U if you didn't replace the O is:

```

P(U without replacing O) = 2/14 = 1/7 ≈ 0.143

```

Therefore, the probability of getting a U if you didn't replace the O is approximately 0.143 or 14.3%.

c. The probability of getting O and then U if you don't replace the O can be calculated as follows:

```

P(O and then U) = P(O on first draw) * P(U without replacing O)

               = 3/15 * 2/14

               = 1/35

               ≈ 0.029

```

Therefore, the probability of getting O and then U if you don't replace the O is approximately 0.029 or 2.9%.

Step-by-step explanation:

Answer:

Profit when producing 70 items = $3,050

Profit when producing 80 items = $1,350

Step-by-step explanation:

To find the profit when producing a certain number of items, we need to subtract the cost from the revenue:

Profit = Revenue - Cost

Let's calculate the profit for producing 70 items:

Revenue when producing 70 items:

R(70) = -0.5(70-120)^2 + 7,200 = $6,800

Cost of producing 70 items:

C(70) = 50(70) + 250 = $3,750

Profit when producing 70 items:

Profit = Revenue - Cost = $6,800 - $3,750 = $3,050

Similarly, let's calculate the profit for producing 80 items:

Revenue when producing 80 items:

R(80) = -0.5(80-120)^2 + 7,200 = $5,600

Cost of producing 80 items:

C(80) = 50(80) + 250 = $4,250

Profit when producing 80 items:

Profit = Revenue - Cost = $5,600 - $4,250 = $1,350

We can see that the profit is less when producing 10 more units because the cost of producing those additional units exceeds the revenue generated from selling them. In other words, the marginal cost (the cost of producing one additional unit) is greater than the marginal revenue (the revenue generated from selling one additional unit) beyond a certain point.

This is an example of the law of diminishing returns, which states that as we increase the quantity of inputs while keeping other inputs constant, the marginal product (output per unit of input) eventually decreases.

Hope this helps!