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I am inviting 32 people for my sweet sixteen. Is that considered as a lot of people? Yes or no?

Answer:  

Dimensions = 20 inches by 20 inches by 80 inches

The floor is 20 inches by 20 inches. The height is 80 inches.

The minimized cost is $2400.

========================================================

Work Shown:

h = height of the box, in inches

x = side length of the square base, in inches

x^2 = area of the floor = area of the ceiling

x^2h = volume of the rectangular box

x^2h = 32000

h = 32000/(x^2)

C = total cost in dollars

C = 0.25*(area of the sides) + 1.00*(area of the floor and ceiling)

C = 0.25*(xh+xh+xh+xh) + 1.00*(x^2+x^2)

C = 0.25*4xh + 1.00*2x^2

C = xh + 2x^2

C = x*(32000/(x^2)) + 2x^2

C = (32000/x) + 2x^2

C = (32000/x) + (x*2x^2)/x

C = (32000/x) + (2x^3)/x

C = (32000+2x^3)/x

The goal is to make C the smallest possible, aka we want to minimize it.

Visually we want the lowest point on the cost curve.

We have two options to get this task done:

I'll assume your teacher hasn't gone over calculus at this point. I'll go for option 1 mentioned above.

Use a graphing tool like GeoGebra to plot out the cost function curve. See the diagram below. The lowest point on this curve is at (20,2400).  I used the "min" function to determine this lowest point. Keep in mind that x > 0.

This lowest point indicates to us that x = 20 causes C(x) to be the smallest at $2400. This is the minimized cost.

Therefore, the square base should be 20 inches by 20 inches. The height should be:

h = 32000/(x^2) = 32000/(20^2) = 80 inches

The box should be 20 inches by 20 inches by 80 inches.

------------------------

Check:

Volume = length*width*height = 20*20*80 = 32000 cubic inches

This helps verify the answer.

Answer:

Width = 20 inches

Length = 20 inches

Height = 80 inches

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{6.7 cm}\underline{Volume of a square-based rectangular box}\\\\$V=x^2h$\\\\where:\\\phantom{ww} $\bullet$ $x$ is the side length of the base.\\\phantom{ww} $\bullet$ $h$ is the height.\\\end{minipage}}[/tex]

Given the volume of the square-based rectangular box is 32,000 in³, substitute this into the equation and rearrange to isolate h:

[tex]\implies x^2h=32000[/tex]

[tex]\implies h=\dfrac{32000}{x^2}[/tex]

[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Surface Area of a square-based rectangular box}\\\\$S=2x^2+4xh$\\\\where:\\\phantom{ww} $\bullet$ $x$ is the side length of the base.\\\phantom{ww} $\bullet$ $h$ is the height.\\\end{minipage}}[/tex]

Given:

Create an equation for the total cost, C, of the materials based on the equation for the surface area:

[tex]\implies C=(1)2x^2+(0.25)4xh[/tex]

[tex]\implies C=2x^2+xh[/tex]

Substitute the expression for h into the equation for cost to create an equation for C in terms of x:

[tex]\implies C=2x^2+x\left(\dfrac{32000}{x^2}\right)[/tex]

[tex]\implies C=2x^2+\dfrac{32000}{x}[/tex]

[tex]\implies C=2x^2+32000x^{-1}[/tex]

To find the value of x that would minimize the cost, differentiate the equation for cost:

[tex]\implies \dfrac{\text{d}C}{\text{d}x}}=4x-32000x^{-2}[/tex]

[tex]\implies \dfrac{\text{d}C}{\text{d}x}}=4x-\dfrac{32000}{x^{2}}[/tex]

[tex]\implies \dfrac{\text{d}C}{\text{d}x}}=\dfrac{4x^3-32000}{x^{2}}[/tex]

Set the differentiated equation to zero and solve for x:

[tex]\implies \dfrac{4x^3-32000}{x^{2}}=0[/tex]

[tex]\implies 4x^3-32000=0[/tex]

[tex]\implies 4x^3=32000[/tex]

[tex]\implies x^3=8000[/tex]

[tex]\implies x=20[/tex]

Therefore, the side lengths of the base of the box that would minimize the cost are 20 inches.

To find the height of the box that would minimize the cost, substitute the found value of x into the expression for height:

[tex]\implies h=\dfrac{32000}{20^2}[/tex]

[tex]\implies h=\dfrac{32000}{400}[/tex]

[tex]\implies h=80\; \rm in\;(2\:d.p.)[/tex]

Therefore, the dimensions of the box that would minimize cost are:

If the height of a right triangular prism is 1 5/6 inches and each side of the triangular base measures 10 inches. The surface area of the solid formed is: 598.33 in².

First step is to convert fraction into simple fraction

Right triangular prism=1 5/6 inches= 11/6 inches

Height of the base=8 2/3 inches = 26/3 inches

Second step is to find the surface area

Surface area = 5(10× 10) + (0.5÷ 10 × 26/3) + 3(10× 11/6)

Surface area = 5(100) +43.33+ 3(18.33)

Surface area = 500+43.33 + 55

Surface area = 598.33 in²

Therefore the surface area of the solid formed is about 598.33 in²

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When the pyramid is cut out of the prism, it has a volume of 36 cubic meters and the same base and height.

It is also known as the object's capacity. The fundamental equation for volume is length, width, and height, as opposed to the fundamental equation for the area of a rectangular shape, which is length, breadth, and height. The math is the same regardless of how you refer to the different dimensions. For example, you can substitute "depth" for "height." Volume is used to describe an object's capacity.  Volume can also be used to describe how much space a three-dimensional object occupies.

given

The ratio of their volumes is always one to three

Since the volume of a prism is 108 cubic meters,

a pyramid's volume is 1/3 * 108, or 36 cubic meters.

When the pyramid is cut out of the prism, it has a volume of 36 cubic meters and the same base and height.

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The constant of variation for the given graph is 1/4.

Slope of a line is the ratio of the change in y coordinates to the change in the x coordinates of two points given.

Slope can also be defined as the tangent of the angle that the line is making with the positive X axis.

Given that the equation of the line in the graph is a direct variation equation.

Direct variation equations are of the form,

y = k x, where k is a constant number called constant of variation.

We have to find the value of k.

y = k x is an equation in slope intercept form where k is the slope and the value of y intercept is 0, since the line passes through origin (0, 0).

It is enough to find the slope of the given line.

Take two points from the graph, let it be (4, 1) and (-4, -1).

Slope = (-1 - 1) / (-4 - 4) = -2 / -8 = 1/4

The value of k is 1/4.

Hence the value of constant of variation is 1/4.

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Answer:

[tex]x^2+7x-9+\dfrac{12}{x-7}[/tex]

Step-by-step explanation:

The detailed explanation is in the attached image. It is a real pain to type this in the brainly equation editor so I just used LateX and printed out the result in case you are interested.

I may have made typos in the detailed explanation, if so, let me know

Answer: To solve this equation, we need to get x on one side of the equation and all the other terms on the other side.

We can start by combining like terms:

three fourths times the quantity x plus 6 end quantity minus one fourth times x equals two fourths times the quantity x plus 9 end quantity

3/4 * x + 3/4 * 6 - 1/4 * x = 2/4 * x + 2/4 * 9

3/4 * x - 1/4 * x + 3/2 = 2/4 * x + 9/2

1/2 * x + 3/2 = x + 9/4

then we can move x to the left side of the equation:

x = x + 9/4 - 3/2

x = 9/4

So the value of x is 9/4.

The answer is A: x = 1.

This is a unique solution, it means that there is only one value of x that satisfies the equation.

Step-by-step explanation:

The surface area of the cylinder is 791.28 sqft.

A three-dimensional object's surface area is the sum of all of its faces. Real-world applications of the concept of surface areas include wrapping, painting, and eventually building things to achieve the best possible design.

The formula for the volume of a hollow cylinder is V = (R2 -r2)h cubic units if "R" denotes the outer radius, "r" denotes the inner radius, and "h" denotes the height.

Given,

radius=9ft

height=5ft

[tex]SA=2\pi rh+2\pi r^2\\SA=2*3.14*8*5+2*3.14*5^2\\SA=791.26 sqft[/tex]

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Answer:

1/4.

Step-by-step explanation:

1/4 is the answer because 25 x 4 = 100. Lets say you had 50/100. That would be equivalent to 2/4 , because 50 x 2 = 100.

25 is often reffered to as a quarter, but if someone says it is lets say a quarter past 5:00, they are trying to say is is 5:15, because 15 x 4 = 60, and 60 minutes is an hour.

Answer: 70

Step-by-step explanation: First, let's find how many are worth each in numbers. Use this formula:

Part Percent * Whole number / 100

Plug in the numbers:

Red = 12 * 250 / 100

3000/100

= 30 red marbles

Blue = 16* 250 / 100

4000 / 100

= 40 blue marbles

40 + 30 = 70

White = 250 - 70

= 180 white marbles

We can check our answer by adding 12 + 16. That's 28. So by multiplying 250 by 28, we get 7000. 7000 divided by 100 = 70

70 = 70

Therefore, both statements are true

I hope this helped!

Answer:

The sequence is geometric.

Step-by-step explanation:

Arithmetic Sequence:

A arithmetic sequence is a sequence of numbers in which the next term is calculated by adding some constant amount to the current term. It can be seen almost as a linear equation, but "x" is always a whole number, that starts at zero and increases by one.

We can check if a sequence is arithmetic or not, by subtracting any term from the next term. This amount should be constant for all terms if a sequence is arithmetic. This constant amount can be seen as the slope, how much it's changing by each term.

Geometric Sequence:

A geometric sequence is a sequence of numbers in which the next term is calculated by multiply by some constant amount and the current term. It can be seen almost as a exponential equation, but "x" is always a whole number, that starts at zero and increases by one.

Using the definition above, it can also be thought of as the current term being equal to the previous term multiplied by a constant amount. This means if we divide any term in the sequence by the previous term, this should be a constant amount, no matter which term we use.

Now that we know what the definitions of a geometric and arithmetic sequence as well as how to check if a sequence is either, we can now apply this knowledge to the problem. Let's start by checking if the sequence is arithmetic.

We can start by using the first term "45" and subtracting it from the second term "15", which gives us "-30". This means we had to "add" "-30" to get the second term. If this sequence is arithmetic, this means we could add this to the second term and get the third term. If we add "-30" to "15" we get "-15" which is not equal to the next term. So the amount that it's changing by is not constant, meaning this sequence is not arithmetic.

Now let's check if the sequence is geometric. Each term can be defined as the previous term multiplied by some constant amount. So if we divide any term by it's previous term we get this amount that it had to be multiplied by which should be constant. We cannot start with the first term, since there is no previous term before the first term. So let's start with the second term: "15", now let's divide it by the previous term, which is the first term: "45" [tex]\frac{15}{45} = \frac{1}{3}[/tex]. If this sequence is geometric, we can multiply the second term by this 1/3 and get the next term, which is the third term. If we multiply "15" (the second term) by "1/3" we get [tex]\frac{15}{3}[/tex] which is equal to "5", which is equal to the third term. So this sequence appears to be geometric.

The order pair solution of the given system of equations is (-5, -6).

Here we will use the substitution method to solve the given equation. In this obtain value of one variable from one equation and substitute the value in another equation to get the value of variables.  

Here we have

3x - 5y = 15 ---- (1)

2x - y = - 4 ---- (2)  

=> y = 2x + 4 --- (3)

From (1) and (3)

=> 3x - 5(2x + 4) = 15

=> 3x - 10x - 20 = 15

=> -7x = 35

=> x = -5

Substitute x = - 5 in 2x - y = - 4  

=> 2(-5) - y = -4

=> -10 - y = -4

=> y = - 6

Therefore,

The order pair solution of the given system of equations is (-5, -6).    

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The approximation of the area of f(x) = x² + 2x, from x = 2 to x = 10, is given as follows:

320 units squared.

The area under a curve is approximated using the definite integral of the function that defines the curve within it's bounds.

As it is specified that the area should be approximated using rectangles, a Left Riemann Sum will be used to approximate the area.

First we must obtain the Delta value, which is the difference between the bounds of the integral, divided by the number of intervals, thus, considering four intervals, we have that:

[tex]\Delta_x = \frac{10 - 2}{4} = 2[/tex]

Then the values of x at which the numeric values are calculated are obtained as follows:

[tex]x_i = a + \Delta_x(i - 1)[/tex]

Thus the values of x are of:

The numeric values are given as follows:

Then the definite integral is obtained as follows:

[tex]\sum_{i = 1}^n \Delta_x f(x_i)[/tex]

Meaning that the result of the integral is of:

2(8 + 24 + 48 + 80) = 320 units squared.

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To write [tex]\dfrac{11}{15}[/tex] as a decimal, you need to divide the numerator by the denominator.

The numerator of the fraction is 11.

The denominator of the fraction is 15.

[tex]11\div15[/tex]

[tex]=\fbox{0.73}[/tex]

Table 1 represents a positive correlation. As the temperature rises, the number of customers also increases.

What is a correlation?

A positive correlation is a relationship between two variables in which an increase in one variable is associated with an increase in the other variable.

Table 1 represents a positive correlation. As the temperature rises, the number of customers also increases.

To plot the set of data points from Table 1, we can use a scatter plot. Each data point in the table would be represented by an (x,y) coordinate, where x is the increase in temperature and y is the number of customers. So, in this case, we would have 8 points in the scatter plot, one for each pair of data in table 1.

The plot would look like a group of points in the coordinate plane with a general upward trend, as the x-values (increase in temperature) increase, the y-values (number of customers) also increase. It's possible to observe a pattern in the points, this pattern confirms that there is a positive correlation between temperature and the number of customers.

Hence, Table 1 represents a positive correlation. As the temperature rises, the number of customers also increases.

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The surface area of the triangular prism is 608 cm².

Given,

base of triangle = 12 cm

height of triangle = 8 cm

length of triangle = 10 + 10 + 12 = 32 cm

Width of the triangle = 16 cm

Area of the rectangle = l x b = 32 x 16 = 512 cm²

Area of triangle = 1/2 x 12 x 8 = 48 cm²

There are 2 triangle, so 48 + 48 = 96 cm²

Add all these values, 512 + 96 = 608 cm²

∴The surface area of triangular prism = 608 cm²

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The sum of the surfaces of the five faces of a triangular prism is its surface area.

Having two triangular bases and three rectangular sides, a triangular prism is a type of polyhedron. It is a three-dimensional object with two base faces and three side faces that are joined by edges.

There are 6 vertices, 5 edges, and 5 faces on it. The other 3 faces have a rectangle-like shape, while the 2 bases are structured like a triangle. Camping tents, chocolate candies, roofs, etc. are a few instances of triangular prisms in the real world.

A solid object called a prism has plane faces enclosing each of its four sides. Faces in a prism come in two different varieties. Bases are used to describe the top and bottom faces, which are identical.

12 cm is the triangle's base.

triangle's height is 8 cm.

Triangle's length is 32 cm (10 + 10 + 10 cm).

Triangle width is 16 cm.

The rectangle's area is 512 cm2 (l x b = 32 x 16).

Triangle area equals 1/2 x 12 x 8 = 48 cm2.

Two triangles are present, therefore 48 + 48 = 96 cm2.

512 + 96 = 608 cm2 after adding all these values.

The triangular prism's surface area is 608 cm2.

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-2.25,-2.5,2.75 I think

Our velocity is said to be 10% of the speed of light. Therefore, we need to multiply by 10% the speed of light. As a result, our speed is three times 10 to 7 m/s. Our diameter is 500 metres.

Thus, our radius is 250 metres. It goes through a volume of space equal to pi r squared times V times T in one second. We then multiply 5.8, 8 times 10 to the 12 m3 by pi times 2, 50 squared times three times 10, and seven times one second. Our density is 10 to 6 atoms per cubic metre, or one atom per cubic centimetre.

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The graph of the new trapezoid is attached

Transformation is the movement of a point in the coordinate plane. Types of transformation are rotation, dilation, reflection and translation.

Dilation is the increase or decrease in size of a figure by a scale factor. The rule for dilation is:

(x, y) ⇒ (kx, ky)

The vertices of the trapezoid is P(1, -2), Q(2, -2), R(2, 1) and S(-2, 1)

If the trapezoid is dilated by a scale factor of 4 about the origin, the new points are:

P'(4, -8), Q'(8, -8), R'(8, 4) and S'(-8, 4)

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B1 should be smaller in order to spread out the explanatory components in order to test a B1 hypothesis or generate confidence intervals around a B1 hypothesis.

Though an explanatory variable is a hypothesis that is predicted to be able to explain the research outcomes, a response variable demonstrates the impact that is anticipated to come from the explanatory variable. The regression line has a slope that is B1.

In other circumstances, the alternative hypothesis is compared against the null hypothesis to determine whether the assertion that the population slope is not equal to Zero is valid. The value of b1 is to be construed as the average result altering when the explanatory variable is increased by one unit.

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There is a 46.7% increase in the number of bracelets made.

A percentage is a number or ratio that can be expressed as a fraction of 100 (100 as denominator). To calculate percent of a number, divide the number by the whole and multiply by 100.

Hence, it means, a part per hundred. The word per cent means per 100. It is represented by the symbol “%”.

Maysol made;

This month = 75 bracelets

Last month = 40 bracelets

Change = 75 - 40

Change = 35

This means there was an increase

% increase = (35/75) x 100

% increase = 46.7%

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The likelihood that the first question will be true is Equally likely and unlikely. That is option C

Probability is defined as the possibility of an outcome of an event which may likely or unlikely occur.

The true and false quiz given by the teacher;

P(true) = 0.5 for each question.

Therefore, False = 0.5 for each question making the probability of the first question being true to be equally likely and unlikely.

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It is TRUE that the given image represents the function.

A function in mathematics from a set X to a set Y allocates exactly one element of Y to each element of X.

The sets X and Y are collectively referred to as the function's domain and codomain, respectively.

Initially, functions represented the idealized relationship between two changing quantities.

A function, according to a technical definition, is a relationship between a set of inputs and a set of potential outputs, where each input is connected to precisely one output.

So, the given image represents a function and it is a kind of many-to-one function.

Therefore, it is TRUE that the given image represents the function.

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In order to estimate the hourly rate of any given babysitter, Ms. Wong can make use of the mean of the sample, because it most closely describes the values of the data in the dataset

The sample mean is the average of the values in the data sample.

Possible part of the question, obtained from a similar question online includes the following table of values;

Babysitter  [tex]{}[/tex]   Rate in dollars per hour

1                         [tex]{}[/tex] 9.6

2                        [tex]{}[/tex] 9.80

3                        [tex]{}[/tex] 15.50

4                        [tex]{}[/tex] 10.6

5                       [tex]{}[/tex]  9.7

6                        [tex]{}[/tex] 10.20

7                        [tex]{}[/tex]  8.80

8                        [tex]{}[/tex]  11.20

9                        [tex]{}[/tex]  8.80

10                        [tex]{}[/tex] 10.10

The estimate of the hourly rate of any given babysitter can be found using the descriptive statistic, such as the mean, median, or mode, which are measures that indicate the possible mid value of a data set.

The mean is the average value of the dataset, which is obtained by adding the individual data points together and dividing the sum by the number of data points as follows;

9.6 + 9.8 + 15.5 + 10.6 + 9.7 + 10.2 + 8.8 + 11.2 + 8.8 + 10.1 = 104.3

The number of datapoints in the data = 10

The mean = 104.3/10 = 10.43

The median is the middle value which is found by arranging the values  in increasing order, then selecting the middle value as follows;

The median values in the dataset that has 10 data points is the average of the 5th and 6th values, which are; 9.8 and 10.1

The median of the dataset is therefore; (9.8 + 10.1)/2 = 9.95

The mode of the dataset is the most common value within the dataset. The mode is therefore 8.8

Based on the above three values, the best estimate that most closely describe the data in the dataset is the mean

Therefore, to estimate the hourly rate of any babysitter, Ms. Wong can make use of the mean, because it most closely describes the values in the dataset.

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The distance between point G and H is 4√2 units51.96 unit²

An equation is a expression that shows the relationship between numbers and variables.

The distance between two points A(x₁, y₁) and B(x₂, y₂) on the origin is:

[tex]AB=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}[/tex]

Let us take point G as out reference point (origin)

Hence:

G = (0, 0) and H = (4, -4)

[tex]GH=\sqrt{(-4-0)^2+(4-0)^2}=4\sqrt{2}\ units[/tex]

The distance is 4√2 units

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Answer: We want to know how many ways we can pay $1.10 with nickels and dimes. Since nickels are worth $0.05 and dimes are worth $0.10, we can use different combinations of these coins to make $1.10.

One way to approach this problem is to use the method of generating functions.

Let x be the number of nickels and y be the number of dimes. Then the generating function for this problem is (x+y) = (x+y)(x^0y^0 + x^1y^0 + x^0y^1 + x^2y^0 + x^1y^1 + x^0y^2...)

We know that the sum of x and y should be 11 because 0.05x + 0.1y = 1.10

The coefficient of x^5y^6 in the expansion of (x+y)^11 gives the number of ways to pay $1.10 with 5 nickels and 6 dimes.

The coefficient of x^5y^6 = 252

So there are 252 ways to pay $1.10 with nickel and dime.

Step-by-step explanation:

The required number line attached below shows the absolute value of the fraction 5/13 is from 0.

Absolute value is defined as the magnitude or size of a number, no negatives are allowed. The distance a number is from 0 on the number line is known as the absolute value of the number.  

The fraction is given in the question, as follows:

-5/13

We have to determine the absolute value of the fraction.

As per the question, we can write the absolute value of the fraction as:

⇒ 5/13

Thus, the number line attached below displays the distance of the fraction from 0.

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Answer:

employee worked 8 hours over 35 hours, for a total of 35 + 8 = 43 hours.

Step-by-step explanation:

(35 hours x $15/hour) + (x hours x $22.50/hour) = $705 (where x is the number of hours worked over 35 hours)

You can then solve for x:

35x15 + x*22.50 = 705

525 + 22.5x = 705

22.5x = 180

x = 8

So the employee worked 8 hours over 35 hours, for a total of 35 + 8 = 43 hours.

Answer:

8) AD = 23

9) m<B = 63

10) G(1,-5)

-------------------------------

8)

AD = BC

Thus, x + 21 = 12x - 1

11x = 22

x = 2

So, AD = 23

9)

consecutive angles are supplementary, sum to 180

y/2 + y - 9 = 180

3/2 y  = 189

y = 126

so, m<B = 63

10)

Diagonals bisect each other. The midpoint of DF = (-1.5, 0.5)

-1.5 = 2+x/2

x = 1

0.5 = 6+y/2
y = -5

G(1,-5)

The optimal production of wind and solar that will maximize overall power output is 3 units/day of solar power and 2 units/day of wind power.

To solve this question, we use the following representations

So, the objective function is to maximize z = x + y, which represents the overall power output.

From the question, we have the following constraints:

Using the Lagrangian function, we have:

L(x,y,λ1,λ2,λ3) = x + y + λ1(6-x) + λ2(y-1) + λ3(2x-y-1)

To find the optimal solution, we need to find the values of x, y, λ1, λ2, and λ3 that will make the partial derivatives of L equal to zero:

∂L/∂x = 1 + λ3 = 0

∂L/∂y = 1 - λ1 + λ2 + λ3 = 0

Using the variables x and y, we have

∂L/∂λ1 = -x + 6 = 0

∂L/∂λ2 = y - 1 = 0

∂L/∂λ3 = 2x - y - 1 = 0

Using a graphing calculator, the points that maximize the objective function are:

x = 3 and y = 2

i.e.

Power output is 3 units/day of solar power and 2 units/day of wind power.

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