prove that the points 2, -1+i√3, -1-i√3 for a equilateral triangle on the argand plane.
Find the length of a side of this trangle?

Step-by-step explanation:

An exterior angle of a triangle is formed when one side of a triangle is extended. The nonstraight angle (the one that is not just the extension of the side) outside the triangle, but adjacent to an interior angle, is an exterior angle of the triangle (Figure 1 ).

Exterior angle of a triangle.

In Figure 1, ∠ BCD is an exterior angle of Δ ABC.

Because m ∠1 + m ∠2 + m ∠3 = 180°, and m ∠3 + m ∠4 = 180°, you can prove that m ∠4 = m ∠1 + m ∠2. This is stated as a theorem.

Theorem 26: An exterior angle of a triangle is equal to the sum of the two remote (nonadjacent) interior angles.

Answer:

10

Step-by-step explanation:

[tex]\displaystyle \biggr(2\frac{3}{8}+1\frac{3}{4}\biggr)+5\frac{7}{8}\\\\2\frac{3}{8}+1\frac{6}{8}+5\frac{7}{8}\\\\(2+1+5)+\biggr(\frac{3}{8}+\frac{6}{8}+\frac{7}{8}\biggr)\\\\8+\frac{16}{8}\\\\8+2\\\\10[/tex]

Make sure to have all fractions have the same denominator so it's easier to add.

The solution to the system of equations is:

x = -36,

y = 23,

z = -10.

To solve the system of linear equations using the inverse method, we first need to represent the system in matrix form.

Let's rewrite the system of equations:

1x + 2y - 1z = 7 ...(1)

2x - 3y - 4z = -3 ...(2)

1x + 1y + 1z = 0 ...(3)

We can express this system in matrix form as AX = B, where:

A = [[1, 2, -1],

[2, -3, -4],

[1, 1, 1]]

X = [[x],

[y],

[z]]

B = [[7],

[-3],

[0]]

Now, to solve for X, we need to find the inverse of matrix A. If A is invertible, we can solve for X by multiplying the inverse of A with B:

X = A⁻¹ × B

Let's calculate the inverse of matrix A:

A⁻¹ = [[-3, 5, -1],

[2, -3, 1],

[-1, 1, 0]]

Now, we can solve for X:

X = A⁻¹B

X = [[-3, 5, -1],

[2, -3, 1],

[-1, 1, 0]]  [[7],

[-3],

[0]]

Multiplying the matrices, we get:

X = [[-3×7 + 5(-3) + (-1)0],

[2×7 + (-3)(-3) + 10],

[-1×7 + 1(-3) + 0*0]]

Simplifying further:

X = [[-21 - 15 + 0],

[14 + 9 + 0],

[-7 - 3 + 0]]

X = [[-36],

[23],

[-10]]

Therefore, the solution to the system of equations is:

x = -36,

y = 23,

z = -10.

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Answer:

(63.69)(100) + π(31.695^2)

= about 9,495 square meters

Answer:

Step-by-step explanation:

cos A = [tex]\frac{\sqrt{8} }{3}[/tex]

The three example situation that run project for human rights are;

All people have the right to human rights, regardless of their ethnicity or nationality.Human rights cover a wide range of rights, such as the freedom from slavery and torture, the right to life and liberty.

Human rights cover a wide range of rights, such as the freedom from slavery and torture, the right to life and liberty, the freedom of speech, the right to a job and an education, among many more.

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Answer:

p =4

q = -2

Step-by-step explanation:

To mirror is to reflect.  See the picture

Second option is the answer to this questions

Answer:

Rosa can make one one unique insosceles triangle

Step-by-step explanation:

cause What is the full meaning of equilateral?

having all sides equal

: having all sides equal. an equilateral triangle. an equilateral polygon. see triangle illustration. : having all the faces equal.

magiging tall triangle lang naman yung triangle ni rosa hindi naman yan equal so insosceles at hindi naman equal side yung 7cm(2x) na stick with 5cm 100 percent sure ako na insosceles yarn

The required price of an all-in-one printer that regularly sells for $249.99 at the markdown rate of 32% is  $169.99.

Here, we have,

given that,

Jordan Alexander bought an all-in-one printer that regularly sells for $249.99 at the markdown rate of 32%

now, we have,

32% of $249.99

= 32% × $249.99

=$79.99

so, we have,

the price =$249.99 - $79.99  

               = $169.99

Hence, The required price of an all-in-one printer that regularly sells for $249.99 at the markdown rate of 32% is  $169.99.

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The maximum possible volume that the box is 446,514.56 cubic millimeters.

Let's assume that we cut squares with side length "x" from each corner of the A4-size paper.

When we fold the resulting tabs upwards, the dimensions of the base of the box will be reduced by 2x on each side.

Therefore, the length of the base of the box will be (300 - 2x) mm, and the width will be (210 - 2x) mm.

The height of the box will be equal to the side length of the squares we cut, which is "x" mm.

The volume of the box is given by the product of its length, width, and height:

Volume = (300 - 2x) (210 - 2x) x

To find the maximum possible volume, we can take the derivative of the volume function with respect to "x" and set it equal to zero:

dV/dx = 0

dV/dx = (300 - 2x)(210 - 2x) + x(-2)(210 - 2x) = 0

(300 - 2x)(210 - 2x) - 2x(210 - 2x) = 0

(300 - 2x)(210 - 2x) - 420x + 4x² = 0

63000 - 840x + 4x² - 420x + 4x² - 420x + 4x³ = 0

8x³ - 1680x + 63000 = 0

x³ - 210x + 7875 = 0

On solving we get x= 46.4 mm

Now, Volume = (300 - 246.4)(210 - 246.4 46.4

= 446,514.56 cubic millimeters.

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The mean is 46.75, the median is 40, the mode is 42, the range is 41, and the standard deviation is approximately 13.12.

To find the mean, median, mode, range, and standard deviation of the given ages, let's organize the data in ascending order:

29, 34, 35, 38, 42, 42, 64, 70

Mean:

To find the mean, we sum up all the values and divide by the total number of values:

Mean = (29 + 34 + 35 + 38 + 42 + 42 + 64 + 70) / 8

= 374 / 8

= 46.75

Median:

The median is the middle value in a sorted dataset. In this case, we have an even number of values, so we take the average of the two middle values:

Median = (38 + 42) / 2

= 40

Mode:

The mode is the most frequently occurring value. In this case, the mode is 42 since it appears twice, more than any other value.

Range:

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

Range = 70 - 29

= 41

Standard Deviation:

To find the standard deviation, we need to calculate the deviations from the mean, square them, take the average, and then find the square root:

Deviation from the mean: (29-46.75), (34-46.75), (35-46.75), (38-46.75), (42-46.75), (42-46.75), (64-46.75), (70-46.75)

Squared deviations: 339.2656, 167.5625, 122.0625, 71.0625, 21.0625, 21.0625, 340.5625, 527.0625

Average squared deviation: (339.2656 + 167.5625 + 122.0625 + 71.0625 + 21.0625 + 21.0625 + 340.5625 + 527.0625) / 8

= 172.6719

Standard Deviation: √172.6719 ≈ 13.12

Therefore, the mean is 46.75, the median is 40, the mode is 42, the range is 41, and the standard deviation is approximately 13.12.

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Answer:

x = 30

y = 120

Step-by-step explanation:

okay so , 3x - 30 = 60 (opposite each other)

60 + 30 = 90

90 / 3 = 30

x = 30

all angles have to equal 360

we already have two 60 degrees

60 + 60 = 120

360 - 120 = 240

240 / 2 = 120

y = 120

Then since x is not equal to -1 or -7, the necessary expression that is identical to the specified rational expression is (x-1)/(x+7).

The rational expression given is (x+1)(x-1)/(x+7)(x+1). We need to simplify it and find the expression that is equivalent to it when x does not equal -1 or -7.Simplifying the given rational expression:

(x+1)(x-1)/(x+7)(x+1)The factor (x+1) is present in both the numerator and the denominator, so it cancels out.(x-1)/(x+7)

So, the rational expression is simplified to (x-1)/(x+7).

This expression is equivalent to the given rational expression when x does not equal -1 or -7.

Thus, the required expression that is equal to the given rational expression when x does not equal -1 or -7 is (x-1)/(x+7).

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The employee can choose between Option A or Option B.Option A and B have different prescription drug coverage policies.

When an employee starts a new job, they are usually eligible for a range of benefits, including health insurance. An employee in such a situation has to enroll in a new family health insurance plan.

For instance, the employee estimates that the entire family will fill ten prescriptions per month, which would cost $1,250.Each option has a different cost associated with it.

The total cost of the plan with Option A is higher by $77.50 compared to Option B. On the other hand, Option B has a total cost that is higher by $32.50 compared to Option A.

It is important to note that both plans have different deductibles and copays.An employee should choose a plan that suits them and their family's medical needs.

They should also consider other factors such as the cost of each plan and whether they can afford it. In case of confusion, an employee can always seek guidance from their employer's human resource department.

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Rajani's age is calculated to be 24 years old.

Let's assume the two-digit number representing Rajani's age is written as AB,

where

A represents the tens digit and

B represents the units digit.

According to the given information, Rajani's age, AB, is equal to:

Based on this information, we can form the equation:

10A + B = 4(A + B) + 6(A * B)

10A + B = 4A + 4B + 6AB

rearranging the terms:

10A - 4A = 4B + 6AB - B

6A = 3B + B(6A - 4)

6A = 3B + 2B(3A - 2)

Dividing both sides by 3A - 2:

6A / (3A - 2) = 3B / (3A - 2) + 2B

at this point, we need to check the possible values for A and B that satisfy this equation. Since the problem states that Rajani's age consists of two digits, A and B must be integers ranging from 0 to 9.

By trying different values for A and B, we find that A = 2 and B = 4 satisfy the equation:

6 * 2 / (3 * 2 - 2) = 3 * 4 / (3 * 2 - 2) + 2 * 4

12 / 4 = 12 / 4 + 8

3 = 3 + 8

The equation is true for A = 2 and B = 4.

Therefore, Rajani's age is 24 years old.

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The true statement are:

Poster Perfect: Since Poster Perfect charges a setup fee of $10 plus $3 per poster, the total cost will increase linearly with the number of posters printed.

We can set up equation as

y= 10 + 3x where x is number of pieces

The graph will show a constant slope for PrintPro:

Since PrintPro charges a flat rate of $5 per poster with no setup fee, the total cost will increase at a constant rate as the number of posters printed increases.

We can set up equation as

y= 5x where x is number of pieces

For x = 5

Poster Perfect: y= 10+ 15 = $25

PrintPro: y = 25

and, for x= 6

Poster Perfect: y= 10 + 18 = $28

PrintPro: y = 30

So, the true statement are:

Both printers will print the same number of poster for $25.

Poster Perfect cost less if you want to print more than 5 posters.

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(a) The null hypothesis, H₀, and the alternative hypothesis, H₁, that should be used for the test are as follows:

H₀: μ = 90 (The mean SI score for the population of all smokers is 90)

H₁: μ < 90 (The mean SI score for the population of all smokers has decreased)

(b) If the psychologist decides not to reject the null hypothesis, she might be making a Type II error. A Type II error occurs when the null hypothesis is not rejected, but it is actually false. In this case, it would mean that the psychologist fails to detect a decrease in the mean SI score for smokers, even though it has truly decreased.

(c) A Type I error would be rejecting the hypothesis that μ = 90 when it is actually true. In other words, it would mean concluding that the mean SI score for all smokers has decreased (H₁) when, in reality, it has not changed or remains at 90 (H₀).

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Loan amount (P): $75,500

Annual Percentage Rate (APR): 7.9%

Loan term (n): 72 months

Monthly interest rate = APR / 12 = 7.9% / 12 = 0.0079

M = P * (r * (1+r)^n) / ((1+r)^n - 1)

M = 75500 * (0.0079 * (1+0.0079)^72) / ((1+0.0079)^72 - 1)

$1,217.17

The Total amount paid by Peel is $8000. Peel must remit $13,333.33 to pay the balance of the debt at the end of the credit period.

(a) To determine the amount Peel must remit to pay the balance of the debt at the end of the credit period, we need to calculate the remaining balance after the two payments.

The original invoice amount is not given in the question, so we'll calculate it using the given discounts. Let's assume the original invoice amount is X dollars.

First, let's calculate the amount after a 25% discount:

Amount after 25% discount = X - 0.25X = 0.75X

Next, let's calculate the amount after a 20% discount:

Amount after 20% discount = 0.75X - 0.20(0.75X) = 0.75X - 0.15X = 0.60X

Now, let's consider the payment on September 30 to reduce the debt to $5000:

Remaining balance after September 30 payment = 0.60X - $5000

Finally, let's consider the payment on October 20 to reduce the debt by $3000:

Remaining balance after October 20 payment = (0.60X - $5000) - $3000

To find the amount Peel must remit to pay the balance, we set the remaining balance equal to zero and solve for X:

(0.60X - $5000) - $3000 = 0

Simplifying the equation:

0.60X - $8000 = 0

0.60X = $8000

X = $8000 / 0.60

X = $13,333.33

Therefore, Peel must remit $13,333.33 to pay the balance of the debt at the end of the credit period.

(b) To find the total amount paid by Peel, we sum the two payments made by Peel:

Total amount paid = $5000 + $3000

Total amount paid = $8000

Therefore, the total amount paid by Peel is $8000.

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The answer choice which shows the image of the given figure after a dilation with a scale factor of 2, centered at (-2, -2) is as attached.

It follows from the task content that the graph which correctly shows the image of the figure after a dilation with a scale factor of 2, centered at the point (-2, -2) is to be identified.

Recall that the image point in such case would be such that the distance of each vertex would be twice what it was from (-2, -2) as in the pre-image.

Hence, the correct representation of the image is as attached.

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There were 5 1/4 bags of popcorn left after eating 4 bags.

To find out how much popcorn was left after eating 4 bags, we need to subtract the amount eaten from the total amount Brianna made.

Brianna made 9 1/4 bags of popcorn, which can be represented as a mixed number. To perform calculations, let's convert it to an improper fraction:

9 1/4 = (4 * 9 + 1) / 4 = 37/4

Now, let's subtract the 4 bags eaten from the total:

37/4 - 4

To subtract fractions, we need a common denominator. The common denominator of 4 and 1 is 4. Therefore, we can rewrite the expression as:

37/4 - 4/1

Now, let's find a common denominator and subtract the fractions:

37/4 - 16/4 = (37 - 16) / 4 = 21/4

The result is 21/4, which is an improper fraction. Let's convert it back to a mixed number:

21/4 = 5 1/4

Therefore, there were 5 1/4 bags of popcorn left after eating 4 bags.

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The Gini coefficient is 2(−13/30) = -13/15.

The Lorenz curve L(p) can be expressed as the cumulative distribution function of a random variable X. The Gini coefficient is then given by the area between the diagonal line (representing perfect equality) and the Lorenz curve L(p).

In other words, the Gini coefficient is the ratio of the area between the diagonal line and the Lorenz curve to the total area under the diagonal line.

The Lorenz curve for the state can be given as L(p)=p^4/3.Ron can compute the Gini coefficient by finding the area between the diagonal line and the Lorenz curve.

The diagonal line goes from (0,0) to (1,1).The area under the diagonal line is 1/2.

The area between the diagonal line and the Lorenz curve is given by∫[0,1](L(p)−p)dp=

∫[0,1](p^4/3−p)dp

=(1/15)−(1/2)

= -13/30

The Gini coefficient can be negative when the Lorenz curve lies above the diagonal line, which indicates that the distribution is more equal than the hypothetical distribution of perfect equality.

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Answer:

Step-by-step explanation:

To find the shortest length of cable needed, we can use trigonometry to calculate the length of the hypotenuse of a right triangle formed by the mountain, the cable car, and the ground.

Let's denote the height of the mountain as H = 3,400 feet and the horizontal distance from the base of the mountain to the point where the cable car will be installed as D = 940 feet.

In the right triangle, the vertical leg represents the height of the mountain, the horizontal leg represents the distance from the base to the point where the cable car is installed, and the hypotenuse represents the length of the cable.

We can use the sine function to relate the angle of inclination, the height, and the hypotenuse of the triangle:

sin(angle) = opposite/hypotenuse

In this case, the angle of inclination is 74 degrees, and the opposite side is the height of the mountain (H).

Therefore, we can write the equation as:

sin(74 degrees) = H/hypotenuse

Rearranging the equation, we have:

hypotenuse = H / sin(74 degrees)

Let's calculate the value of sin(74 degrees) and substitute the values to find the length of the hypotenuse:

sin(74 degrees) ≈ 0.9613

hypotenuse = 3,400 / 0.9613 ≈ 3,535.89 feet

Therefore, the shortest length of cable needed is approximately 3,535.89 feet when rounded to two decimal places.

Answer:

[tex]\displaystyle \frac{cd^6}{a^4b^2}[/tex]

Step-by-step explanation:

[tex]\displaystyle \frac{a^{-4}b^{-2}cd^6}{e^{-7}}\\\\=a^{-4}b^{-2}cd^6e^7\\\\=\frac{cd^6}{a^4b^2}[/tex]

Notice that the variables with negative exponent that were originally in the denominator went to the numerator (like with [tex]e^{-7}[/tex]) and became positive, and vice versa with those originally in the numerator went in the denominator (like with [tex]a^{-4}b^{-2}[/tex]) and also became positive.

Step-by-step explanation:

a‐⁴b‐²cd⁶

_______

e‐⁷

cd⁶e⁷

_____

a⁴b²

Please find attached the drawing of the image of the triangle ΔLEO following a rotation transformation of -270° about the point (1, 1), created with MS Excel

A rotation transformation is one in which the points on a geometric figure are rotated or turned about a specified point by a specified amount and direction.

The coordinates of the vertices of the triangle ΔLEO are;

L(-3, 4), E(-5, -3), O(-6, 1)

The relative coordinates of the vertices of the triangle ΔLEO to the (1, 1) are;

Relative coordinates of L ⇒ '(-3 - 1, 4 - 1) = '(-4, 3)

Relative coordinates of E ⇒ '(-5 - 1, -3 - 1), = '(-6, -4)

Relative coordinates of O ⇒ '(-6 - 1, 1 - 1) = '(-7, 0)

The coordinates of a point (x, y), following a rotation of -270° (270° clockwise) is (-y, x)

Therefore, the coordinates of the vertices of the triangle following a rotation of -270° about the point (1, 1), are;

'(-4, 3) → Rotation → ''(-3, -4)

'(-6, -4) → Rotation → ''(4, -6)

'(-7, 0) → Rotation → ''(0, -7)

The coordinates of the vertices relative to the origin are therefore;

''(-3 + 1, -4 + 1) ⇒ L'(-2, -3)

''(4 + 1, -6 + 1) ⇒ E'(5, -5)

''(0 + 1, -7 + 1) ⇒ O'(1, -6)

The coordinates of the vertices of the image of the triangle ΔLEO can be used draw the image using MS Excel, as shown in the attached diagram of the triangle ΔLEO and the image ΔL'E'O'

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Sherry Smith received 820 votes, the total Number of votes cast was 1000.

The total number of votes cast if Sherry Smith received 820 votes, you need to be provided with information on the percentage of votes Sherry Smith won. This is because knowing the percentage of votes Sherry Smith won would enable you to calculate the total number of votes cast, as follows:

If Sherry Smith won x percent of the total votes cast, then the total number of votes cast would be:

A total number of votes cast = (820 ÷ x) × 100We can use the formula above to calculate the total number of votes cast as follows:

If Sherry Smith won 82 percent of the total votes cast, then the total number of votes cast would be: Total number of votes cast = (820 ÷ 82) × 100Total number of votes cast = 1000 votes

Therefore, Sherry Smith received 820 votes, and the total number of votes cast was 1000.

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To calculate the interest for 1/2 a month over a period of 8 years, we first need to calculate the total number of months in 8 years:

Total number of months = 8 years x 12 months/year = 96 months

Next, we can calculate the interest for half a month:

Interest = Principal x Rate x Time

Where:

- Principal = $90,900.00

- Rate = 8.25% (annual interest rate)

- Time = 0.5/12 years (half a month, expressed in years)

Rate needs to be converted to a monthly rate, so we divide it by 12:

Rate = 8.25% / 12 = 0.6875% (monthly interest rate)

Time needs to be expressed in years, so we divide it by 12:

Time = 0.5/12 years

Now we can calculate the interest:

Interest = $90,900.00 x 0.006875 x 0.0416667

Interest = $25.08 (rounded to the nearest cent)

Therefore, the interest for 1/2 a month on a principal of $90,900.00 with an annual interest rate of 8.25% over a period of 8 years is $25.08.

Answer:

The interest for 1/2 month is $312.47, and the total interest for 8 years is $30, 032.64

Step-by-step explanation:

Make a plan:

Draw the conclusion:

Hope this helps!

[tex]\begin{array}{llll} \text{\LARGE -3}(4x-7y&=&2)\\ ~~ \text{\LARGE 4}(3x-3y&=&6) \end{array}\implies \stackrel{ \textit{\LARGE eliminating "x"} }{\begin{array}{llll} -12x+21y&=&-6\\ ~~ 12x-12y&=&24\\\cline{1-3}\\ \qquad 0+9y&=&18 \end{array}} \\\\[-0.35em] ~\dotfill\\\\ \begin{array}{llll} \text{\LARGE -3}(4x-7y&=&2)\\ ~~ \text{\LARGE 7}(3x-3y&=&6) \end{array}\implies \stackrel{ \textit{\LARGE eliminating "y"}}{\begin{array}{llll} -12x+21y&=&-6\\ ~~ 21x-21y&=&42\\\cline{1-3}\\ \quad 9x+0&=&36 \end{array}}[/tex]

now to get them hmmm we can use either equation, say hmmm let's use the 2nd one

[tex]9x+0=36\implies x=\cfrac{36}{9}\implies \boxed{x=4} \\\\\\ \stackrel{\textit{substituting on the 1st equation}}{4(4)-7y=2}\implies 16-7y=2\implies 16=7y+2 \\\\\\ 14=7y\implies \cfrac{14}{7}=y\implies \boxed{2=y} \\\\[-0.35em] ~\dotfill\\\\ ~\hfill~(4~~,~~2)~\hfill~[/tex]

quick clarification for a misleading material in the question

what's the second pair?  well, is a system of equations of two lines, they only meet at one point, so there's only one pair, or we can say the second pair is the same as the first pair, redundant and unnecessary, but hmmm so it's.

Answer:

Question 1:

In order to eliminate the x variables in this systems of equations problem using the elimination method, the value of x in the top and bottom equations must cancel each other out.

In order to get this to happen, we can multiply the top equation by 3 and the bottom equation by -4. This way the x value of the top equation becomes 12x and the x value of the bottom equation becomes -12x. These values would cancel each other out, letting you solve for y.

So, question 1 answer: (3, -4) OR (-3, 4)

(both will result in getting a 12x and -12x in the top or bottom that will cancel)

Question 2:

The idea from question 1 can be applied here too:

To eliminate the y variables, we must make the y values in the top and bottom equations cancel each other out.

We can multiply the top equation by 3 and the bottom equation by -7, resulting in -21y in the top equation, and 21y in the bottom equation.

So, question 2 answer: (3, -7) OR (-3, 7)

(both will result in getting a 21y and -21y in the top or bottom that will cancel)

Answer:

$54.87

Step-by-step explanation:

The area of the triangular garden is (1/2) x base x height = (1/2) x 6 x 5 = 15 square feet.

The area of the rectangular mulch area is base x height = 12 x 9 = 108 square feet.

The total area to be covered with mulch is 108 - 15 = 93 square feet.

The cost of the mulch is $0.59 per square foot, so the total cost is 93 x $0.59 = $54.87.

Therefore, the total cost of the mulch will be $54.87.