1 Express 12 + 5i in polar form (i.e in form of \[z=r\cos\theta + i\sin\theta\]
A. [13(\cos 22.6 - i\sin 22.6)\]
B [13(\cos 22.6+i\sin 22.6)\]
C. [13(\cos 23.5 - i\sin 23.5)\]
D. [13(\cos 23.6 - i\sin 23.6​

a. The distribution of X is a normal distribution.

b. The median giraffe height is also 19 feet.

c. The Z-score for a giraffe that is 22 foot tall is 3.

d. The probability that a randomly selected giraffe will be shorter than 18.9 feet tall is less than -0.1.

f. The 80th percentile represents the value below which 80% of the data falls.

a. The distribution of X is a normal distribution (or Gaussian distribution) with a mean of 19 feet and a standard deviation of 1 foot. This can be denoted as X ~ N(19, 1).

b. The median of a normal distribution is equal to its mean.

c. To find the Z-score for a giraffe that is 22 feet tall, we can use the formula: Z = (X - μ) / σ, where X is the observed value, μ is the mean, and σ is the standard deviation. Plugging in the values, we get Z = (22 - 19) / 1 = 3.

d. To find the probability that a randomly selected giraffe will be shorter than 18.9 feet tall, we need to calculate the area under the normal curve to the left of 18.9. This can be done using the Z-score and a standard normal distribution table or a calculator.

Alternatively, we can use the Z-score formula from the previous question. The Z-score for 18.9 feet can be calculated as Z = (18.9 - 19) / 1 = -0.1. We can then look up the corresponding probability in the standard normal distribution table or use a calculator to find the probability that Z is less than -0.1.

e. To find the probability that a randomly selected giraffe will be between 18.6 and 19.5 feet tall, we need to calculate the area under the normal curve between these two values.

Again, we can use the Z-score formula to standardize the values and then find the corresponding probabilities using a standard normal distribution table or a calculator.

f. To find the height at the 80th percentile, we can use the standard normal distribution table or a calculator to find the Z-score that corresponds to the 80th percentile.

Once we have the Z-score, we can use the formula Z = (X - μ) / σ to solve for X. Rearranging the formula, we have X = Z * σ + μ. Plugging in the values for Z (obtained from the percentile) and μ (mean) and σ (standard deviation), we can calculate the height at the 80th percentile.

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The approximate solution to the equation 1/x-1 = |x-2| after three iterations of successive approximation is x ≈ 5/2 or x ≈ 2.5.

To solve the equation 1/x-1 = |x-2| using three iterations of successive approximation, we will start with an initial guess and refine it using an iterative process.

Given that the equation involves absolute value, we will consider two cases:

Case 1: x - 2 ≥ 0

In this case, |x-2| simplifies to x-2, and the equation becomes 1/(x-1) = x-2.

Case 2: x - 2 < 0

In this case, |x-2| simplifies to -(x-2), and the equation becomes 1/(x-1) = -(x-2).

Now, let's perform the successive approximation:

Iteration 1:

Let's start with an initial guess, x = 2.

Case 1: When x - 2 ≥ 0,

1/(2-1) = 2-2,

1/1 = 0,

which is not true.

Case 2: When x - 2 < 0,

1/(2-1) = -(2-2),

1/1 = 0,

which is not true.

Since our initial guess did not satisfy the equation in either case, we need to choose a different initial guess.

Iteration 2:

Let's try x = 3.

Case 1: When x - 2 ≥ 0,

1/(3-1) = 3-2,

1/2 = 1,

which is not true.

Case 2: When x - 2 < 0,

1/(3-1) = -(3-2),

1/2 = -1,

which is not true.

Again, our guess did not satisfy the equation in either case.

Iteration 3:

Let's try x = 2.5.

Case 1: When x - 2 ≥ 0,

1/(2.5-1) = 2.5-2,

1/1.5 = 0.5,

which is true.

Case 2: When x - 2 < 0,

1/(2.5-1) = -(2.5-2),

1/1.5 = -0.5,

which is not true.

Our guess of x = 2.5 satisfies the equation in Case 1.

Therefore, the approximate solution to the equation 1/x-1 = |x-2| after three iterations of successive approximation is x ≈ 5/2 or x ≈ 2.5.

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The hypotenuse (x) of the right triangle is 10 units

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

The right triangle

The hypotenuse (x) of the right triangle can be calculated using the following sine equation

sin(30) = 5/x

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

x = 5/sin(30)

Evaluate

x = 10

Hence, the hypotenuse of the right triangle is 10

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5 whole numbers are written in order. 5,8,x,y,12 The mean and median of the five numbers are the same then the values of x and y are:[tex]$$\boxed{x=8, \ y=3}$$[/tex] OR [tex]$$\boxed{x=12, \ y=53}$$[/tex].

let's first calculate the median of the given numbers.

Median of the given numbers is the middle number of the ordered set.

As there are five numbers in the ordered set, the median will be the third number.

Thus, the median of the numbers = x.

The mean of a set of numbers is the sum of all the numbers in the set divided by the total number of items in the set.

Let the mean of the given set be 'm'.

Then,[tex]$$m = \frac{5+8+x+y+12}{5}$$$$\Rightarrow 5m = 5+8+x+y+12$$$$\Rightarrow 5m = x+y+35$$[/tex]

As per the given statement, the median of the given set is the same as the mean.

Therefore, we have,[tex]$$m = \text{median} = x$$[/tex]

Substituting this value of 'm' in the above equation, we get:[tex]$$x= \frac{x+y+35}{5}$$$$\Rightarrow 5x = x+y+35$$$$\Rightarrow 4x = y+35$$[/tex]

Also, as x is the median of the given numbers, it lies in between 8 and y.

Thus, we have:[tex]$$8 \leq x \leq y$$[/tex]

Substituting x = y - 4x in the above inequality, we get:[tex]$$8 \leq y - 4x \leq y$$[/tex]

Simplifying the above inequality, we get:[tex]$$4x \geq y - 8$$ $$(5/4) y \geq x+35$$[/tex]

As x and y are both whole numbers, the minimum value that y can take is 9.

Substituting this value in the above inequality, we get:[tex]$$11.25 \geq x + 35$$[/tex]

This is not possible.

Therefore, the minimum value that y can take is 10.

Substituting y = 10 in the above inequality, we get:[tex]$$12.5 \geq x+35$$[/tex]

Thus, x can take a value of 22 or less.

As x is the median of the given numbers, it is a whole number.

Therefore, the maximum value of x can be 12.

Thus, the possible values of x are:[tex]$$\boxed{x = 8} \text{ or } \boxed{x = 12}$$[/tex]

Now, we can use the equation 4x = y + 35 to find the value of y.

Putting x = 8, we get:

[tex]$$y = 4x-35$$$$\Rightarrow y = 4 \times 8 - 35$$$$\Rightarrow y = 3$$[/tex]

Therefore, the values of x and y are:[tex]$$\boxed{x=8, \ y=3}$$[/tex]  OR [tex]$$\boxed{x=12, \ y=53}$$[/tex]

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The output value f(3) in the functions f( x ) = 3x + 5, f( x ) =  [tex]\frac{1}{2}x^2-1.5[/tex] and f( x ) = [tex]\frac{3}{2x}[/tex] is 14, 3 and 1/2 respectively.

Given the functions in the question:

f( x ) = 3x + 5

f( x ) =  [tex]\frac{1}{2}x^2-1.5[/tex]

f( x ) = [tex]\frac{3}{2x}[/tex]

To evaluate each function at f(3), we simply replace the variable x with 3 and simplify.

a)

f( x ) = 3x + 5

Replace x with 3:

f( 3 ) = 3(3) + 5

f( 3 ) = 9 + 5

f( 3 ) = 14

b)

f( x ) =  [tex]\frac{1}{2}x^2-1.5[/tex]

Replace x with 3:

[tex]f(3) = \frac{1}{2}(3)^2 - 1.5\\\\f(3) = \frac{1}{2}(9) - 1.5\\\\f(3) = 4.5 - 1.5\\\\f(3) = 3[/tex]

b)

f( x ) = [tex]\frac{3}{2x}[/tex]

Replace x with 3:

[tex]f(3) = \frac{3}{2(3)} \\\\f(3) = \frac{3}{6} \\\\f(3) = \frac{1}{2}[/tex]

Therefore, the output value of f(3) is 1/2.

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Answer:

0

Step-by-step explanation:

[tex]\displaystyle \frac{F(b)-F(a)}{b-a}\\\\=\frac{F(\pi)-F(-\pi)}{\pi-(-\pi)}\\\\=\frac{\sin(4\pi)-\sin(-4\pi)}{2\pi}\\\\=\frac{0-0}{2\pi}\\\\=0[/tex]

Not sure how the correct answer is stated as 45/28, but the answer is definitely 0.

Answer:

c

Step-by-step explanation:

assume base 10

-logy = x

[tex] \frac{1}{ log(y) } = x[/tex]

log base x y = 5 turns into the format x^ 5 = y

implement that to get c

The slope of a line that is perpendicular to the given line x + 6y = 3 is 6.

To determine the slope of a line that is perpendicular to the given line, we need to find the negative reciprocal of the slope of the given line.

The equation of the given line is x + 6y = 3.

To find the slope of the given line, we can rearrange the equation into slope-intercept form (y = mx + b), where m represents the slope:

x + 6y = 3

6y = -x + 3

y = (-1/6)x + 1/2

From the equation y = (-1/6)x + 1/2, we can see that the slope of the given line is -1/6.

To find the slope of a line that is perpendicular, we take the negative reciprocal of -1/6.

The negative reciprocal of -1/6 can be found by flipping the fraction and changing its sign:

Negative reciprocal of -1/6 = -1 / (-1/6) = -1 * (-6/1) = 6

Therefore, the slope of a line that is perpendicular to the given line x + 6y = 3 is 6.

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The approximate age of the rock outcrop is 1.2 billion years.

To approximate the age of the rock outcrop, we can use the concept of radioactive decay and the half-life of the U-238 isotope.

The half-life of U-238 is 4.5 billion years, which means that after each half-life, the amount of U-238 remaining is reduced by half.

We are given that the rock outcrop has 89.00% of its parent U-238 isotope remaining.

This means that the remaining fraction is 0.8900.

To find the number of half-lives that have elapsed, we can use the following formula:

Number of half-lives = log(base 0.5) (fraction remaining)

Using this formula, we can calculate:

Number of half-lives = log(base 0.5) (0.8900)

≈ 0.1212

Since each half-life is 4.5 billion years, we can find the approximate age of the rock outcrop by multiplying the number of half-lives by the half-life duration:

Age of the rock outcrop = Number of half-lives [tex]\times[/tex] Half-life duration

≈ 0.1212 [tex]\times[/tex] 4.5 billion years

≈ 545 million years

Therefore, the approximate age of the rock outcrop is approximately 545 million years.

Based on the answer choices provided, the closest option to the calculated value of 545 million years is 757 million years.

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Answer:

A: $51.00

B: $54.57

Step-by-step explanation:

Let amount = $68.00

Part A:

Since the coupon is for 25%, the family pays 75% of $68.00

75% of $68.00 = 0.75 × $68.00 = $51.00

The new price is $51.00

Part B:

The tax is 7% of $51.00

7% of $51.00 = $3.57

The total price is the sum of $51.00 and the amount of tax, $3.57

Total price = $51.00 + $3.57 = $54.57

The number of votes needed in an election can vary depending on various factors such as the type of election, voting rules, and specific requirements.

Without additional context or information about the specific election, it is challenging to provide an exact number of votes needed.The number of votes needed in an election is typically determined by factors such as the majority threshold, minimum vote requirement, or any specific criteria outlined in the election rules.

For example, in some elections, a candidate may need a simple majority (more than half) of the votes cast to win, while in others, a candidate may need a specific number or percentage of votes to secure victory.To determine the number of votes needed, it is essential to refer to the specific guidelines or rules established for that particular election.

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The cross section of the geometric solid is (d) rectangle

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

The geometric solid

Also, we can see that

The geometric solid is a cylinder

And the cylinder is divided vertically

The resulting shape from the division is a rectangle

This means that the cross section of the geometric solid is (d) rectangle

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The shape of the cross-section for the geometric solid given in the diagram is a rectangle.

The cross section of the geometric solid represents the shape which extends beyond the actual geometric solid which is a cylinder.

A rectangle has opposite side being equal. This means that the width and and length are of different length.

Therefore, the shape of the cross-section is a rectangle.

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Answer:

The answer is, x= 145

Step-by-step explanation:

Since line BD passes through the center E of the circle, then the angle must be a right angle or a 90 degree angle.

Hence angle DAB must be 90 degrees

or,

[tex]angle \ DAB = 0.3(2x+10) = 90\\90/0.3 = 2x+10\\300 = 2x+10\\300-10=2x\\290=2x\\\\x=145[/tex]

Hence the answer is, x= 145

Juan could have a maximum of 3 quarters.

1. Let's assume Juan has x quarters.

2. The value of x quarters in dollars would be 0.25x.

3. We are given that Juan has 18 dimes, which have a value of 0.10 * 18 = $1.80.

4. The combined value of Juan's dimes and quarters should be less than or equal to $2.75.

5. We can write the equation: 0.25x + 1.80 ≤ 2.75.

6. Subtracting 1.80 from both sides of the equation, we have: 0.25x ≤ 2.75 - 1.80.

7. Simplifying, we get: 0.25x ≤ 0.95.

8. Dividing both sides of the inequality by 0.25, we have: x ≤ 0.95 / 0.25.

9. Evaluating the expression, we find: x ≤ 3.8.

10. Since Juan cannot have a fraction of a quarter, the maximum number of quarters he could have is 3.

11. However, we need to check if the combined value of the dimes and quarters is at least $0.20.

12. If Juan has 3 quarters (0.25 * 3 = $0.75) and 18 dimes ($1.80), the combined value is $0.75 + $1.80 = $2.55.

13. Since $2.55 is less than $2.75, Juan can have 3 quarters.

14. Therefore, the maximum number of quarters Juan could have is 3.

Note: There are no possible solutions where Juan has more than 3 quarters and still satisfies the conditions.

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The pattern between the numbers 3, 20, 110, and 1715 can be found using a mathematical method. In order to get the following number from each, there is a sequence that must be applied.Let's take a look at the sequence that was used to generate these numbers:

The first number is multiplied by 2 and then increased by 14 to get the second number. For example:

3 x 2 + 14 = 20

Then, the second number is multiplied by 3 and 20, and 110 is added.

20 x 3 + 110 = 170

The third number is multiplied by 4 and then increased by 110.

110 x 4 + 110 = 550

Finally, the fourth number is multiplied by 5 and then increased by 110.

550 x 5 + 110 = 2825

Therefore, using the above formula, the next number in the sequence can be calculated:

1715 x 6 + 110 = 10400

As a result, the sequence of numbers 3, 20, 110, 1715, 10400 can be calculated using the mathematical formula stated above.

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Cameron has 1 book left over after sorting it into group of 5

To determine the number of book left,

we need to first divide 56 by 5         [56 ÷ 5]

Quotient = 11

then find the remainder which is 1

therefore there os only 1 book left over

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The area of the obtuse triangle is 34 square feet with a base of 10 ft and height of 6.8 ft.

To find the area of the obtuse triangle, we can use the formula A = (1/2) * base * height. Let's denote the unknown part of the base as x.

In the given triangle, we have the width of the obtuse angle triangle as 10 ft, the height (perpendicular) as 6.8 ft, and the unknown part of the base as x.

Using the formula, we can calculate the area as:

A = (1/2) * (10 + x) * 6.8

Simplifying this expression, we get:

A = 3.4(10 + x)

Now, we need to determine the value of x. From the given information, we know that the width of the obtuse angle triangle is 10 ft. This means the sum of the two parts of the base is 10 ft. Therefore, we can write the equation:

x + 10 = 10

Solving for x, we find:

x = 0

Since x = 0, it means that one part of the base has a length of 0 ft. Therefore, the entire base is formed by the width of the obtuse angle triangle, which is 10 ft.

Now, substituting this value of x back into the area formula, we have:

A = 3.4(10 + 0)

A = 3.4 * 10

A = 34 square feet

Hence, the area of the obtuse triangle is 34 square feet.

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Answer:

a. Measure of skewness

Step-by-step explanation:

Skewness is a measure of the asymmetry of a probability distribution. It quantifies the extent to which a dataset's values deviate from a symmetric distribution. Various measures of skewness exist, including the Pearson's skewness coefficient, the Bowley skewness coefficient, and the moment coefficient of skewness. These measures provide a numerical indication of the skewness present in the dataset.

This flexibility of a money market account makes it an ideal option for people who need to save money without risking their inheritance.

After inheriting $25,000 from a distant uncle, you would like to save it for ten years before doing anything with it. Since you want to save the money, there are several options for keeping it safe and earning interest, including a savings account, a certificate of deposit (CD), and a money market account.

Money market accounts, in my opinion, would be the best place to keep the inheritance. The money market account is a low-risk account with high-interest rates, making it an attractive option for someone who wants to save their money. As a result, it would be reasonable to place the inheritance into a money market account that pays a daily compounded rate of 3%.There are several reasons for choosing this option.

Firstly, the daily compounded interest will generate a higher return over the ten-year period than the simple interest or monthly compounded interest offered by other accounts. Second, the account is FDIC-insured, which means that the account holder is guaranteed to receive their money in the event of a bank failure.

Furthermore, the money market account provides easy access to the account holder's money while still earning interest. Most money market accounts have a limit on the number of withdrawals a person can make per month. Still, the account holder can easily transfer funds into a checking account or withdraw money from an ATM if needed.

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The numbers to complete the pythagorean triple in the equation (n² - 1)² + k² = (n² + 1)² are B. 35 and 37

A pythagorean triple is a set of 3 numbers that obey the pythagorean theorem.

Given the equation

(n² - 1)² + k² = (n² + 1)², we need to find the remaining numbers generated by the equation when k = 12. So, we proceed as follows/

Since we have the equation

(n² - 1)² + k² = (n² + 1)²

Subtracting (n² + 1)², from both sides, we have that

(n² - 1)² - (n² + 1)² + k² = (n² + 1)² - (n² + 1)²

(n² - 1)² - (n² + 1)² + k² = 0

Now, subtracting k from both sides, we have that

(n² - 1)² - (n² + 1)² + k² = 0

(n² - 1)² - (n² + 1)² + k² - k² = 0 - k²

(n² - 1)² - (n² + 1)² + 0 = - k²

(n² - 1)² - (n² + 1)² = - k²

Using the difference of two squares a² - b² = (a + b)(a - b)

(n² - 1)² - (n² + 1)² = - k²

(n² - 1 + n² + 1)[n² - 1 - (n² + 1)] = - k²

(n² + n² - 1 + 1)[n² - n² - 1 - 1)] = - k²

(2n² + 0)[0 - 2)] = - k²

(2n²)(- 2) = - k²

-4n² = - k²

4n² = k²

n² = k²/4

Taking square root of both sides, we have that

n = √(k²/4)

n = k/2

Since k = 12, we have that

n = 12/2

n = 6

So, the first number is (n² - 1) = (6² - 1)

= 36 - 1

= 35

The second number is (n² + 1) = (6² + 1)

= 36 + 1

= 37

So, the numbers are B. 35 and 37

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The function representing the situation is f(x) = 2(x + 1)^2 + 2. Option C.

To determine which function represents the given situation, we can use the information provided about the minimum point and the point on the parabola.

We are given that the minimum of the graph of the quadratic function is located at (-1, 2). This means that the vertex of the parabola is at (-1, 2).

The standard form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) represents the vertex.

In this case, h = -1 and k = 2, so the equation becomes f(x) = a(x + 1)^2 + 2.

Additionally, we know that the point (2, 20) lies on the parabola. We can substitute these coordinates into the equation to solve for the value of a:

20 = a(2 + 1)^2 + 2

20 = 9a + 2

18 = 9a

a = 2

Substituting the value of a back into the equation, we have:

f(x) = 2(x + 1)^2 + 2

So the function that represents the given situation is f(x) = 2(x + 1)^2 + 2, which is option C.

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Note the complete question is

The minimum of the graph of a quadratic function is located at (–1, 2). The point (2, 20) is also on the parabola. Which function represents the situation?

A) f(x) = (x + 1)2 + 2

B) f(x) = (x – 1)2 + 2

C) f(x) = 2(x + 1)2 + 2

D) f(x) = 2(x – 1)2 + 2

Answer: False

Step-by-step explanation:

Spherical geometry, on the other hand, is a type of non-Euclidean geometry that is specifically concerned with studying the properties of curved surfaces, such as spheres.

Hope this help! Have a good day!

The length of line segment IG of the circle using the chord-chord power theorem is 6.

Chord-chord power theorem simply state that "If two chords of a circle intersect, then the product of the measures of the parts of one chord is equal or the same as the product of the measures of the parts of the other chord".

From the figure:

Line segment FG = 12

Line segment GH = 4

Line segment GJ = 8

Line segment IG = ?

Now, usig the chord-chord power theorem:

Line segment FG × Line segment GH = Line segment GJ × Line segment IG

Plug in the values:

12 × 4 = 8 × Line segment IG

48 = 8 × Line segment IG

Line segment IG = 48/8

Line segment IG = 6

Therefore, the line segment IG measures 6 units.

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The midpoint of the line segment joining the pair of points (8, -8) and (-5, -1) is (1.5, -4.5).

To find the midpoint, we need to take the average of the x-coordinates and the average of the y-coordinates of the two given points.

For the x-coordinate: (8 + (-5))/2 = 3/2 = 1.5

For the y-coordinate: (-8 + (-1))/2 = -9/2 = -4.5

Thus, the midpoint of the line segment is (1.5, -4.5).

In more detail, the midpoint formula is derived by averaging the x-coordinates and y-coordinates separately. For a line segment with endpoints (x1, y1) and (x2, y2), the midpoint (xm, ym) is given by:

xm = (x1 + x2)/2

ym = (y1 + y2)/2

In this case, the x-coordinates are 8 and -5, and their average is (8 + (-5))/2 = 1.5. The y-coordinates are -8 and -1, and their average is (-8 + (-1))/2 = -9/2 = -4.5.  

Therefore, the midpoint of the line segment joining the two given points is (1.5, -4.5).

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The height of the isosceles triangle is approximately 9.70 meters.

To find the height (y) of an isosceles triangle given its slant height and two angles, we can use trigonometry. Here are the steps to solve this problem:

Start by drawing a sketch of the isosceles triangle. Label the base as "b," the height as "y," and the slant height as "s."

Since the triangle is isosceles, the two base angles are congruent, meaning each angle measures 57 degrees. Label one of these angles as "θ."

We know that the slant height (s) is given as 11.5 m.

Apply the sine function to relate the slant height (s) to the angle (θ) and the height (y) of the triangle. The sine of an angle is defined as the ratio of the opposite side (y) to the hypotenuse (s). So, we have sin(θ) = y/s.

Substitute the given values into the equation: sin(57 degrees) = y/11.5.

Solve the equation for y by multiplying both sides by 11.5: y = 11.5 * sin(57 degrees).

Use a calculator to find the value of sin(57 degrees) and multiply it by 11.5 to obtain the height (y) of the triangle.

Performing the calculation, y = 11.5 * sin(57 degrees) ≈ 9.70 meters (rounded to two decimal places).

Therefore, the height of the isosceles triangle is approximately 9.70 meters.

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