which table represents g(x) when f(x) = 3x-4 and g(x)=f(2x)

Answer:

  210

Step-by-step explanation:

You want the product of x and y given similar triangles with side ratios ...

Corresponding sides in similar triangles are proportional. This means ...

  x/10 = 21/y

  xy = 210 . . . . . . . cross multiply

The product of x and y is 210.

__

Additional comment

Sides in the larger triangle are 3/2 times those in the smaller triangle. This means x = 15, y = 14.

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The missing length of the right triangle is 8√3

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

The right triangle

The missing length of the right triangle can be calculated using the following Pythagoras theorem

x² = difference of squares of the other sides

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

x² = (16)² - (8)²

Evaluate

x² = 192

Take the square roots

x = 8√3

Hence, the missing length of the right triangle is 8√3

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According to the information we can infer that the correct match is screwdriver: 250mm, height of a kitchen worktop: 0.9m, the overall length of a JCB 3cx Backhoe loader digger: 5620mm, and height of a door: 2m

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The value of [tex]x_5[/tex], rounded to the nearest decimal place using Newton's method of approximation is 1.466.

The correct answer is option B.

To find the value of [tex]x_5[/tex]using Newton's method of approximation, we need to iterate the following formula:

[tex]x_(_n_+_1_) = x_n - f(x_n) / f'(x_n)[/tex]

where f'(x) represents the derivative of the function f(x). Let's calculate the values step by step.

Given function: f(x) = [tex]x^3 - x^2 - 1[/tex]

Step 1: Find the derivative of f(x)

f'(x) = [tex]3x^2 - 2x[/tex]

Step 2: Initialize the starting value

[tex]x_0[/tex]= 1

Step 3: Calculate [tex]x_1[/tex]

[tex]f(x_0) = f(1) = (1^3) - (1^2) - 1 = 1 - 1 - 1 = -1[/tex]

[tex]f'(x_0) = f'(1) = (3(1^2)) - (2(1)) = 3 - 2 = 1[/tex]

[tex]x_1 = x_0 - f(x_0) / f'(x_0)[/tex]

   = 1 - (-1) / 1

   = 2

Step 4: Calculate [tex]x_2[/tex]

[tex]f(x_1) = f(2) = (2^3) - (2^2) - 1 = 8 - 4 - 1 = 3[/tex]

[tex]f'(x_1) = f'(2) = (3(2^2)) - (2(2)) = 12 - 4 = 8[/tex]

[tex]x_2 = x_1 - f(x_1) / f'(x_1)[/tex]

   = 2 - 3 / 8

   = 1.625

Step 5: Repeat the process until we reach [tex]x_5[/tex]

Performing the calculations for [tex]x_3, x_4, andx_5[/tex], we find:

[tex]x_3[/tex] ≈ 1.465

[tex]x_4[/tex] ≈ 1.466

[tex]x_5[/tex]≈ 1.466

After applying Newton's method of approximation to the function f(x) = [tex]x^3 - x^2 - 1,[/tex]starting with,[tex]x_0 = 1,[/tex] we iteratively calculated the values of [tex]x_1, x_2, x_3, x_4, and x_5[/tex]. The final approximation, [tex]x_5[/tex], is approximately 1.466 when rounded to the nearest decimal place. This aligns with option B as the correct answer.

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To solve the question, we need to consider the transformation described and determine the function that represents the stretched graph.

The absolute value parent function is [tex]\displaystyle\sf f(x) = |x|[/tex]. According to the question, this parent function is horizontally stretched by a factor of 5 to create the graph of [tex]\displaystyle\sf g(x)[/tex].

A horizontal stretch by a factor of 5 means that the x-values on the graph are multiplied by 5. In other words, the transformation can be represented as [tex]\displaystyle\sf g(x) = |5x|[/tex].

Therefore, the correct function that represents the graph of [tex]\displaystyle\sf g(x)[/tex], the stretched version of [tex]\displaystyle\sf f(x) = |x|[/tex], is [tex]\displaystyle\sf g(x) = |5x|[/tex].

The correct answer is option D: [tex]\displaystyle\sf g(x) = |5x|[/tex].

The equation y = 10x represents a proportional relationship. The constant of proportionality is 10.

A proportional relationship means for every change in x, y also changes proportionately. Let us consider an example

If y = 20

then proportionality constant say k = 10

then x will be 2.

if y = 30

then proportionality constant remains the same ie 10

x will be 3

Here it is evident that as you increase x, y also increase. As you decrease x, y also decreases in a proportionate manner.

k here links x and y variables in a proportionate relationship. Therefore k= 10 is the ratio between y and x.

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The measure that changes the most when the value of 82° is changed to 94° is the Median, which becomes 84°.

To determine which measure changes the most when the value of 82° is changed to 94°, we need to calculate the measures for both the original and new data.

Given the average high temperatures: 58, 61, 71, 77, 91, 100, 105, 102, 95, 82, 66, 57.

First, let's calculate the original measures:

Mean: The mean is the average of all the values.

Mean = (58 + 61 + 71 + 77 + 91 + 100 + 105 + 102 + 95 + 82 + 66 + 57) / 12 = 81.4°.

Median: The median is the middle value when the data is arranged in ascending order.

Arranging the data in ascending order: 57, 58, 61, 66, 71, 77, 82, 91, 95, 100, 102, 105.

The median is the 6th value, which is 82°.

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

Range = Highest value - Lowest value

= 105 - 57

= 48°.

IQR (Interquartile Range): The IQR is the difference between the third quartile (Q3) and the first quartile (Q1).

Calculating Q1 and Q3:

Q1 = 66, Q3 = 95.

IQR = Q3 - Q1

= 95 - 66

= 29°.

Now, let's calculate the new measures after changing the value of 82° to 94°:

Mean: The mean is unaffected by changing a single value.

Median: Since we only changed one value, the new median will still be the same as the original median, which is 82°.

Range: The range is unaffected by changing a single value.

IQR (Interquartile Range): The IQR is unaffected by changing a single value.

Therefore, the measure that changes the most when the value of 82° is changed to 94° is the Median, which becomes 84°.

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Answer:

Step-by-step explanation:

if x = -1,

     a-b+c = 0 (eq1)

if x = 1,

    a+b+c = 0(eq2)

from eq 1 and 2;

a+c = 0;

so, a = -c,

if a+c = 0, b = 0

Ans : a=-c, b=0

The value of constants a and b are -2 and 0 respectively.

Given: x=-1 and x=1 are both the solution to equation ax²+bx+c=0.

To find: the constant a and b

Solution:

To obtain this, first, we have to simplify the equation by putting x=1 into the given equation, we get;

a+b=-2

afterward, we have to put x=-1 in the given equation, and we get;

a-b=-2

Now, we will add the two obtained equations, we get;

2a=-4 i.e., a=-2

Now, a+b=0 which means the value of b is 0.

Hence, the constants a and b are -2 and 0 respectively.

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The total cost of 21 bottles of cola at £1.59 each is £33.39.

1. Determine the cost per bottle: Each bottle of cola costs £1.59.

2. Multiply the cost per bottle by the number of bottles: £1.59 x 21 = £33.39.

3. Therefore, the total cost of 21 bottles of cola is £33.39.

In the given scenario, each bottle of cola is priced at £1.59. To calculate the total cost of 21 bottles, we need to multiply the cost per bottle by the number of bottles. By multiplying £1.59 by 21, we get £33.39 as the final answer.

Certainly! Here's an expanded explanation in 100 additional words:

To calculate the total cost of 21 bottles of cola, we need to multiply the cost per bottle by the number of bottles. In this case, each bottle costs £1.59. By multiplying £1.59 by 21, we can determine the total cost.

To perform the calculation, we can use the distributive property of multiplication. We multiply the cost per bottle (£1.59) by the number of bottles (21) individually, and then sum up the results.

First, we multiply the ones place: 9 multiplied by 21, which equals 189. Since 9 multiplied by 1 equals 9, we write down the 9 in the ones place and carry over the 18 to the tens place.

Next, we multiply the tens place: 5 multiplied by 21, which equals 105. Adding the carried-over 18, we get 123. We write down the 3 in the tens place and carry over the 12 to the hundreds place.

Finally, we multiply the hundreds place: 1 multiplied by 21, which equals 21. Adding the carried-over 12, we get 33.

Combining the results, we have £33.39 as the total cost of 21 bottles of cola.

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Answer:

The correct answer is A. triangles.

Step-by-step explanation:

Answer:

A)triangles

Step-by-step explanation:

CPCTC is a theorem that states that if two triangles are congruent, then their corresponding parts are also congruent. This means that if you have two triangles that are identical in shape and size, you can conclude that their corresponding angles and sides are equal.

The polar form of (2cis60°)(5cis30°) is √(75 - 15√3) cis(π/4). The rectangular form of (2cis60°)(5cis30°) is (-5 + 5√3)/2 + (15√3 + 5)i/2.

Polar form:

(2cis60°) and (5cis30°) can be expressed in polar form as follows:

2cis60° = 2(cos60° + isin60°) = 2(1/2 + i√3/2) = 1 + i√3

5cis30° = 5(cos30° + isin30°) = 5(√3/2 + i/2) = (5√3)/2 + (5i)/2

Thus, (2cis60°)(5cis30°) = (1 + i√3)((5√3)/2 + (5i)/2)

To multiply these complex numbers, we can use FOIL method as we do with algebraic expressions to find:

= [(1)(5√3)/2 + (1)(5i)/2 + (i√3)(5√3)/2 + (i√3)(5i)/2]

= [(5√3)/2 + (5i)/2 + (5i√3)/2 − (15/2)]

= [(5√3)/2 − (15/2) + (5i)/2 + (5i√3)/2]

Now, let's determine the magnitude and argument of this complex number:

Magnitude:

|z| = √[(5√3)/2 − (15/2)]² + [(5/2)√3]²

= √[75/4 - 15√3 + 75/4]

= √(75 - 15√3)

Argument:

θ = tan^⁻1((5/2)√3 / [(5√3)/2 − (15/2)])

= tan^⁻1(1)

= π/4

Rectangular form:

To find the rectangular form, we can use the magnitude and argument we just found, and the following formula:

z = r(cosθ + isinθ)

So, we have:

(2cis60°)(5cis30°) = √(75 - 15√3) cis(π/4)

= √(75 - 15√3)(cos(π/4) + isin(π/4))

= (√(75 - 15√3)/2)(1 + i) + (√(75 - 15√3)/2)(√3 - i)

= ([(5√3)/2 − (15/2)] + [(5/2)√3]) + ([(5√3)/2 − (15/2)]√3 + (5/2))

= (-5 + 5√3) + (15√3 + 5)i/2

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Step-by-step explanation:

The rectangular form of a cis equation is

[tex]a + bi[/tex]

Given that we have

[tex]2cis60 \times 5cis30[/tex]

We can just multioly the magnitudes

[tex]2 \times 5 = 10[/tex]

And add the angles

[tex]60 + 30 = 90[/tex]

And that gives us our new equestion

[tex]10cis(90)[/tex]

Remember that

cis is basically

[tex] \cos( \alpha ) + i \sin( \alpha ) [/tex]

So our answer in polar form is

[tex]10( \cos(90) + i \sin(90) )[/tex]

To convert to rectangular form, distribute the 10

[tex]10 \cos(90) = a[/tex]

[tex]10i \sin(90) = bi[/tex]

So

[tex]a = 0[/tex]

[tex]b = 10[/tex]

So, our equation in rectangular form is

[tex]10i[/tex]

9.  42 chocolate lava cakes and 15 caramel apple tarts were sold.

10.Brian needs to sell approximately 14,286 meals to break even.

9. Let's denote the number of chocolate lava cakes as x and the number of caramel apple tarts as y. We can set up a system of equations based on the given information:

x + y = 57 (sum of the two desserts is equal to 57)

x - y = 27 (difference of the two desserts is 27)

We can solve this system of equations using the method of substitution or elimination. Let's use the method of elimination:

Add the two equations:

(x + y) + (x - y) = 57 + 27

2x = 84

x = 42

Substitute the value of x into one of the equations to find y:

42 - y = 27

y = 42 - 27

y = 15

Therefore, 42 chocolate lava cakes and 15 caramel apple tarts were sold.

10. To determine the number of meals Brian needs to sell to break even, we can calculate the break-even point by comparing the total cost and the total revenue.

Total cost = $250,000 (start-up cost) + ($17.50 per meal) * N (number of meals)

Total revenue = ($35.00 per meal) * N (number of meals)

To break even, the total cost should be equal to the total revenue:

$250,000 + ($17.50 per meal) * N = ($35.00 per meal) * N

Simplifying the equation:

$250,000 = $35.00N - $17.50N

$250,000 = $17.50N

N = $250,000 / $17.50

N ≈ 14,285.71

Since we cannot sell a fraction of a meal, we round the number of meals up to the nearest whole number.

Therefore, Brian needs to sell approximately 14,286 meals to break even.

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Answer:

The opportunity cost of attending summer school in this scenario is $4,000. Opportunity cost is the value of the next best alternative that must be given up in order to pursue a certain action. In this case, by choosing to attend summer school, you are giving up the opportunity to earn $3,000 from your usual summer job. Additionally, you have to pay $1,000 for tuition, bringing the total opportunity cost to $3,000 + $1,000 = $4,000.

Step-by-step explanation:

The opportunity cost of attending summer school in this scenario is $4,000, which includes both the foregone earnings from the summer job and the cost of tuition.

The potential earnings from a job that you choose not to do are considered an opportunity cost in economics. In this situation, the opportunity cost of attending summer school is both the $3,000 you lose from not working your summer job and the tuition fee of $1,000 you pay for the school. Economic theory suggests that we consider both of these costs because both are expenses that are a direct result of your decision to attend summer school. Therefore, the total opportunity cost in this scenario is $4,000 ($3,000 + $1,000).

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Answer:

170, 171, 172

Step-by-step explanation:

x + x + 1 + x + 2 = 513

3x + 3 = 513

3x = 510

x = 170

x + 1 = 171

x + 2 = 172

The range of g(x) is [8, ∞).Thus, the function g(x) = -2√x + 8 has the same range as f(x) = -2√(x - 3) + 8.

Let f (x) = -2√(x - 3) + 8. The function g(x) has the same range as f (x) if the range of f(x) is equal to the range of g(x). Let's determine the range of f(x).Range of f(x):The domain of f(x) is all non-negative real numbers because the square root of a negative number is not a real number. In other words, x - 3 ≥ 0 ⇒ x ≥ 3. This implies that f(x) is a decreasing function because as x increases, the value of f(x) decreases.

Furthermore, the minimum value that f(x) can take is 8, which occurs when x = 3. As x approaches infinity, the value of f(x) approaches negative infinity, but never quite reaches it. Therefore, the range of f(x) is [8, ∞).In order for g(x) to have the same range as f(x), its range must also be [8, ∞). Therefore, g(x) must also be a decreasing function that is bounded from below by 8.

A possible function that satisfies these conditions is g(x) = -2√x + 8. Let's check the range of g(x):Range of g(x):As with f(x), the domain of g(x) is all non-negative real numbers. g(x) is also a decreasing function because as x increases, the value of g(x) decreases. Furthermore, the minimum value that g(x) can take is 8, which occurs when x = 0. As x approaches infinity, the value of g(x) approaches negative infinity, but never quite reaches it

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The rational expression that is undefined when x = 0 is (8x + 6)/(-x^2 + 1).

To determine which rational expression is undefined when x = 0, we need to find the values of x that make the denominator equal to zero. When the denominator of a rational expression is zero, the expression is undefined because division by zero is not possible.

Let's analyze each of the given expressions:

(5x^2 - 4)/(3x - 3):

To check if this expression is undefined when x = 0, we substitute x = 0 into the denominator:

3(0) - 3 = -3

Since the denominator is not zero, this expression is defined when x = 0.

(6x^2)/(9x - 3):

Similarly, we substitute x = 0 into the denominator:

9(0) - 3 = -3

Again, the denominator is not zero, so this expression is defined when x = 0.

(8x + 6)/(-x^2 + 1):

For this expression, we substitute x = 0 into the denominator:

-(0)^2 + 1 = -0 + 1 = 1

The denominator is not zero, indicating that this expression is defined when x = 0.

(4x - 8)/(-2x^2):

When we substitute x = 0 into the denominator:

-2(0)^2 = -2(0) = 0

The denominator becomes zero when x = 0. Therefore, this expression is undefined when x = 0.

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In this triangle, the product of sin B and tan C is[tex]b^2/(ac)[/tex]  , and the product of sin C and tan B is [tex]c^2/(ab)[/tex].

In a right-angled triangle ABC, where angle A is 90 degrees, we have the following side lengths:

AB = c (base)

BC = a (hypotenuse)

AC = b (perpendicular)

We need to calculate the products of sin B and tan C, and sin C and tan B.

First, let's calculate sin B and tan C:

sin(B) = opposite/hypotenuse = AC/BC = b/a

tan(C) = opposite/adjacent = AC/AB = b/c

The product of sin B and tan C is sin(B) * tan(C) = (b/a) * (b/c) = [tex]b^2[/tex]/(ac).

Next, let's calculate sin C and tan B:

sin(C) = opposite/hypotenuse = AB/BC = c/a

tan(B) = opposite/adjacent = AB/AC = c/b

The product of sin C and tan B is sin(C) * tan(B) = (c/a) * (c/b) = [tex]c^2[/tex]/(ab).

Therefore, in the given right-angled triangle ABC, the product of sin B and tan C is[tex]b^2[/tex]/(ac), and the product of sin C and tan B is [tex]c^2[/tex]/(ab).

These formulas hold true for any right-angled triangle, where the base is AB, the hypotenuse is BC, and the perpendicular is AC.

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The question probable may be:

In this triangle, the product of sin B and tan C is _____ , and the product of sin C and tan B is _______.

Part A: The mean and median for Bay Side School are 3.8 and 5, respectively, while for Seaside School they are 4.93 and 5.

Part B: The range and IQR for Bay Side School are 8 and 4, respectively, while for Seaside School they are 8 and 2.

Part C: Bay Side School is a better choice for smaller class sizes due to its smaller mean, smaller range, and larger IQR.

Part A: Measures of Center

Bay Side School:

Mean = 3.8 (The sum of all class sizes divided by the number of classes)

Median = 5 (The middle value when class sizes are arranged in ascending order)

Seaside School:

Mean = 4.93 (The sum of all class sizes divided by the number of classes)

Median = 5 (The middle value when class sizes are arranged in ascending order)

Part B: Measures of Variability

Bay Side School:

Range = 8 (The difference between the maximum and minimum class sizes)

Interquartile Range (IQR) = 4 (The difference between the first quartile and third quartile)

Seaside School:

Range = 8 (The difference between the maximum and minimum class sizes)

Interquartile Range (IQR) = 2 (The difference between the first quartile and third quartile)

Part C: Conclusion and Reasoning

If you are interested in a smaller class size, Bay Side School is a better choice. Bay Side School has a smaller mean (3.8) compared to Seaside School (4.93), indicating a generally smaller class size. Additionally, the median class size is the same for both schools, but the range and IQR are smaller for Bay Side School.

This suggests that class sizes at Bay Side School have less variation and are more consistently smaller. Therefore, based on the provided data, Bay Side School is the better choice for smaller class sizes.

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The correct answer is option e. The joint probability of two statistical dependent events Y and Z can be written as P(Y and Z) = P(Y) * P(Z|Y).

The concept of joint probability is utilized in probability theory, a statistical tool that investigates the probability of two events happening simultaneously. It is an estimation of the probability of two or more events occurring simultaneously. Joint probability is a method of defining the probability of two events happening at the same time.The joint probability of two statistical dependent events Y and Z can be written as P(Y and Z) = P(Y) * P(Z|Y).This implies that the possibility of Z happening given Y has happened multiplied by the probability of Y happening is the joint probability of Y and Z. P(Z|Y) refers to the probability of Z occurring when Y is true or has already occurred. It is conditional probability and is often used to calculate joint probabilities.The formula for conditional probability is P(A|B) = P(A and B)/P(B), which can be simplified to P(A and B) = P(A|B) * P(B). This formula is applicable to P(Y and Z), where P(Y and Z) = P(Z|Y) * P(Y).

The correct answer is option e.

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Answer:

Factorizing each expression:

x^2 + 8x + 15 = (x+3)(x+5)

x^2 - 2x - 15 = (x-5)(x+3)

x^2 + 9x - 36 = (x+12)(x-3)

x^2 - 12x - 45 = (x-15)(x+3)

So the factorized expressions are:

- x^2 + 8x + 15 = (x+3)(x+5)

- x^2 - 2x - 15 = (x-5)(x+3)

- x^2 + 9x - 36 = (x+12)(x-3)

- x^2 - 12x - 45 = (x-15)(x+3)

The calculated value of y when x = -10 is -30

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

y varies directly with x

This means that

y = kx

Where

k = constant of variation

So, we have

24 = 8k

k = 3

This also means that

y = 3x

When x = -10, we have

y = 3 * -10

Evaluate

y = -30

Hence, the value is -30

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Question

Suppose y varies directly with x and y=24 when x=8.

What is the value of y when x = -10?

Algebra is introduced to primary school learners in a basic form to provide them with a foundation for mathematical thinking and problem-solving.

Here are some reasons why algebra is taught to primary school learners and the importance of teaching it at an early age:

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The mean score is  = 53.16.the average of the two middle values:  53.5. there is no repeated value, so there is no mode.

To describe Michael's scores on the math tests, we can consider different measures of central tendency, namely the mean, median, and mode. Each measure provides a different perspective on the typical or representative value of the data.

The mean is calculated by summing up all the scores and dividing by the number of scores. It is affected by extreme values and can be skewed if there are outliers. In this case, Michael's scores are 24, 28, 21, 79, 84, and 93. The mean score is (24 + 28 + 21 + 79 + 84 + 93) / 6 = 53.16.

The median is the middle value when the scores are arranged in ascending or descending order. It is less affected by extreme values or outliers compared to the mean. To find the median, we sort the scores: 21, 24, 28, 79, 84, 93. Since there is an even number of scores, we take the average of the two middle values: (28 + 79) / 2 = 53.5.

The mode represents the most frequently occurring value in the data set. In this case, there is no repeated value, so there is no mode.

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The value of x for measure of the angle m∠CED subtended by the arc CD at the circumference is equal to 0.4 to the nearest tenth.

The angle subtended by an arc of a circle at it's center is twice the angle it substends anywhere on the circles circumference. Also the arc measure and the angle it subtends at the center of the circle are directly proportional.

arc CD = 2(m∠CED)

Also arc CD = 84°

84° = 2(12x + 37)°

12x + 37 = 84/2 {divide through by 2}

12x + 37 = 42

12x = 42 - 37 {collect like terms}

12x = 5

x = 5/12 {divide through by 12}

x = 0.4167

Therefore, the value of x for measure of the angle m∠CED subtended by the arc CD at the circumference is equal to 0.4 to the nearest tenth.

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Answer: 6 feet

Step-by-step explanation:

The height of the man and the length of his shadow are proportional.

This means that we can use proportions to find the height of the man if we know the length of his shadow.

We can set up a proportion as follows: height of man/length of shadow = (which includes the triangle formed by the man, his shadow, and the man/length constant.

Let's call the constant "k". We don't know the value of "k" yet, but we do know that it will be the same for all similar triangles ground).

Using the measurements given in the problem, we get 6 / 12 = simplifying the left side of the equation gives 1 / 2 = kNow we know that k = 1/2.

We can use this to find the height of the man. Let's call the height "h". We can set up another proportion using the value we just found:h / 12 = 1 / 2

Cross-multiplying gives us: 2h = 12

Simplifying further: h = 6 Therefore, the height of the man is 6 feet.

Step-by-step explanation:

You didn't include a diagram, but a dodecagon is a 12 sided polygon.

The SUM of the interior angles =   (n-2) * 180    where n =12  

(n-2)* 180

(12-2) * 180 =   1800    <====this is the SUM of the 12 interior angles

         so EACH angle is   1800 / 12 = 150 degrees