To solve the quadratic equation 3(x−7)2=27 , Diego and Mai wrote the following: Diego 3(x−7)2=27 (x−7)2=9 x2−72=92 x2−49=81 x2=130 x=130−−−√ and x=−130−−−√ Mai 3(x−7)2=27 (x−7)2=9 x−7=9 x=16 Identify the mistake(s) each student made. Solve the equation and show your reasoning.

There are 27 chickens and 13 Rabbits in the pen on your grandmother's farm, based on the given information of 40 heads and 106 legs.

a) The unknowns in the problem are the number of chickens and rabbits in the pen. We don't know the exact quantities of each animal, which is what we need to determine.

b) Let's denote the number of chickens as "C" and the number of rabbits as "R." To represent the total number of animals in the pen, we can write the equation: C + R = total number of animals.

c) The total number of legs can be represented by the equation: 2C + 4R = total number of legs. This equation accounts for the fact that chickens have two legs, while rabbits have four.

d) To find the solution, we can use the given information: 40 heads and 106 legs. Since each animal has one head, the total number of animals is equal to the total number of heads, which is 40.

Using the equation C + R = 40, we can solve for one variable in terms of the other. For example, if we solve for C, we get C = 40 - R.

Substituting this value of C into the equation 2C + 4R = 106, we can solve for R:

2(40 - R) + 4R = 106

80 - 2R + 4R = 106

2R = 26

R = 13

Therefore, there are 13 rabbits in the pen.

To find the number of chickens, we can substitute the value of R into the equation C = 40 - R:

C = 40 - 13

C = 27

Hence, there are 27 chickens in the pen.

In conclusion, there are 27 chickens and 13 rabbits in the pen on your grandmother's farm, based on the given information of 40 heads and 106 legs.

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By connecting these points with a straight line, we can graph the line y = 3x - 8.

The line should have a positive slope, rising from left to right.

To graph the line y = 3x - 8, we can use the slope-intercept form of a linear equation, which is y = mx + b. In this equation, m represents the slope of the line, and b represents the y-intercept.

Comparing the given equation with the slope-intercept form, we can see that the slope, m, is 3, and the y-intercept, b, is -8.

To plot the graph, we'll start by plotting the y-intercept, which is the point (0, -8)T.

his point represents where the line intersects the y-axis.

Next, we can use the slope to find additional points on the line. The slope of 3 means that for every increase of 1 in the x-coordinate, the y-coordinate increases by 3.

Starting from the y-intercept (0, -8), we can move 1 unit to the right and 3 units up to reach the next point (1, -5).

We can continue this pattern to plot more points.

Using this information, we can plot multiple points and then connect them to form the line:

Point 1: (0, -8)

Point 2: (1, -5)

Point 3: (2, -2)

Point 4: (-1, -11)

Point 5: (-2, -14)

By connecting these points with a straight line, we can graph the line y = 3x - 8.

The line should have a positive slope, rising from left to right.

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The observation of the pattern is discussed below.

When we calculate the squares of the numbers from 1 to 10 and observe the digits at the one's place of each square, we notice a pattern:

1²=1

2²=4

3²=9

4²=16

5²=25

6²=36

7²=49

8²=64

9²=81

10²=100

If we focus on the digits at the one's place, we can see that they form a repeating pattern: 1, 4, 9, 6, 5, 6, 9, 4, 1, 0. This pattern repeats itself as we calculate the squares of higher numbers.

This observation is known as the "digit at one's place pattern" or the "units digit pattern." It occurs because the square of any number depends only on the digit at the one's place of that number. Hence, when we square numbers that have the same digit at the one's place, we get the same digit at the one's place in the result.

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The total volume of the 21 bowling balls is 6048π cubic inches.

We have,

To find the total volume of the 21 bowling balls, we first need to calculate the volume of a single ball and then multiply it by the number of balls.

The volume of a sphere can be calculated using the formula:

V = (4/3) x π x r³

where V is the volume and r is the radius of the sphere.

Given that the diameter of the bowling ball is 12 inches, the radius can be calculated as half of the diameter:

r = 12/2 = 6 inches

Now we can calculate the volume of a single ball:

V_ball = (4/3) x π x (6³) = (4/3) x π x 216 = 288π cubic inches

Since there are 21 balls, we can calculate the total volume by multiplying the volume of a single ball by the number of balls:

Total volume = 21 x 288π = 6048π cubic inches

Thus,

The total volume of the 21 bowling balls is 6048π cubic inches.

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TanA value is 22/3 and TanB is 3/2 from the given triangle ABC.

We have to find the values of Tan A and TanB.

We know that tan function is a ratio of opposite side and adjacent side.

To find TanA, we have to take opposite side of vertex A has opposite side.

TanA=18/27

TanA=2/3

Now let us find TanB , which is 27/18

TanB =3/2

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The function A(x) = 100 + 4x describes the scenario of an ant colony that starts with an initial population of 100 and experiences a constant rate of increase of 4 ants per year.

We have,

The function A(x) = 100 + 4x represents a linear relationship between the variable x (representing time in this case) and the variable A(x) (representing the population of the ant colony).

The term 100 in the function represents the initial population of the ant colony.

It indicates that at the starting point (x = 0), the population is 100.

The term 4x in the function represents the rate at which the population increases over time. Since the coefficient of x is positive (4), it indicates that the population is increasing.

For every unit increase in x (in this case, for every year that passes), the population increases by 4.

Therefore,

The function A(x) = 100 + 4x describes the scenario of an ant colony that starts with an initial population of 100 and experiences a constant rate of increase of 4 ants per year.

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The calculated value of the price per cup is $1.16

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

A 3-quart jug of water costs $3.48

The price per cup is calculated as

Unit rate = Total cost/Size of the cup

Substitute the known values in the above equation, so, we have the following representation

Unit rate = 3.48/3

Evaluate

Unit rate = 1.16

Hence, the price per cup is $1.16

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The circumference of the circle is 62.8cm.

The Circumference of the circle is given by the formula,

Circumference = 2*[tex]\pi[/tex]*r cm.........equation 1

Given, r = 10cm

On substituting the radius value in equation 1

Circumference = 2*[tex]\pi[/tex]*10

Circumference = 62.8 cm

Therefore, the circumference of the circle is 62.8cm.

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The mean of the given set of numbers is 11.

To find the mean (average) of a set of numbers, we sum up all the numbers and divide the sum by the total count of numbers.

Given the set of numbers: 8, 14, 22, 7, 2, 11, 25, 7, 5, 9

To find the mean, we add up all the numbers:

8 + 14 + 22 + 7 + 2 + 11 + 25 + 7 + 5 + 9 = 110

Next, we divide the sum by the total count of numbers, which is 10:

110 / 10 = 11

Therefore, the mean of the given set of numbers is 11.

The mean is a measure of central tendency and represents the average value of the data set.

In this case, it indicates that, on average, the numbers in the set tend to cluster around the value of 11.

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Step-by-step explanation:

PLot two pioints....connect the points

Use the y -intercept given   y = -1 when x = 0     Plot that

when   y = 0   x = 1/2      plot that

draw line through these points :

Answer:

Step-by-step explanation:

The two transformations applied to pentagon ABCDE to create A''B''C''D''E'' are rotation and reflection.

To determine the two transformations applied to pentagon ABCDE to create A''B''C''D''E'', we need more specific information about the transformations or the corresponding points.

Transformations can include translations, rotations, reflections, dilations, or combinations of these.

Without knowing the specific transformations applied or having additional information, it is challenging to provide an accurate answer.

However, I can briefly explain some common transformations that could potentially be applied to a pentagon:

Translation: A translation involves moving the entire figure in a specific direction without changing its shape or orientation.

This transformation can be described by shifting the points of the pentagon horizontally or vertically.

Rotation: A rotation involves rotating the pentagon about a fixed point. The point of rotation could be inside or outside the pentagon, and the angle of rotation would determine the final position of the points.

Reflection: A reflection involves flipping the pentagon across a line, called the line of reflection.

Each point of the pentagon would be reflected across the line, resulting in a mirrored image.

Dilation: A dilation involves scaling the size of the pentagon by a certain factor.

This transformation can result in an enlarged or reduced version of the original pentagon.

Without specific details or information about the corresponding points in A''B''C''D''E'', it is not possible to determine the exact transformations applied.

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The required sample size to estimate the population proportion with 98% confidence and a margin of error of 1.5% is approximately 4,847.60.

To find the sample size required to estimate a population proportion with a 98% level of confidence, we will use the formula:

n = (z² * p * q) / E²

where:

n = sample size

z = z-value associated with the desired confidence level

p = population proportion - estimated

q = 1 - p (complement of the estimated proportion)

E = desired margin of error

We, first of all, will calculate the required sample size:

Given:

z = z-value corresponding to a 98% confidence level

E = 1.5% = 0.015

p = 28% = 0.28

q = 1 - p = 1 - 0.28 = 0.72

Then, we use a standard normal distribution table or a calculator to find the z-value. For a 98% confidence level, the z-value is approximately 2.326.

Putting the values into the formula:

n = (2.326² * 0.28 * 0.72) / 0.015²

n = (5.410 * 0.2016) / 0.000225

n≈ 1.091/0.000225

n ≈ 4,847.60

Hence, to estimate the population proportion with a 98% confidence level and a margin of error of 1.5%, a sample size of approx. 4,848.60 is required.

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Answer:

The three costs that contribute to the total cost of a mortgage are the down payment, mortgage tax and monthly interest.

Step-by-step explanation:

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Answer:

Step-by-step explanation:

The correct statement is:

C. The function has a y-intercept at (0, -2).

Let's analyze each statement:

A. As the value of x increases, the value of f(x) moves toward a constant.

This statement is false.

The function f(x) = (1/5)ˣ - 2 is an exponential function with a base of 1/5. As x increases, the exponential term (1/5)ˣ becomes smaller and approaches zero, causing f(x) to approach -2.

However, it does not approach a constant value.

B. The domain of the function is (-2, ∞).

This statement is false.

The function f(x) = (1/5)ˣ - 2 is defined for all real numbers.

There are no restrictions on the domain, so the correct statement is that the domain is (-∞, ∞).

C. The function has a y-intercept at (0, -2).

This statement is true.

To find the y-intercept, we set x = 0 in the function and solve for f(x):

f(0) = (1/5)⁰ - 2

= 1 - 2

= -1

Therefore, the y-intercept of the function is (0, -2).

D. The function is increasing.

This statement is true.

An exponential function with a positive base, such as (1/5)ˣ, is always decreasing.

However, when we subtract 2 from the exponential term, it shifts the graph vertically downward by 2 units.

This transformation makes the function f(x) = (1/5)ˣ - 2 increasing.

So, the correct statement is:

C. The function has a y-intercept at (0, -2).

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Topic Sentence: In a business setting, concepts from this course can be applied when optimizing customer service operations and improving customer satisfaction.

Supporting Detail: One concept from the course that can be applied is the use of queuing theory to manage customer wait times and reduce service bottlenecks. By analyzing arrival rates, service rates, and queue lengths, businesses can optimize staffing levels and allocate resources efficiently.

Worked Example: One practical application of queuing theory is in call centers. With the aid of this theory, the center can analyze the number of customer service reps required in different periods of the day. The objective is to keep waiting times at their lowest and deliver timely service to customers.

Supporting Detail: Another concept that can be applied is the concept of customer lifetime value (CLV). By understanding CLV, businesses can identify their most valuable customers, prioritize their needs, and tailor personalized marketing strategies to enhance customer retention.

Worked Example: An e-commerce business may use information on customer buying patterns, typical order amounts, and the rate of customer loss to determine the overall lifetime value of a customer. By utilizing this data, the organization has the opportunity to execute tailored loyalty initiatives and customized suggestions, ultimately enhancing customer contentment and optimizing overall revenue for the long haul.

Conclusion:

The application of course concepts in a business environment can result in better customer service, shorter wait times, higher customer approval, and increased customer allegiance, thereby fueling the expansion and profitability of a business.

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Answer:

79

Step-by-step explanation:

add this you have easily find answer

The cross section will be of rectangle in shape .

Given,

A cylinder with a circular base is sliced through the center vertically, perpendicular to the base.

Now,

A vertical cross section is a cross section in which the intersecting plane is perpendicular to the base of the solid.

If the intersecting plane is perpendicular to the base of cylinder then the shape obtained will be of rectangular in nature .

Therefore, vertical cross section of a cylinder is a rectangle.

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Let's denote crew A's work per week as A and crew B's work per week as B.

We are given that A = 58 ft per week and B = 59 ft per week.

We have to calculate how many weeks it will take both crews to complete the tunnel.

585 feet have already been dug from the south end. So, the total tunnel length that needs to be dug is 7605 - 585 = 7020.

Now, let's assume that both the crews continue to work at their respective rate of A = 58 ft/week and B = 59 ft/week.

Therefore, total tunnel length that needs to be dug in weeks = (7020 ft) / (A + B)

                       =  (7020 ft) / (58 ft/week + 59 ft/week)

                       =  120.20 weeks

Hence, it will take a total of 120.20 weeks for both crews to complete the tunnel.

Answer:

D

Step-by-step explanation:

A. a^1/8 x a^1/8 = a^2/8 = a^1/4

B. a^3/4 / a^1/2 = a^3/4 / a^2/4 = a^1/4

C. (a^1/8)^2 = a^2/8 = a^1/4

D. √a = a^1/2

The reason that is NOT typically used in a proof is two angles being congruent. Option C.

Among the options A, B, C, and D, the reason that is NOT typically used in a proof is option C: two angles being congruent.

Congruence is a fundamental concept in geometry that refers to the equality of shape and size. When two angles are congruent, it means they have the same measure. While congruence is an important concept used in geometric proofs, it is typically not used as a standalone reason in a proof.

In geometric proofs, various properties, theorems, and postulates are used to establish relationships and make deductions. Let's briefly discuss the other options:

A. Definition of supplementary angles: This is a valid reason that is commonly used in proofs involving supplementary angles. The definition states that if the sum of two angles is 180 degrees, they are considered supplementary.

B. Substitution property: The substitution property is a logical property used in algebraic proofs. It allows us to replace one expression with another expression that is equal to it. This property is often employed when simplifying equations or substituting known values.

D. Parallel lines: The concept of parallel lines is frequently utilized in geometric proofs, particularly in proving angles and angle relationships. Properties such as alternate interior angles, corresponding angles, and vertical angles are established based on the parallel lines. Option C is correct.

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The value of x in the given figure is 8.

In the given figure there are two lines which are parallel.

We have to find the value of x.

In the straight line the sum of angles is 180 degrees.

8x+1+115=180

8x+116=180

Subtract 116 from both sides:

8x=180-116

8x=64

Divide both sides by 8:

x=64/8

x=8

Hence, the value of x in the given figure is 8.

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A value of a that would make the expression 5 1/3 ÷ 4/a equivalent to a whole number include the following: C. 9.

In order to use the given expressions to determine the value of a (a-value) that makes the expression equivalent to a whole number, we would have to substitute the values of a (a-value or domain) into each of the expressions and then evaluate as follows;

First of all, we would re-write the mixed fraction as an improper fraction and then evaluate the given expression;

16/3 ÷ 4/a

16/3 × a/4

When a = 5, we have the following:

16/3 × a/4

16/3 × 5/4 = 20/3 (not a whole number).

When a = 7, we have the following:

16/3 × a/4

16/3 × 7/4 = 28/3 (not a whole number).

When a = 5, we have the following:

16/3 × a/4

16/3 × 9/4 = 36/3

36/3 = 12 (a whole number).

When a = 11, we have the following:

16/3 × a/4

16/3 × 11/4 = 44/3 (not a whole number).

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Complete Question:

Which value of a would make the expression 5 1/3 ÷ 4/a equivalent to a whole number?

5

7

9

11

To calculate the amount of trace element collected by Robert, we need to find the volume of the cylinder-shaped container. Rounded to the nearest tenth, Robert has collected approximately 211.4 grams of the trace element in the second sample.

To calculate the amount of trace element collected by Robert, we need to find the volume of the cylinder-shaped container and then multiply it by the concentration of the trace element.

The volume of a cylinder is given by the formula:

V = π * r^2 * h

where V is the volume, π is approximately 3.14159, r is the radius, and h is the height.

Given:

Radius (r) = 8.7 cm

Height (h) = 17.3 cm

Concentration of trace element = 0.052 grams/cm³

Substituting these values into the volume formula:

V = 3.14159 * (8.7 cm)^2 * 17.3 cm

V ≈ 4068.57196 cm³

Now, we can calculate the amount of trace element collected by multiplying the volume by the concentration:

Amount of trace element = V * concentration

Amount of trace element ≈ 4068.57196 cm³ * 0.052 grams/cm³

Amount of trace element ≈ 211.42941792 grams

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To find the common roots between two functions, we need to find the roots (or solutions) of each function individually and then identify the shared solutions.

For the function f(x) = x^2 + x - 6, we can find the roots by setting the function equal to zero and solving for x:

x^2 + x - 6 = 0

To factorize this quadratic equation, we need to find two numbers that multiply to -6 and add up to 1 (the coefficient of x). The numbers that satisfy these conditions are 3 and -2:

(x + 3)(x - 2) = 0

Setting each factor equal to zero:

x + 3 = 0 or x - 2 = 0

Solving for x in each equation:

x = -3 or x = 2

Therefore, the function f(x) = x^2 + x - 6 has two roots: x = -3 and x = 2.

To find the common roots between this function and another function, we would need to know the second function. If you provide the second function, I can help determine if there are any shared roots.

Given statement solution is :- The given expression √(-8 + 6i) does not have a real solution in the form w = a + bi, where a and b are real numbers.

To find the complex number in the form w = a + bi, where a and b are real numbers, we can solve the given expression.

Let's solve √(-8 + 6i) step by step:

First, we can express -8 + 6i in the form a + bi:

-8 + 6i = -8 + 6i + 0i = -8 + 6i + [tex]0i^2[/tex]

Now, we can take the square root of -8 + 6i:

√(-8 + 6i) = √((-8) + 6i + [tex]0i^2[/tex])

Since [tex]i^2[/tex] = -1, we can simplify the expression further:

√(-8 + 6i) = √(-8 - 6 + 6i)

= √(-14 + 6i)

Now, we want to express -14 + 6i in the form a + bi:

-14 + 6i = -14 + 6i + 0i = -14 + 6i +[tex]0i^2[/tex]

Taking the square root, we have:

√(-14 + 6i) = √((-14) + 6i + [tex]0i^2[/tex])

Since [tex]i^2[/tex] = -1, we can simplify the expression further:

√(-14 + 6i) = √(-14 - 6 + 6i)

= √(-20 + 6i)

Now, we want to express -20 + 6i in the form a + bi:

-20 + 6i = -20 + 6i + 0i = -20 + 6i + [tex]0i^2[/tex]

Taking the square root, we have:

√(-20 + 6i) = √((-20) + 6i + [tex]0i^2[/tex])

Since [tex]i^2[/tex] = -1, we can simplify the expression further:

√(-20 + 6i) = √(-20 - 6 + 6i)

= √(-26 + 6i)

Now, we want to express -26 + 6i in the form a + bi:

-26 + 6i = -26 + 6i + 0i = -26 + 6i +[tex]0i^2[/tex]

Taking the square root, we have:

√(-26 + 6i) = √((-26) + 6i + [tex]0i^2[/tex])

Since [tex]i^2[/tex] = -1, we can simplify the expression further:

√(-26 + 6i) = √(-26 - 6 + 6i)

= √(-32 + 6i)

We continue this process until we reach a point where the expression simplifies. However, in this case, we encounter an expression that cannot be simplified further, as the square root of a negative number is not defined in the set of real numbers.

Therefore, the given expression √(-8 + 6i) does not have a real solution in the form w = a + bi, where a and b are real numbers.

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For the given triangles:

(1) x = 5.4

(2)  x  =  √23

(3) θ = 25.84 degree

(4)   x = 9.5

(5) n = 41

(1) In the given triangle,

One angle = 16 degree

Perpendicular = x

Hypotenuse    = 20

Since we know that

Sinθ = opposite side of θ/hypotenuse

Therefore,

⇒ sin 16 = x/20

⇒   0.27 = x/20

⇒         x = 5.4

(2)  In the given triangle,

Hypotenuse = 12 km

Base = 11 km

perpendicular = x

We know that the Pythagoras theorem for a right angled triangle:

⇒  (Hypotenuse)²= (Perpendicular)² + (Base)²

⇒ (12)²= (x)² + (11)²

⇒ 144 = (x)² + 121

⇒   x² =  23

Taking square root both sides we get,

Hence,

⇒   x  =  √23

(3) In the given,

Base = 20

Perpendicular = 42

We know that the Pythagoras theorem for a right angled triangle:

⇒  (Hypotenuse)²= (Perpendicular)² + (Base)²

⇒ (Hypotenuse)² =  (42)² + (20)²

⇒ (Hypotenuse)² =  2164

Taking square root both sides,

⇒ (Hypotenuse)  = 46.51

⇒ cosθ = Adjacent/hypotenuse

             = 42/46.51

             = 0.90

Taking inverse of cosθ,

⇒ θ = 25.84 degree

(4) In the given triangle,

One angle = 30 degree

Base = x

Hypotenuse    = 11

Since we know that

cosθ = Adjacent/hypotenuse

Therefore,

⇒ cos 30  = x/11

⇒   √3/2 = x/11

⇒         x = 9.5

(4) In the given triangle,

Base = 40

Perpendicular = 9

Hypotenuse = n

We know that the Pythagoras theorem for a right angled triangle:

⇒  (Hypotenuse)²= (Perpendicular)² + (Base)²

⇒  n² = 9² + 40²

⇒  n² = 81 + 1600

⇒  n² = 1681

⇒  n = 41

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The volume of the prisms are:

1. 432 yd³

2. 36in³

3. 252 m³

4. 240 ft³

5. 576 mm³

6. 144 cm³

7. 343 m³

8. 120 yd³

9. 150 in³

The formula for calculating the volume of a rectangular prism is expressed as;

V = lwh

such that;

Now, substitute the value for each of the prisms, we have;

1. Volume = 6 × 6 ×12

Multiply

Volume = 432 yd³

2. Volume = 2 ×9 × 2

Multiply

Volume = 36in³

3. Volume = 9 × 4 × 7

Multiply

Volume = 252 m³

4. Volume = 10 × 8 × 3

Multiply

Volume = 240 ft³

5. Volume = 4 × 12 × 12

Multiply the values

Volume = 576 mm³

6. Volume = 6 × 8 × 3

Volume = 144 cm³

7. Volume = 7 × 7 ×7

Volume = 343 m³

8. Volume = 8 × 3 × 5

Volume = 120 yd³

9. Volume = 5 × 6 × 5

Volume = 150 in³

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The central angle of Bathing = 108 degrees.

In the given pie chart,

Since we know,

A circle is a closed, two-dimensional object where every point in the plane is equally spaced from a central point. The line of reflection symmetry is formed by all lines that traverse the circle. Additionally, every angle possesses rotational symmetry around the center.

The whole circle is 360 degrees

which represents 100%.

It is given that,

Bathing = 30%

So have need to find 30% of 360 degrees.

Therefore,

Bathing = (30/100)x360

            =  (30/10) x 36

            = 3x36

            = 108

Hence,

The central angle = 108 degrees.

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The complete question is:

To the nearest degree, what is the measure of the central angle for bathing?

The area of the figure as shown in the diagram is 30.64 cm².

The area of a figure is the number of unit squares that cover the surface of a closed figure. Area is measured in square units like cm² and m². Area of a shape is a two dimensional quantity.

To calculate the area of the figure, we use the formula below

Formula:

Where:

From the diagram,

Given:

Substitute these values into equation 1

Learn more about area here: https://brainly.com/question/28470545

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