brody and amanda canoed 1 1/2 hours before stopping to fish at 11:15 pm. at what time did amanda start canoeing.

Answer:

The 90% confidence interval to estimate the mean breaking weight for this type cable is between 751.38 lb and 803.42 lb.

Step-by-step explanation:

We are in posession of the sample's standard deviation, so we use the student t-distribution to solve this question.

The first step to solve this problem is finding how many degrees of freedom, we have. This is the sample size subtracted by 1. So

df = 47 - 1 = 46

90% confidence interval

Now, we have to find a value of T, which is found looking at the t table, with 46 degrees of freedom(y-axis) and a confidence level of [tex]1 - \frac{1 - 0.9}{2} = 0.95[/tex]. So we have T = 1.6787

The margin of error is:

M = T*s = 1.6787*15.5 = 26.02 lb

In which s is the standard deviation of the sample.

The lower end of the interval is the sample mean subtracted by M. So it is 777.4 - 26.02 = 751.38

The upper end of the interval is the sample mean added to M. So it is 777.4 + 26.02 = 803.42

The 90% confidence interval to estimate the mean breaking weight for this type cable is between 751.38 lb and 803.42 lb.

Answer:

[tex]777.4-1.679\frac{15.5}{\sqrt{47}}=773.60[/tex]    

[tex]777.4+1.679\frac{15.5}{\sqrt{47}}=781.20[/tex]    

We are 90% confident that the true mean for the breaking weigth for this typpe of cable is between 773.60 and 781.20

Step-by-step explanation:

Data provided from the problem

[tex]\bar X=777.4[/tex] represent the sample mean for the breaking weights

[tex]\mu[/tex] population mean

s=15.5 lb represent the sample standard deviation

n=47 represent the sample size  

Confidence interval

The confidence interval for the true population mean is given by:

[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]   (1)

The degrees of freedom, given by:

[tex]df=n-1=47-1=46[/tex]

The Confidence level is 0.90 or 90%, the significance would be [tex]\alpha=1-0.9=0.1[/tex] and [tex]\alpha/2 =0.05[/tex] and the critical value for this case is [tex]t_{\alpha/2}=1.679[/tex]

Replacing into the confidence formula we got:

[tex]777.4-1.679\frac{15.5}{\sqrt{47}}=773.60[/tex]    

[tex]777.4+1.679\frac{15.5}{\sqrt{47}}=781.20[/tex]    

We are 90% confident that the true mean for the breaking weigth for this typpe of cable is between 773.60 and 781.20

Answer:

$150

Step-by-step explanation:

Annual Real Estate Tax Rate =1.8%

Value of Bonnie's House = $100,000

Annual Tax= 1.8 % × $100,000 =0.018 × $100,000 = $1,800

Therefore, tax payment on a monthly basis

= $1,800÷12 Months

= $150 per month.

Real estate tax will add $150 per month to Bonnie's mortgage payment.

Final answer:

The real estate taxes on Bonnie's $100,000 house at a 1.8 percent tax rate add $150 to her monthly mortgage payment, after calculating the annual tax and dividing by 12 months.

Explanation:

To calculate the real estate taxes that will add to Bonnie's monthly mortgage payment, we need to apply the tax rate to the value of the home and then divide by 12 to find the monthly amount.

The annual real estate taxes would be calculated as follows:

Therefore, the real estate taxes will add $150 to Bonnie's monthly mortgage payment.

The number at the end of the equation shifts the graph up or down.

Changing the 1/2 to a -2 would shift the graph down which would change the y-intercept.

The answer is D.

Answer:

Step-by-step explanation:

You can write the linear system model as follows. Let t, l, c represent acres of tomatoes, lettuce, and carrots, respectively. The we have ...

  t + l + c ≤ 120 . . . . . constraint on available land

  5t +4l +2c ≥ 480 . . requirement for spending on fertilizer

  4t +2l +2c ≤ 600 . . constraint on available labor

  t ≥ 0; l ≥ 0; c ≥ 0 . .  requirement for non-negative acres

Then the objective function (profit) is ...

  p = 3000t +1400l +400c

The linear programming problem is to maximize p subject to the above constraints.

__

Any of a variety of solvers can find the solution to this problem. That solution is ...

  (t, l, c) = (120, 0, 0) and p = 360,000

In summary ...

  tomatoes acre(s) = 120 (all available acres are used)

  lettuce acre(s) = 0

  carrots acre(s) = 0

  profit $ = $360,000

_____

Additional comments

This solution suggests that it can be found simply by examining the profit associated with each unit of resource.

  profit per acre is maximized for tomatoes, at $3000 per acre

  profit per fertilizer dollar is maximized for tomatoes, at $600 per dollar

  profit per labor hour is maximized for tomatoes, at $750 per hour

That is, the profit per acre is maximized for tomatoes, regardless of the resource being considered. Thus it make sense to put all of the acreage in tomatoes. At $5 per acre for fertilizer, we use $600 worth of fertilizer. At 4 hours per week per acre, we use 480 hours of labor, so not all available labor is used.

To maximize profits, use linear programming to find the number of acres for each crop. The total farm acres used depends on the constraints.

To maximize total profits, the number of acres of each crop should be determined. Let's assign the variables t, l, and c to represent the number of acres of tomatoes, lettuce, and carrots, respectively. We have the following constraints:

The objective is to find the values of t, l, and c that satisfy the constraints and maximize the profit. This is a linear programming problem that can be solved using methods such as the simplex method or graphical method.

To determine if all 120 acres will be used, we need to check if t + l + c = 120 under the constraints.

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Answer:

By the Empirical Rule, the approximate percentage of cars that remain in service between 69 and 74 months is 13.5%.

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed(bell-shaped) random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 64

Standard deviation = 5.

Using the empirical rule, what is the approximate percentage of cars that remain in service between 69 and 74 months?

69 = 64 + 1*5

So 69 is one standard deviation above the mean.

74 = 64 + 2*5

So 74 is two standard deviations above the mean.

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

This means that:

95 - 68 = 27% between 1 and 2 standard deviations of the mean.

The normal distribution is symmetric, which means that 27/2 = 13.5% of those are below the mean(within 1 and 2 standard deviations below the mean) and 13.5% of those are above the mean(between 69 and 74).

So

By the Empirical Rule, the approximate percentage of cars that remain in service between 69 and 74 months is 13.5%.

Answer:

V = (10*6*12)/3

V = 720/3

V = 240 cm^3

Step-by-step explanation:

V = LWH/3

L = base length

W = base width

H = pyramid height

Answer:

The students have eaten 72 ounces of fruit snacks.

X = 72 ounces

Step-by-step explanation:

There are total 20 packs of fruit snacks

The weight of each pack is 6 ounces.

Let us first find the total weight of the snacks.

Total weight = 6*20

Total weight = 120 ounces

Let X represents the ounces of fruit snacks that students have eaten.

Let Y represents the 48 ounces of fruit snacks left in the box

Total weight = X + Y

120 = X + 48

X = 120 - 48

X = 72 ounces

Therefore, the students have eaten 72 ounces of fruit snacks.

Answer:

x = 72 ounces

The number of ounces of fruit snacks the students ate is 72 ounces

Step-by-step explanation:

Given;

Let x represent the number of ounces of fruit the students ate;

The total number of ounces of fruit initially in the box is;

20 packs × 6 ounces/pack = 120 ounces

The remaining ounces of fruit is;

48 ounces

Therefore;

120 - x = 48

Solving for x;

x = 120 - 48

x = 72 ounces

The number of ounces of fruit snacks the students ate is 72 ounces

Answer: 1/2

Step-by-step explanation:

Answer:

14.1 u^2

Step-by-step explanation:

A=pi*6^2*45/360=3.14*36*45/360=3.14*45/10≈14.1

Answer: 4) 50 minutes walking , 50 minutes basketball and 6) 70 minutes walking, 60 minutes basketball

Joshua can distribute his time among walking and basketball in a variety of ways.

Without any constraints on total time spent, all provided options, from 40 minutes each for walking and basketball to 70 minutes walking and 60 minutes basketball, are considered valid solutions.

The problem seems to deal with the way Joshua divides his time between two activities, which are walking and basketball. As the question does not state a limit to the total time Joshau can spend on activities, we can assume each mentioned time division can be considered a possible solution.

Therefore, under this assumption, all provided options are valid as Joshua can spend:

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Fractions not equivalent to 7/8 are those that do not share the same 7 to 8 ratio when simplified. Multiplying by the reciprocal is key to understanding fraction division, and it is important to comprehend fraction operations for proficiency in mathematics. Educational systems need to improve the teaching of fractions, as evidenced by standardized test results.

The fractions that are not equivalent to 7/8 are any fractions where the numerator and denominator do not have the same ratio as 7 to 8. For example, the fraction 14/16 is equivalent to 7/8 because if you divide both 14 and 16 by the same number, which is 2, you will get 7/8. However, fractions like 1/2, 6/8, or 9/10 are not equivalent to 7/8. To determine if fractions are equivalent, you can simplify them to their lowest terms or cross-multiply to compare ratios.

Turning to common misconceptions about fractions, remember that multiplication and division are intimately linked: dividing by 8 is the same as multiplying by its reciprocal (1/8), and vice versa. Understanding this concept helps to build a strong foundation for working with fractions. Furthermore, not all fractions of a total are equal, and understanding the mechanics of adding, subtracting, multiplying, and dividing fractions is essential for developing proficiency in mathematics.

As illustrated in the example with 1,000 and 1/8, recognizing the patterns and relationships between fractions and other numbers aids in solving problems quickly and accurately. Educational systems evidently need to better equip students with these essential skills for handling fractions and percentages, as highlighted by the results of the High School Exit Exam.

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Fractions that are not equivalent to 7/8 include 3/4, 1/2, 2/3, 1/3, and 1/5. These fractions represent different values when simplified.

To determine which fractions are not equivalent to 7/8, we need to understand the concept of equivalent fractions. Two fractions are equivalent if they represent the same part of a whole even if their numerators and denominators are different.

For example, 14/16 is an equivalent fraction to 7/8 because if you simplify 14/16, you divide both the numerator and the denominator by 2, which equals 7/8. However, some fractions that are not equivalent to 7/8 include:

These fractions do not simplify to 7/8 and therefore represent different values.

Answer:

16% of the apples have diameters that are less than 6.82cm

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed(bell-shaped) random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 7.25cm

Standard deviation = 0.43cm

The normal distribution is symmetric, which means that 50% of the measures are below the mean and 50% are above.

Using the empirical rule, what percentage of the apples have diameters that are less than 6.82cm?

6.82 = 7.25 - 1*0.43

So 6.82 is one standard deviation below the mean.

Of the 50% of the diameters below the mean, 68% are within 1 standard deviation of the mean.

So 100-68 = 32% are not within 1 standard deviation of the mean.

0.5*0.32 = 0.16

16% of the apples have diameters that are less than 6.82cm

Answer:

This study is an experiment.

Step-by-step explanation:

This is a case of experiment, as the visitors are assigned randomly to one of the two advertisements. The researcher intervene introducing a variable (the type of advertisement) and then study the results.

If it was an observational study, the researcher would observe, for example, the behaviour of the visitor without defining or controlling the exposure to one of the advertisement for a group of visitors.

Answer:(x-5)^2+(y-4)^2=49

Step-by-step explanation:

equation of a circle is:

(x-h)^2+(y-k)^2=r^2

(x-5)^2+(y-4)^2=7^2

(x-5)^2+(y-4)^2=7x7

(x-5)^2+(y-4)^2=49

The equation of the circle of center (5, 4) and radius 7, is:

(x - 5)^2 + (y - 4)^2  = 49

The general equation for a circle of radius R centered at the point (a, b) is:

(x - a)^2 + (y - b)^2 = R^2

In this case, the center is (5, 4) and the radius is 7, so we need to replace:

a = 5

b = 4

R = 7

Then the equation is:

(x - 5)^2 + (y - 4)^2 = 7^2 = 49

If you want to learn more about circles, you can read:

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Answer:

C - 7

Step-by-step explanation:

A^2-B^2

Let A = 3 and B=-4

(3)^2 - (-4)^2

9 - 16

-7

Answer:

19

Step-by-step explanation:

We'll begin by calculating the mean of the data. This is illustrated below:

Mean = summation of data / number of data

Data from the question: 9, 13, 43, 55

Mean = (9 + 13 + 43 + 55)/4

Mean = 120/4

Mean = 30.

Now, we can calculate the mean absolute deviation (MAD) as follow:

MAD = summation of the absolute deviation from the mean / number of data.

MAD = [ |9–30| + |13–30| + |43–30| + |55–30| ] /4

MAD = [ 21 + 17 + 13 + 25] /4

MAD = 76/4

MAD = 19.

Therefore, the mea absolute deviation (MAD) of their ages is 19

Answer:

19

Step-by-step explanation:

Final answer:

The desired exponential function that goes through the points (0,18) and (2,288) is y=18 × 4^x.

Explanation:

To write an exponential function in the form y = ab^x that passes through the points (0,18) and (2,288), we first use the given points to find the values of a and b.

When x = 0, the equation simplifies to y = a because b0 equals 1. Therefore, from the point (0,18), we can conclude that a equals 18.

Next, substituting x = 2 and y = 288 into the equation and using the value of a, we have 288 = 18 × b^2. To find b, divide both sides by 18 to get b^2 = 16, which means b = 4 since b is positive in the growth of an exponential function.

So the desired exponential function is y = 18 × 4^x.

Answer:

(4, -1)

Step-by-step explanation:

The point of intersection of two lines is also the solution to the system of equations of the two lines

Step-by-step explanation:

The answer to this question is where both of the lines intersect.  For this problem, the answer is (4, -1)

Look at the attachment for clarification:

Answer:

5

Step-by-step explanation:

2 log₃(x + 4) − log₃9 = 2

2 log₃(x + 4) − log₃3² = 2

2 log₃(x + 4) − 2 = 2

2 log₃(x + 4) = 4

log₃(x + 4) = 2

3^(log₃(x + 4)) = 3^2

x + 4 = 9

x = 5

The solutions to the logarithmic equation 2 log₃(x + 4) - log₃(9) = 2 are x = 5 and x = -13.

To solve the logarithmic equation 2 log₃(x + 4) - log₃(9) = 2, we can use logarithmic properties to simplify the equation and then solve for x.

First, let's apply the properties of logarithms:

2 log₃(x + 4) - log₃(9) = 2

Using the power rule of logarithms, we can rewrite 2 log₃(x + 4) as log₃((x + 4)²):

log₃((x + 4)²) - log₃(9) = 2

Next, we can combine the logarithms using the quotient rule:

log₃((x + 4)² / 9) = 2

Now, we can eliminate the logarithm by rewriting the equation in exponential form:

3² = (x + 4)² / 9

9 = (x + 4)² / 9

Multiplying both sides of the equation by 9:

81 = (x + 4)²

Taking the square root of both sides:

±√81 = ±√((x + 4)²)

±9 = x + 4

Solving for x, we have two possible solutions:

x + 4 = 9

x = 9 - 4

x = 5

or,

x + 4 = -9

x = -9 - 4

x = -13

Therefore, the solutions to the logarithmic equation 2 log₃(x + 4) - log₃(9) = 2 are x = 5 and x = -13.

Learn more about logarithmic equation click;

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Answer:

Step-by-step explanation:

Hello!

The variable of interest is

X: Systolic blood pressure of a U.S. adult. (mmHg)

X~N(μ;σ²)

μ= 122 mmHg

σ= 20 mmHg

For all calculations you have to work under the standard normal distribution, because it is tabulated. It is easier and faster to standardize or "translate" all values of X into values of Z and look for the corresponding probabilities in the table than to manually calculate them using the density function of the normal distribution.

The standard normal distribution is derived from the normal distribution. Considering a random variable X with normal distribution, mean μ and variance δ², the variable Z =(X-μ)/δ ~N(0;1) is determined.

Any value of any random variable X with normal distribution can be "converted" by subtracting the variable from its mean and dividing it by its standard deviation.

Since the distribution is centered in zero, there are two entries for its table, the left entry shows the cumulated probabilities corresponding to negative values of Z: P(Z≤z)=α and the right entry show the cumulated probabilities corresponding to positive values of Z: P(Z≤z)= 1 - α

a)

You need to calculate the percentage/ proportion of U.S. adults that have a systolic pressure greater than 142 mmHg, symbolically:

P(X>142) = 1 - P(X≤142)

To standardize this value od the variable you have to do the following calculation:

Z =(X-μ)/δ= (142-122)/20= 1

Now you look in the right entry for the cumulative probability until z= 1.00

(Remember, the first column of the table shows you the integer and first digit, the first row of the table shows you the second integer of the z value)

P(Z≤1)= 0.84134

Now you calculate the asked value:

P(X>142) = 1 - P(X≤142)= 1 - P(Z≤1)= 1 - 0.84134= 0.15866

b)

The sample mean is derived from a random variable with a normal distribution it shares that distribution with the exception that its variance is directly affected by the sample size:

X[bar]~N(μ;σ²/n)

μ= 122 mmHg

σ/√n= 20/√29= 3.71 mmHg

c)

The claim is that the mean systolic pressure of the employees is higher than the national average, symbolically μ> 122

The statistical hypotheses are:

H₀: μ≤ 122

H₁: μ> 122

d)

t= 5.495 p-value: 0.000003591

In this example the test statistic depends on the mean and the p-value is 3.591*10⁻⁶. This value indicates that 0.0003591% of the samples with size n=29 taken from a population with mean 122 mmHg, will produce a mean that provides evidence as (or stronger) than the current sample that μ is not at most 122 mmHg.

e)

The type II error is the scenario when you fail to reject the null hypothesis when the hypothesis is false. In this case, it is to fail to reject that the average systolic pressure of the company's employees is at most as the national average.

f)

The variable "X: Age of an employee" was recorded and linear regression of the systolic blood pressure as a function of the employee's ages estimated.

^Y= 97.0771 + 0.9493Xi

0.9493mmHg/years is the modification of the estimated average systolic blood pressure of the company's employees when their age increases one year.

g)

To determine the type of linear regression between the two variables, you have to analyze the slope of the equation:

As you can see in the graphic, the slope of the regression is positive, which means there is a positive regression between the systolic pressure and the age of the employees. I.e. each time the age of the employee increases, his systolic pressure also increases.

To determine the strength of the regression between these two variables you have to analyze the coefficient of determination R²:

The coefficient of determination gives you an idea of how much of the variability of the dependent variable (Y) is due to the explanatory variables under the estimated regression. It takes values between 0 to 1 or 0 to 100% if expressed in percentage. The closer the coefficient is to zero, the weaker the relationship between these two variables.

The closer the coefficient is to 100%, the stronger the relationship between these two variables.

R²= 0.712

71.2% of the variability of the systolic pressure is explained by the age fo the employees under this estimated model ^Y= 97.0771 + 0.9493Xi

The relationship between these variables is strong enough to consider the regression.

I hope this helps!

Answer: The answer is C (about 47%) because the total amount of Mountain View residents is 0.38, 0.18 people chose Olunloyo, which is about 47 percent of residents.

To find out what percentage of the Mountain View polled supported Olunloyo, divide the number of Olunloyo supporters by the total number of poll participants, and multiply by 100. This reflects the views of the polled sample, not the entire population.

To determine what percentage of the Mountain View polled supported Olunloyo, you would first need to know the total number of people who participated in the poll from Mountain View. Then, find out the number of these participants who indicated their support for Olunloyo. Divide the number of supporters by the total number of participants, and multiply the result by 100. This will give you the percentage of Mountain View residents in the poll who supported Olunloyo. Remember that this only applies to the sample of the population that participated in the poll and may not reflect the entire population of Mountain View.

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Answer:

$676

Step-by-step explanation:

26 percent of 2600 is 676.

2600 x .26

Answer:585 cubic inch

Step-by-step explanation:

volume=15 x 10 x 3 + 9 x 3 x 5

Volume=450 + 135

Volume=585

Answer:

C -9a - 15

Step-by-step explanation:

bionomial means two terms

Answer:

The answer is  " [tex]C= \frac{5}{2x^2}+ 4xy - 2y^4.[/tex]"

Step-by-step explanation:

It given that, [tex]\frac {\partial(5x + 4y)}{\partial y} = \frac{\partial(4x- 8y^3)} {\partial x} \\\\[/tex] it is the differential equation.

Solve the above equation.

[tex]\frac {\partial M (x, y)}{\partial y} = (5x+ 4y)[/tex]

So, [tex]M (x, y) = \frac{5}{2x^2}+4xy +\varphi (y).[/tex]

[tex]\frac{\partial M (x, y)}{ \partial y }= 4x- 8y^3 \ and \ \ 4x + \varphi ' (y) = 4x- 8y^3 \\\\\ so, \ \\varphi ' (y) = -8y^3[/tex]

or [tex]\varphi (y) = -2y^4+c[/tex]

To find the general solution

[tex]M(x, y )= c \\C= \frac{5}{2x^2}+ 4xy - 2y^4.[/tex]

Answer:

Step-by-step explanation:

Hello!

To study time perception, 8 students measured one minute (counted to 60), the given data show the actual time (in seconds) it took each student to measure one minute:

53, 52, 75, 62, 68, 58, 49, 49

8 observations make this sample, n=8

You can summarize the observed data using descriptive statistics:

Central tendency measures:

Median (Me):

The median is the value that separates the sample in a half ( the bottom 50% from the top 50%). To calculate it you have to calculate its position:

(for even samples) PosMe: n/2= 8/2= 4

Then you arrange the data from least to greatest and identify the 4th value:

49, 49, 52, 53, 58, 62, 68, 75

Me= 53s

Mode (Md):

The mode is the most observed value, i.e. the value with the most absolute frequency:

In this data set, the only value that was observed more than once is 49, so that is the value of the mode

Md= 49s

Mean (X[bar]):

Is the average or expected value of the data set, it is always within the range of definition of the variable but it doesn't necessarily correspond to and observation. You calculate it as:

X[bar]= ∑X/n= (53+52+75+62+68+58+49+49)/8= 466/8= 58.25s

Measures of dispersion:

Range (R):

The is the difference between the minimum and maximum observations of the sample.

R= max - min = 75 - 49= 26s

Standard deviation (S):

It shows the variability between the values of the sample. A low standard deviation tells that the observations are close to the meanwhile a high standard deviation show that the values are further away from the mean.

∑X= 466

∑X²= 27772

[tex]S= \sqrt{(\frac{1}{n-1})[sumX^2-\frac{(sumX)^2}{n} ]= \sqrt{(\frac{1}{7} )[27772-\frac{(466)^2}{8} ]} = \sqrt{\frac{1255}{14} } = 9.47s[/tex]

S= 9.47s

I hope this helps!

Answer:

6 días (6 days)

Step-by-step explanation:

El número de horas-hombre es directamente proporcional a la cantidad de sillas producidas. Entonces: (The number of man-hours is inversely proportional to the quantity of manufactured chairs. Then:)

[tex]\frac{(16\,p)\cdot \left(9\,\frac{h}{d\cdot p} \right)\cdot \left(12\,d\right)}{60\,c} = \frac{(40\,p)\cdot \left(8\,\frac{h}{d\cdot p} \right)\cdot x}{60\,c}[/tex]

La cantidad de sillas producidas es (The quantity of manufactures chairs is):

[tex]x = \frac{\left(16\,p\right)\cdot \left(9\,\frac{h}{d\cdot p} \right)\cdot \left(12\,d\right)}{(40\,p)\cdot \left(8\,\frac{h}{d\cdot p} \right)}[/tex]

[tex]x = 5.4\,days[/tex]

Lo cual significa que se necesitan 6 días (Which mean that 6 days are needed).

Answer:

The value of x is equal to 26 cm.

Step-by-step explanation:

We can use the Pythagorean Theorem to solve this problem.

[tex]a^2+b^2=c^2[/tex]

[tex]a=10\\b=24\\c=?[/tex]

[tex]10^2+24^2=c^2[/tex]

[tex]100+576=c^2[/tex]

[tex]676=c^2[/tex]

[tex]\sqrt{676} =\sqrt{c^2}[/tex]

[tex]26=c[/tex]

Answer:

Polygons - Hexagons. So, the sum of the interior angles of a hexagon is 720 degrees. All sides are the same length (congruent) and all interior angles are the same size (congruent).

Step-by-step explanation:

Search google

Final answer:

Two regular hexagons are congruent when they have equal side lengths, making them identical in shape and size. The definition of a regular hexagon as having equal sides and angles, paired with the criteria for congruence, confirms this. However, without equal side lengths, hexagons may only be similar, not congruent.

Explanation:

The question of whether two regular hexagons are always congruent comes down to understanding the definition of both a regular hexagon and congruence in geometry. A regular hexagon is a six-sided polygon where all sides and angles are equal. Congruent figures, on the other hand, are those that are identical in shape and size, meaning their corresponding sides and angles are equal. Thus, two regular hexagons that have the same lengths of sides are indeed congruent because they will have identical shapes and sizes, fulfilling the criteria for congruence.

However, it's important to note that while all regular hexagons will be similar (having the same shape but not necessarily the same size), they are only congruent if their side lengths are equal. This distinction is key in understanding the conditions under which two regular hexagons can be considered congruent. Without the specification of size, two hexagons can still be similar but not congruent.

In summary, two regular hexagons are congruent when they have the same side lengths, hence matching in both shape and size. This specific condition must be met for congruence to hold between any two geometric figures, including regular hexagons.