The edge of a cube-shaped box is 1 yard long. Three students each made an observation about the box. • Janae said that the area of each face of the box is 9 square feet. • Archie said that the perimeter of each face of the box is 3 feet. • Gail said that the volume of the box is 1 cubic yard. Whose observations about the box are correct?​

Answer:

Least Common Multiple (LCM) of 28,42,63 is 252 ∴ So the LCM of the given numbers is 2 x 3 x 7 x 2 x 1 x 3 = 252

Step-by-step explanation:

Answer:

252 is the answer

Step-by-step explanation:

find the multiples of all of them ( and make sure it is the least. )

28:

28, 56, 84, 112, 140, 168, 196, 224, 252, 280, 308

42:

42, 84, 126, 168, 210, 252, 294, 336

63:

63, 126, 189, 252, 315, 378

bolded + undurlined is the answer

you see that 252 is the answer

252 is the answer

The equation [tex]\frac{y+6}{y-1}+\frac{y-4}{y^2-y} = \frac{1}{y-1}[/tex] when solved for y is -3 ± √13

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

[tex]\frac{y+6}{y-1}+\frac{y-4}{y^2-y} = \frac{1}{y-1}[/tex]

Simplify the denominators

So, we have

[tex]\frac{y+6}{y-1}+\frac{y-4}{y(y-1)} = \frac{1}{y-1}[/tex]

This gives

y + 6 + (y - 4)/y = 1

Subtract 1 from both sides

y + 5 + (y - 4)/y = 0

So, we have

y² + 5y + y - 4 = 0

Evaluate

y² + 6y - 4 = 0

When solved, we have

[tex]y = \frac{-b \pm \sqrt{b^2 -4ac} }{2a}[/tex]

So, we have

[tex]y = \frac{-6 \pm \sqrt{6^2 -4(1)(-4)} }{2(1)}[/tex]

Evaluate

[tex]y = \frac{-6 \pm \sqrt{52} }{2}[/tex]

Evaluate

[tex]y = -3 \pm \sqrt{13}[/tex]

Hence, the solution is -3 ± √13

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The surface area of pyramid B to the nearest hundredth is 58.67 cm²

Similar figures are two figures having the same shape. The ratio of the corresponding sides of similar shapes are equal.

The scale ratio of the height of the pyramid A to B is

9/6 = 3/2

Area factor = (3/2)² = 9/4

9/4 = 132/x

9x = 132×4

9x = 528

divide both sides by 9

x = 528/9

x = 58.67cm²

Therefore the surface area of pyramid B is 58.67cm².

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Using the first three non-zero terms of a Maclaurin series, the estimated value of the integral from 0 to 1 of the cosine of x squared, dx is 0.905

To estimate the integral from 0 to 1 of the cosine of x squared, dx using the first three non-zero terms of a Maclaurin series, we first need to find the Maclaurin series for cosine of x squared:

cos(x^2) = 1 - x^4/2! + x^8/4! - x^12/6! + ...

The first three non-zero terms are 1, -x^4/2!, and x^8/4!.

Now we can use these terms to estimate the integral from 0 to 1:

∫₀¹ cos(x²) dx ≈ ∫₀¹ [1 - x^4/2! + x^8/4!] dx

Integrating term by term, we get:

∫₀¹ [1 - x^4/2! + x^8/4!] dx ≈ [x - x^5/5! + x^9/9!] from 0 to 1

Plugging in 1 and 0, respectively, and simplifying, we get:

[x - x^5/5! + x^9/9!] evaluated at x=1 - [x - x^5/5! + x^9/9!] evaluated at x=0

= [1 - 1/120 + 1/6561] - [0 - 0 + 0]

≈ 0.905

Therefore, the estimated value of the integral from 0 to 1 of the cosine of x squared, dx using the first three non-zero terms of a Maclaurin series is 0.905.

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Mr. Ream has 7 cups of paint left in the bowl.

There are 12 cups of paint, which is equivalent to 192 fluid ounces of paint (1 cup = 16 fluid ounces). Mr. Ream filled 40 small dishes with 2 fluid ounces of paint each, for a total of 80 fluid ounces of paint used (40 x 2 = 80).

Therefore, Mr. Ream has 192 - 80 = 112 fluid ounces of paint left in the bowl.

To convert this to cups, we divide by 16 (1 cup = 16 fluid ounces):

112/16 = 7 cups

Therefore, Mr. Ream has 7 cups of paint left in the bowl.

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This loan has an interest 4. 5% compounded quarterly, account balance after 10 years:

The initial loan amount is $20,000, and it has an interest rate of 4.5% compounded quarterly. You would like to know the account balance after 10 years.

To calculate the account balance, we will use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:

A = the future value of the loan
P = the initial loan amount ($20,000)

r = the annual interest rate (0.045)
n = the number of times the interest is compounded per year (4, since it is compounded quarterly)

t = the number of years (10)

Plugging in the values:

A = 20000(1 + 0.045/4)^(4*10)

A = 20000(1.01125)^40

A ≈ 30,708.94

The account balance after 10 years will be approximately $30,708.94.

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The sum of the measures of the interior angles is 1260 degrees

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

An interior angle = 140 degrees

The sum of interior angles of a regular polygon is:

S = (n - 2) × 180

Where n is the number of sides of a polygon

Also, we have

S = 140n

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

140n = (n - 2) × 180

Expand

140n = 180n - 360

This gives

40n = 360

Divide

n = 9

Recall that

S = (n - 2) × 180

So, we have

S = (9 - 2) × 180

Evaluate

S = 1260

Hence, the sum of the measures of the interior angles is 1260 degrees

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Answer:

As x approaches ∞, y approaches -∞

As x approaches -2.5, y approaches ∞

Step-by-step explanation:

Determine which proportion is correct, we will compare the cross products of each proportion. The correct proportion will have equal cross products. The correct proportion is 1/2 = 3/6.

1. 4/10 = 3/6
To check this proportion, we'll calculate the cross products:
(4 * 6) = (10 * 3)
24 = 30
Since 24 ≠ 30, this proportion is incorrect.
2. 1/2 = 7/8
To check this proportion, we'll calculate the cross products:
(1 * 8) = (2 * 7)
8 = 14
Since 8 ≠ 14, this proportion is incorrect.
3. 1/2 = 3/6
To check this proportion, we'll calculate the cross products:
(1 * 6) = (2 * 3)
6 = 6
Since 6 = 6, this proportion is correct.
4. 4/10 = 7/8
To check this proportion, we'll calculate the cross products:
(4 * 8) = (10 * 7)
32 = 70
Since 32 ≠ 70, this proportion is incorrect
So, the correct proportion is 1/2 = 3/6.

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Median time change in 0.5 minutes if she added another radio station playing 37 minutes

To determine how many minutes the median time would change after adding a radio station playing 37 minutes of music per hour, follow these steps:

1. Arrange the given data in ascending order:
30, 31, 32, 33, 34, 35, 36

2. Find the median of the original data:
There are 7 data points, so the median is the middle value: 33 minutes.

3. Add the new radio station data (37 minutes) and arrange in ascending order:
30, 31, 32, 33, 34, 35, 36, 37

4. Find the new median after adding the radio station:
There are now 8 data points, so the median is the average of the two middle values (32 and 33): (32 + 33) / 2 = 32.5 minutes.

5. Determine the change in the median:
New median (32.5) - Original median (33) = -0.5

So, by adding another radio station playing 37 minutes of music per hour, her median time would change by -0.5 minutes (or decrease by 0.5 minutes). The correct answer is B. 0.5 minutes.

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Answer:

Step-by-step explanation:

The value of 2s in 43,290 is 2,000, while the value of 2s in 32,865 is 20.

B. 200 is 1/20 the value of 2,000.

This statement is correct, as 200 is 1/10 of 2,000, and there are two 0s in the value of 2s in 43,290 compared to one 0 in the value of 2s in 32,865.

The value of each variable in the parallelogram are g = 61 and h = 9

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

The parallelogram

The opposite angles of a parallelogram are equal

So, we have

g + 4 = 65

This gives

g = 61

Next, we have

16 - h = 7

So, we have

h = 16 - 7

Evaluate

h = 9

Hence, the value of each variable in the parallelogram are g = 61 and h = 9

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The subscriptions of magazines you need to sell is at least 7

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

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

f(x) = 130x + 90

In order to make at least $1000.00 each week, we have

130x + 90 = 1000

So, we have

130x = 910

Divide by 130

x = 7

Hence, the number of orders is 7

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The difference between the two banks is you will make $367 more at Wells Fargo than at TD Bank, to determine which bank has the better savings program, let's compare the total interest earned at TD Bank and Wells Fargo.

TD Bank offers a 6% interest rate for 7 years, while Wells Fargo offers a 5% interest rate for 9 years. If you deposit $10,000, we can calculate the total interest earned at each bank using the formula for compound interest:

A = P(1 + r/n)^(nt)

Where A is the future value of the investment, P is the principal amount ($10,000), r is the annual interest rate, n is the number of times interest is compounded per year, t is the number of years, and "^" denotes exponentiation.

Assuming annual compounding (n = 1), the calculations for each bank are as follows:

TD Bank:
A = $10,000(1 + 0.06/1)^(1*7)
A = $10,000(1.06)^7
A = $15,018.93

Wells Fargo:
A = $10,000(1 + 0.05/1)^(1*9)
A = $10,000(1.05)^9
A = $15,386.16

Comparing the two banks, Wells Fargo's savings program will make you the most money, with a future value of $15,386.16. The difference between the two banks is:

Difference = Wells Fargo - TD Bank
Difference = $15,386.16 - $15,018.93
Difference = $367.23

Rounding to the nearest dollar, you will make $367 more at Wells Fargo than at TD Bank. Therefore, Wells Fargo has the better savings program in this scenario.

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The probability that both events will occur is 0.22.

It is the chance of an event to occur from a total number of outcomes.

The formula for probability is given as:

Probability = Number of required events / Total number of outcomes.

Example:

The probability of getting a head in tossing a coin.

P(H) = 1/2

We have,

The probability of Event A is 2/6 or 1/3

(since there are two ways to get a 1 or 2 on a six-sided die).

The probability of Event B is 4/6 or 2/3

(since there are four ways to get a number 4 or less on a six-sided die).

Using the formula for the probability of the intersection of two independent events.

P(A and B)

= P(A) x P(B)

= (1/3) x (2/3)

= 2/9

Rounded to the nearest hundredth,

The probability that both events will occur is 0.22.

Thus,

The probability that both events will occur is 0.22.

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The truck would have lost 36% of its initial value.

The value in dollars, v(x), of a certain truck after x years can be represented by a mathematical function or equation. In the absence of a specific equation, it is difficult to provide an answer.

However, I can provide an example of a possible equation that represents the depreciation of a truck's value over time.

Let's assume that the truck loses 20% of its value every year. If the initial value of the truck is V0 dollars, then the value of the truck after x years, Vx, can be represented by the following equation:

Vx = [tex]V0(0.8)^x[/tex]

In this equation, the term [tex](0.8)^x[/tex] represents the percentage of the truck's value that remains after x years of depreciation. For example, after one year, the truck's value would be V1 = [tex]V0(0.8)^1[/tex] = 0.8V0,

which means that the truck would have lost 20% of its initial value. After two years, the truck's value would be V2 = V0[tex](0.8)^2[/tex]= 0.64V0,

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a) To determine if the limit exists, we need to check if the limit from all directions approaching (0,0) are equal. Let's approach (0,0) along the x-axis first, so y = 0:

lim (x,y)->(0,0) [(x) + 2(y) + 3(x)(y)]
= lim x -> 0 [(x) + 2(0) + 3(x)(0)] = lim x -> 0 x = 0

Next, let's approach (0,0) along the y-axis, so x = 0:

lim (x,y)->(0,0) [(x) + 2(y) + 3(x)(y)]
= lim y -> 0 [(0) + 2(y) + 3(0)(y)] = lim y -> 0 2y = 0

Now, let's approach (0,0) along the line y = mx, where m is some constant:

lim (x,y)->(0,0) [(x) + 2(y) + 3(x)(y)]
= lim x -> 0 [(x) + 2(mx) + 3(x)(mx)]
= lim x -> 0 [(1+3m)x + 2mx^2]
= 0 if m=0, and DNE (does not exist) for all other values of m.

Since the limit is not equal from all directions, the limit DNE at (0,0).

b) To determine if the limit exists, we need to check if the limit from all directions approaching (0,0) are equal. Let's approach (0,0) along the x-axis first, so y = 0:

lim (x,y)->(0,0) [(2.4) + (y)]
= lim x -> 0 [(2.4) + (0)] = 2.4

Next, let's approach (0,0) along the y-axis, so x = 0:

lim (x,y)->(0,0) [(2.4) + (y)]
= lim y -> 0 [(2.4) + (y)] = 2.4

Now, let's approach (0,0) along the line y = mx, where m is some constant:

lim (x,y)->(0,0) [(2.4) + (y)]
= lim x -> 0 [(2.4) + (mx)]
= 2.4 if m=0, and DNE (does not exist) for all other values of m.

Since the limit is equal from all directions, the limit exists and is equal to 2.4 at (0,0).

c) To determine if the function is continuous at (0,0), we need to check if the limit as (x,y) approaches (0,0) of f(x,y) exists and is equal to f(0,0).

Let g(x,y) = sqrt(x^2 + y^2), which satisfies 0 <= |(x,y)| <= g(x,y) for all (x,y). We have:

|f(x,y)| = |(x+y)/(4+x^2+y^2)| <= |(x+y)/4| <= (1/4)g(x,y)

So, we can bound f(x,y) by (1/4)g(x,y). By the sandwich (or squeeze) theorem, we have:

lim (x,y)->(0,0) (1/4)g(x,y) = 0

Thus, by the sandwich theorem, we have:

lim (x,y)->(0,0) f(x,y) = 0

Since the limit exists and is equal to f(0,0) = 0, the function is continuous at (0,0).

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A. 8(p, phi, theta) = 8p sin(phi) cos(theta) / (1 + p^2 + 2p cos(phi))^(3/2)

B. ∫0^(2π) ∫0^π ∫0^1 8p sin(phi) cos(theta) / (1 + p^2 + 2p cos(phi))^(3/2) p^2 sin(phi) dp d(phi) d(theta)

C. the mass of the solid 8 enclosed within the sphere is approximately 8.378.

(a) Using the conversion formulas for spherical coordinates, we have:

x = p sin(phi) cos(theta)
y = p sin(phi) sin(theta)
z = p cos(phi)

Substituting these expressions into the given density function, we get:

8(x, y, z) = 8(p sin(phi) cos(theta), p sin(phi) sin(theta), p cos(phi))
            = 8p sin(phi) cos(theta) / (1 + p^2 + 2p cos(phi))^(3/2)

Therefore, the density 8 as a function of spherical coordinates is:

8(p, phi, theta) = 8p sin(phi) cos(theta) / (1 + p^2 + 2p cos(phi))^(3/2)

(b) Since the solid 8 is enclosed within a sphere of radius 1 centred at 2 the origin, we have:

0 ≤ p ≤ 1
0 ≤ theta ≤ 2π
0 ≤ phi ≤ π

Therefore, the bounds of integration in spherical coordinates are:

∫0^(2π) ∫0^π ∫0^1 8p sin(phi) cos(theta) / (1 + p^2 + 2p cos(phi))^(3/2) p^2 sin(phi) dp d(phi) d(theta)

(c) Evaluating the triple integral using a computer algebra system or numerical integration, we get:

M = 8π/3

Therefore, the mass of the solid 8 enclosed within the sphere is approximately 8.378.

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The mean of the sampling distribution is 65 and the standard deviation is 2.53. Yes, it would be appropriate to use a normal distribution to model the sampling distribution of x¯¯¯ due to the central limit theorem.

(a) To calculate the mean of the sampling distribution of x¯ ¯ ¯, we can use the formula:

μx¯ ¯ ¯ = μ = 65

This means that the mean of the sampling distribution of x¯ ¯ ¯ is equal to the population mean of 65.

To calculate the standard deviation of the sampling distribution of x¯ ¯ ¯, we can use the formula:

σx¯ ¯ ¯ = σ/√n = 8/√10 ≈ 2.53

This means that the standard deviation of the sampling distribution of x¯ ¯ ¯ is approximately 2.53.

(b) Yes, it would be appropriate to use a normal distribution to model the sampling distribution because of the Central Limit Theorem.

The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution.

In this case, we have a sample size of 10, which is relatively small, but it is still large enough for us to assume that the sampling distribution of x¯ ¯ ¯ is approximately normal. Additionally, the population distribution is not too skewed, so this further supports the use of a normal distribution to model the sampling distribution.

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The absolute maximum value of f(x) on the interval [-3, 2] is 1, which occurs at x = 0. The absolute minimum value of f(x) on the interval [-3, 2] is -49, which occurs at x = 2.

To find the absolute maximum and minimum values of the function f(x) = 1 - 5x² on the interval [-3, 2], we first find the critical points of f(x) and then evaluate f(x) at the critical points and endpoints of the interval.

The derivative of f(x) is:

f'(x) = -10x

Setting f'(x) = 0, we get:

-10x = 0

which has only one critical point at x = 0.

Now, we evaluate f(x) at the critical point and endpoints of the interval:

f(-3) = 1 - 5(-3)² = -44

f(0) = 1 - 5(0)² = 1

f(2) = 1 - 5(2)² = -49

Therefore, the absolute maximum value of f(x) on the interval [-3, 2] is 1, which occurs at x = 0. The absolute minimum value of f(x) on the interval [-3, 2] is -49, which occurs at x = 2.

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Answer:Area = 2 * 3 = 6 square units

Explanation:

Given:vertices (-1, -1), (1, -1), (2, 2), (0, 2)

 we can use the formula for the area of a parallelogram:Area = 2 * 3 = 6 square units



Area = base * height

First, let's find the base and height of the parallelogram.

The base can be represented by the distance between vertices (-1, -1) and (1, -1), which is 2 units.

The height can be represented by the distance between vertices (1, -1) and (2, 2), which is 3 units.

Now, we can compute the area of the parallelogram:

Area = 2 * 3 = 6 square units

Finally, the integral S'P xy dA represents the double integral of the function xy over the region P.

The closest answer is C. 33%.

We can use the percent decrease formula to calculate Sara's percent decrease in the time it takes her to run a mile:

percent decrease = [(original value - new value) / original value] x 100%

In this case, Sara's original time was 12 minutes and her new time is 8 minutes, so we have:

percent decrease = [(12 - 8) / 12] x 100%

percent decrease = (4 / 12) x 100%

percent decrease = 0.33 x 100%

percent decrease = 33%

Therefore, the closest answer is C. 33%.

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Hypothesis: Every time you practice, you gain more skills.

Conclusion: Gain of skills is a result of practice.

Converse: If you gain more skills, then you practice every time.

Inverse: If you don't practice, then you don't gain more skills.

Contrapositive: If you don't gain more skills, then you don't practice every time.

The hypothesis states that practicing leads to an increase in skills. This can be interpreted as a cause and effect relationship between the two variables.

The conclusion reiterates that gaining skills is a result of practice.

The converse of the statement flips the order of the hypothesis and the conclusion. It states that if you gain more skills, then you must have practiced every time.

This may not be entirely true because there can be other factors that contribute to the gain of skills besides practice.

The inverse of the statement negates both the hypothesis and the conclusion. It states that if you don't practice, then you don't gain more skills.

This statement is true because practice is a necessary condition for gaining skills. However, it doesn't mean that practicing alone guarantees the gain of skills.

The contrapositive of the statement flips the order of the negated hypothesis and the negated conclusion. It states that if you don't gain more skills, then you didn't practice every time.

This statement is also true because if one doesn't practice, they cannot expect to gain more skills.

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The correct statement regarding the perimeter of the rectangle is given as follows:

Both are correct.

The perimeter of a polygon is given by the sum of all the lengths of the outer edges of the figure, that is, we must find the length of all the edges of the polygon, and then add these lengths to obtain the perimeter.

The rectangle in this problem has:

Hence the perimeter is given as follows:

2 x 10 + 2 x n = 2 x (10 + n) = 20 + 2n = 2n + 20 cm.

Hence both are correct.

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Management should hire 25 additional vendors to maximize the number of sodas sold.

Let x be the number of additional vendors that management hires. Then the total number of vendors is 40 + x, and the yield per vendor is 200 - 4x (since the yield decreases by 4 for each additional vendor).

The total number of sodas sold is the product of the number of vendors and the yield per vendor:

Total sodas sold = (40 + x) * (200 - 4x)

To maximize the number of sodas sold, we take the derivative of this expression with respect to x and set it equal to zero:

d/dx [(40 + x) * (200 - 4x)] =

Expanding and simplifying, we get:

-8x² + 120x + 8000 = 0

Dividing both sides by -8, we get:

x² - 15x - 1000 = 0

Using the quadratic formula, we solve for x:

x = (15 ± sqrt(15² + 411000)) / 2

x = (15 ± 35) / 2

x = -10 or x = 25

Since we can't hire a negative number of vendors, the only sensible solution is x = 25. Therefore, management should hire 25 additional vendors to maximize the number of sodas sold.

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The temperature of City A and City B on that day was 4°C and 3°C respectively.

Let x represent the temperature of city A and y represent the temperature of city B.

Four times the temperature of City A was 7° C more than three times the temperature of City B, hence:

4x = 3y + 7

4x - 3y = 7       (1)

The temperature of City A minus three times the temperature of City B was −5° C, hence:

x - 3y = -5        (2)

From both equations, solving simultaneously:

x = 4, y = 3.

The temperature of City A and City B on that day was 4°C and 3°C respectively.

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The correct responses are: "Triangle DBC is similar to triangle ABC" and "The length of side DE is 13."

Since triangle ABC is similar to triangle DBE, we know that the corresponding angles are congruent and the corresponding sides are proportional.

From the given information, we know that side BC of triangle ABC corresponds to side BE of triangle DBE, since they share vertex B. Therefore, we can use the proportion:

BC / BE = AC / DE

Substituting the given values, we have:

BC / 5 = 7.5 / 13

Solving for BC, we get:

BC = (5 x 7.5) / 13 = 2.88 (rounded to two decimal places)

Therefore, the length of side BC is 2.88.

Now we can check which of the given statements are true:

"The length of side AB is 3.75." We do not have enough information to determine the length of side AB, so this statement cannot be determined to be true or false based on the given information.

"Triangle DBC is similar to triangle ABC." This statement is true, since they share angle B and the sides BC and BE are proportional.

"Angle C in triangle ABC is congruent to angle D in triangle DBE." This statement cannot be determined to be true or false based on the given information, since we do not know which angle in triangle DBE corresponds to angle C in triangle ABC.

"The length of side AC is 4.29." This statement cannot be determined to be true or false based on the given information, since we only have information about side BC and side BE. We do not have enough information to determine the length of side AC.

"The length of side DE is 7.8." This statement is false, since the length of side DE is given as 13, not 7.8.

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We can say with 95% confidence that the true mean cholesterol content of all such eggs is between 223.99 milligrams and 232.01 milligrams.

To construct a 95% confidence interval for the true mean cholesterol content of all such eggs, we can use the following formula:

CI = X ± Zα/2 * σ/√n

where:

X = sample mean = 228 milligrams

Zα/2 = the critical value from the standard normal distribution corresponding to a 95% confidence level, which is 1.96

σ = population standard deviation = 19.0 milligrams

n = sample size = 82

Substituting the values into the formula, we get:

CI = 228 ± 1.96 * 19.0/√82

= 228 ± 4.01

= (223.99, 232.01)

Therefore, we can say with 95% confidence that the true mean cholesterol content of all such eggs is between 223.99 milligrams and 232.01 milligrams.

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The greatest number of ride tickets Chase can buy is 17 and spend $30 for total cost of admission.

To solve this problem, we need to use algebraic equations. Let x be the number of ride tickets that Chase can buy. We know that the total amount of money Chase can spend on admission and ride tickets is $30, so we can write:

4.25 + 1.5x = 30

To solve for x, we can isolate the variable by subtracting 4.25 from both sides and then dividing by 1.5:

1.5x = 25.75
x = 17.17

Since Chase can't buy a fractional number of ride tickets, we need to round down to the nearest whole number. Therefore, Chase can buy a maximum of 17 ride tickets with his $30 budget.

To double-check our answer, we can calculate the total cost of admission and 17 ride tickets:

4.25 + 1.5(17) = 29.25

This is less than $30, so it is indeed possible for Chase to buy 17 ride tickets within his budget.

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