1. The data sel below represents the number of animals in different exhibits at a zoo.
48, 86, 15, 27, 18, 52, 103
a. Write the data from least to greatest.
h. What is the minimum number of animals?
c. What is the maximum number of animals?
d. What is the median number of animals?
e. What is the median of the first half of the data? (first quartile)
f. What is the median of the second half of the data? (third quartile)
g. What is the interquartile range?

The given statement "In an equation with two x’s, the solution is the number that makes the two sides equalwhen put in for both x’s is false because  in an equation with two x's, the solution is the number that makes the two sides equal when put in for one or both of the x's.

For example, consider the equation 2x + 3 = 5x - 1. To find the solution, we need to find the value of x that makes both sides of the equation equal. We can do this by simplifying the equation:

2x + 3 = 5x - 1

2x - 5x = -1 - 3

-3x = -4

x = 4/3

So the solution to this equation is x = 4/3. Notice that we only substituted the value of x once in the equation, but we still found the solution.

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if the test has 25 questions, the number of multiple choice questions plus the number of short answer questions is 25. The correct answer is c.

To explain, let x be the number of multiple choice questions and y be the number of short answer questions. We know that there are 25 questions in total, so x + y = 25.

We also know that each multiple choice question is worth 2 points, and each short answer question is worth 4 points. If we let M be the total number of points from the multiple choice questions, and S be the total number of points from the short answer questions, we can set up the equation:

M + S = 80

We can also express M and S in terms of x and y:

M = 2x
S = 4y

Substituting these equations into the first equation, we get:

2x + 4y = 80

Dividing both sides by 2:

x + 2y = 40

Now we have two equations with two variables:

x + y = 25
x + 2y = 40

Subtracting the first equation from the second, we get:

y = 15

Substituting this into the first equation, we get:

x + 15 = 25

x = 10

Therefore, there are 10 multiple choice questions on the test.

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o, the correct sequence of transformations that would result in figure M' is: A) Reflection over the y-axis, then translation down 5 units.

In mathematics, a transformation is a process that changes the position, size, or shape of a geometric figure. Transformations are often used to study the properties of geometric objects and to solve problems in various areas of mathematics, such as geometry, algebra, and calculus.

Here,

Looking at the coordinates of the vertices of M and M', we can see that M' is obtained from M by reflecting it over the y-axis and then translating it down 5 units. So, the correct sequence of transformations that would result in figure M' is:

A) Reflection over the y-axis, then translation down 5 units.

This means that we first reflect M over the y-axis to obtain a new rectangle, which we can call M''. Then, we translate M'' down 5 units to obtain M'. Therefore, the sequence of transformations is:

Reflection over the y-axis

Translation down 5 units

This sequence of transformations will map M onto M'.

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The test statistic is large and the corresponding P-value is small, we reject the null hypothesis that the main effects of piston ring type are all equal to 0.

a) The degree of freedom for the effect of oil type is 3 (number of levels of oil type minus 1).

b) The degrees of freedom for the effect of piston ring type is 2 (number of levels of piston ring type minus 1).

c) The degrees of freedom for interactions is (3-1) * (2-1) = 2 (product of the degrees of freedom for oil and piston ring types).

d) The degrees of freedom for error is (3 * 2 * 4) - (3 * 2) = 18 (total number of observations minus the number of treatments).

e) The ANOVA table is as follows:

Source Sum of Squares Degrees of Freedom Mean Square F-Statistic P-value

Oil 1.0926 3 0.3642 F1 P1

Piston Ring 0.9340 2 0.4670 F2 P2

Interaction 0.2485 2 0.1243 F3 P3

Error 1.7034 18 0.0946  

Total 3.9785 25  

Note: The F-statistics and P-values will need to be calculated using the appropriate formulas.

f) To test the plausibility of the additive model, we can compare the residual mean square from the ANOVA table to the mean square for error. If the residual mean square is much smaller than the mean square for error, this indicates that there may be additional sources of variation in the data that are not explained by the additive model. The test statistic for this is:

F = (mean square for error) / (residual mean square)

If the test statistic is large and the corresponding P-value is small, we reject the additive model.

g) To test the plausibility that the main effects of oil type are all equal to 0, we can use the F-test:

F1 = (mean square for oil type) / (mean square for error)

If the test statistic is large and the corresponding P-value is small, we reject the null hypothesis that the main effects of oil type are all equal to 0.

h) To test the plausibility that the main effects of piston ring type are all equal to 0, we can use the F-test:

F2 = (mean square for piston ring type) / (mean square for error)

If the test statistic is large and the corresponding P-value is small, we reject the null hypothesis that the main effects of piston ring type are all equal to 0.

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The expected number of defective items and the probability of at least 4 are defective is equal to 3.6 and 0.370 or 37.0%.

Total number of items 'n' = 18

Probability of an item being defective 'p' =20%

                                                                    = 0.2  

Expected number of defective items,

Use the formula for the expected value of a binomial distribution,

E(X) = np

where X is the number of defective items.

Plug in the values we have,

E(X) = 18 x 0.2

      = 3.6

Expect average items out of 18 to be defective = 3.6  .

Probability that at least 4 items are defective,

Calculate the probability of 4, 5, 6, ..., 18 defective items

Use the complement rule to simplify it,

P(at least 4 defective)

= 1 - P(less than 4 defective)

Using the CDF function,

'binomcdf' is the binomial cumulative distribution function.

18 is the number of trials,

0.2 is the probability of success,

And 3 is the maximum number of successes

P(less than 4 defective)

= binomcdf (18, 0.2, 3)

= P(X <= 3)

=[tex]\sum_{x=0}^{3}[/tex] ¹⁸Cₓ × (0.2)^x × (0.8)^(18-x)

= ¹⁸C₀× (0.2)^0 × (0.8)^(18-0) + ¹⁸C₁× (0.2)^1 × (0.8)^(18-1) + ¹⁸C₂× (0.2)^2 × (0.8)^(18-2) + ¹⁸C₃× (0.2)^3 × (0.8)^(18-3)

= (0.8)^(18) + 18× (0.2) × (0.8)^(17) + 153 × (0.04) × (0.8)^(16) + 1632× (0.008) × (0.8)^(15)

= 0.630

Plug in the values,

P(at least 4 defective)

= 1 - 0.630

= 0.370

Therefore, the expected items to be defective and probability that at least 4 items out of 18 are defective is equal to 3.6 and 0.370 or 37.0%.

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The domain of (g o f o g)(x),  where "o" denotes composition of functions is the set of all x in F such that f(x) is in the domain of g, and g(f(x)) is in the domain of g.

To determine the domain of the function (g o f o g)(x), we need to consider the domains of the functions g(x) and f(x), as well as the composition of these functions.

Let's assume that the domain of g(x) is G and the domain of f(x) is F.

Since (g o f o g)(x) means that we apply the function g to f(x), and then apply g to the result again, we need to ensure that the output of f(x) is a valid input for g(x).

Therefore, the domain of (g o f o g)(x) is the set of all values of x in F such that g(f(x)) is in G, and g(y) is in G for all y in the range of g(f(x)).

In other words, the domain of (g o f o g)(x) is the set of all x in F such that f(x) is in the domain of g, and g(f(x)) is in the domain of g.

Thus, the domain of (g o f o g)(x) is the set of all x in F such that f(x) is in the domain of g, and g(f(x)) is in the domain of g.

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The vector describing the cat's final position is <7,0.5>.

The cat first travels along the vector <-1,2> and then chases the ball along the vector <2,-6>. So the cat's position after these two movements is:

<-1,2> + <2,-6> = <1,-4>

The cat's position after this movement is:

1.5 * <4,3> = <6,4.5>

Finally, we add this vector to the cat's previous position to find its final position:

<1,-4> + <6,4.5> = <7,0.5>

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The p-value for this hypothesis test is approximately 0.1317.

To calculate the p-value for the hypothesis test, we need to perform a one-sample proportion test using the sample data provided.

Let's define the null and alternative hypotheses:

Null hypothesis (H₀): The proportion of Marriott customers carrying two Wi-Fi devices is equal to or less than 0.60.

Alternative hypothesis (H₁): The proportion of Marriott customers carrying two Wi-Fi devices exceeds 0.60.

Sample size (n) = 120

Number of customers with two Wi-Fi devices (x) = 78

To test the hypothesis, we can use the normal approximation to the binomial distribution since the sample size is reasonably large.

First, calculate the sample proportion:

[tex]\hat{p}[/tex] = x / n = 78 / 120 = 0.65

Next, calculate the test statistic (z-score):

z = [tex]\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n} } }[/tex]

= (0.65 - 0.60) / √((0.60 * (1 - 0.60)) / 120)

= 0.05 / √(0.24 / 120)

= 1.1180

Now, we can find the p-value corresponding to the calculated test statistic using a standard normal distribution table or a statistical calculator.

In this case, the p-value for a one-sided test (since we are testing if the proportion exceeds 0.60) is approximately 0.1317.

Therefore, the p-value for this hypothesis test is approximately 0.1317.

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Note that this graph only shows the behavior of the function for positive values of x, as the natural logarithm is not defined for x ≤ 0.

What is Function?

Function can be defined in which it relates an input to output.

To graph the function g(x) = ln(x)÷4, we can start by creating a table of values:

x g(x) = ln(x)÷4

1 0

2 0.173

10 0.575

100 0.921

1000 1.146

Next, we can plot these points on a coordinate plane and connect them to create a smooth curve:

Therefore, Note that this graph only shows the behavior of the function for positive values of x, as the natural logarithm is not defined for x ≤ 0.

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The phrases that imply correlation without causation are:

These correlations do not imply a causal relationship, meaning that an increase or decrease in one variable does not directly cause a corresponding change in the other variable.

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Looking at the plot, we can see that the stems range from 60 to 89, with each stem representing a group of ten pounds. The leaves represent the remaining single digits, indicating the exact weight of each tuna. There are 4 tuna that weigh less than 90 pounds

Based on the stem-and-leaf plot of the weights of yellowfin tuna caught during a fishing contest, we can count the number of tuna that weigh less than 90 pounds.

To determine the number of tuna that weigh less than 90 pounds, we need to look at the stems that are less than 9. This includes stems 6, 7, and 8. The leaves associated with these stems show the weights of the tuna that are less than 90 pounds. We can count the number of leaves associated with these stems to determine the number of tuna that weigh less than 90 pounds.

In this case, there are 4 tuna that weigh less than 90 pounds. Two of them weigh 88 pounds and the other two weigh 87 pounds. Therefore, we can conclude that there are 4 tuna that weigh less than 90 pounds in the fishing contest based on the stem-and-leaf plot.

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The financing option that will likely lead to the lowest monthly payment is:

$3000 down payment, 6% interest, 84 months

The longer loan term (84 months) will spread out the payments over a longer period of time, resulting in a lower monthly payment. The down payment will also help to reduce the monthly payment amount.

The 6% interest rate is relatively low, so it won't have a significant impact on the monthly payment compared to the loan term.

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To administer a daily dose of 375 mg of naproxen using 250 mg scored tablets, the patient would need to take 1.5 tablets, rounded up to 2 tablets of 250 mg each.

To administer 375 mg of naproxen using 250 mg scored tablets, we need to divide 375 by 250 to determine how many tablets to administer.

375 mg / 250 mg per tablet = 1.5 tablets

Therefore, the dosage of 375 mg of naproxen would require 1.5 tablets.

Since tablets cannot be divided into halves, the patient would need to take 2 tablets of 250 mg each to achieve the prescribed dosage of 375 mg.

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The correct option is (a)  270 degree clockwise rotation because they both result in the same final orientation, just in opposite directions.

A 90-degree counter-clockwise rotational  symmetry  is the same as a 270-degree clockwise rotation. This is because rotations are measured in degrees, and a full circle is 360 degrees. Therefore, rotating something 90 degrees counter-clockwise means turning it a quarter of a full circle, while a 270-degree clockwise rotation means turning it three-quarters of a full circle.

To visualize this, imagine a square with its sides parallel to the x and y axes. If we rotate this square 90 degrees counter-clockwise, its sides will now be parallel to the y and negative x axes.

However, if we rotate the same square 270 degrees clockwise, its sides will also be parallel to the y and negative x axes, as it has been turned three-quarters of the full circle in the opposite direction.

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Answer: x = -1, y = 6

Step-by-step explanation:

lets substitute the value of y from the first equation into the y in the second equation.

-2x + 4 = -3x + 3

4 - 3 = -3x + 2x

1 = -1x

x = -1

we know from before that y = -2x + 4

so y = -2(-1) + 4

y = 6

To synthesize information regarding when books were written, Alex should create a timeline.  So, the correct option is A.

A timeline is a visual representation of chronological events, which can be used to arrange information in order based on their time of occurrence. In this case, Alex can plot the publication dates of the books on a timeline, allowing him to see how the dates relate to each other and to other events. This will help him to identify patterns and trends in the publication history of the books.

A Venn diagram, on the other hand, is a tool used to compare and contrast two or more sets of information. It is not well-suited for presenting chronological information.

A chart may be useful in presenting data in a visual manner, but it may not be as effective as a timeline in showing the order of events over time.
Therefore, the best answer from the choices provided is A. timeline.

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A graph of the image of ABC under a dilation whose center is P and scale factor is 3 is shown in the image below.

In Geometry, dilation is a type of transformation which typically changes the size of a geometric shape, but not its shape. This ultimately implies that, the size of the geometric shape would be increased (enlarged) or decreased (reduced) based on the scale factor applied.

In this exercise, we would use an online graphing calculator to plot the image of ABC after a dilation by a scale factor of 3 centered at P.

Based on the image (see attachment), we can logically deduce that each vertex is 3 times as far from center P as the original vertex and each segment is 3 times as long as the original

Side length AC = (3.0)3 = 9.0

Side length AB = (5.0)3 = 15.0

Side length CB = (4.0)3 = 12.0

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The expression that is equivalent to 3v+2v is:

D. v+v+v+v+v

Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we substitute the same value(s) for the variable(s).

To find the expressions that are equivalent to 3v+2v, we need to find the expression which when simplified will give the same expression as 3v+2v. That is: 3v + 2v = 5v

v*5 = 5v

v+5 = v + 5

v+5v = 6v

v+v+v+v+v = 5v

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Answer:

18:19

Step-by-step explanion:
To solve this problem, we need to first calculate the time it takes for the plane to fly from Singapore to Sydney:

Time = Distance ÷ Speed

Time = 6310 km ÷ 757.2 km/h

Time ≈ 8.34 hours

This is the time it takes to fly from Singapore to Sydney in Singapore time. However, we need to convert this time to Sydney time, which is 2 hours ahead of Singapore time. Therefore, the local time when the plane arrives in Sydney is:

Time in Sydney = Singapore time + 2 hours + Flight time

Time in Sydney = 07:45 + 2 hours + 8.34 hours

Time in Sydney = 18:19

Therefore, the local time when the plane arrives in Sydney is 18:19 using the 24-hour clock.

The equation for the sphere centered at (-8, 8, -9) with radius [tex]\sqrt(689)[/tex] and passing through the point (9, -8, -1).

The equation for a sphere with center (a, b, c) and radius r is given by:

[tex](x - a)^2 + (y - b)^2 + (z - c)^2 = r^2[/tex]

In this case, the center of the sphere is (-8, 8, -9) and the point (9, -8, -1) is on the sphere.

Let's plug these values into the equation and solve for the radius:

[tex](9 - (-8))^2 + (-8 - 8)^2 + (-1 - (-9))^2 = r^2[/tex]

[tex](17)^2 + (-16)^2 + (8)^2 = r^2[/tex]

[tex]r^2 = 689[/tex]

Now that we have the center and the radius, we can write the equation of the sphere as:

[tex](x + 8)^2 + (y - 8)^2 + (z + 9)^2 = 689[/tex]

This is the equation for the sphere centered at (-8, 8, -9) with radius [tex]\sqrt(689)[/tex] and passing through the point (9, -8, -1).

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A region is bounded by the curves  y = sinπ x ,  y = 4 x − 1 , and the x-axis. determine the area of the region. use the area formula for a triangle to expedite the calculation and show all your work.

To find the area of the region bounded by the curves y = sin(πx), y = 4x - 1, and the x-axis, we need to first find the points of intersection of the curves.

Setting y = sin(πx) and y = 4x - 1 equal to each other, we get:

sin(πx) = 4x - 1

Solving for x is difficult algebraically, so we can use numerical methods or graphing to estimate the solutions. A graph of the two curves shows that they intersect at approximately x = 0.161 and x = 1.239.

Next, we can find the area of the region by breaking it up into two parts: a triangle and a region bounded by the curve y = sin(πx), the x-axis, and the vertical lines x = 0.161 and x = 1.239.

The triangle has base 1.239 - 0.161 = 1.078 and height 4(1.239) - 1 = 3.956. Using the formula for the area of a triangle, we get:

Area of triangle = (1/2) * base * height

= (1/2) * 1.078 * 3.956

= 2.148

To find the area of the region bounded by y = sin(πx), the x-axis, and the vertical lines x = 0.161 and x = 1.239, we can use integration:

∫ from 0.161 to 1.239 of sin(πx) dx = [-cos(πx)/π] from 0.161 to 1.239 = [-cos(π(1.239))/π] - [-cos(π(0.161))/π] = (1/π) * (cos(0.161π) - cos(1.239π))

Using a calculator, we get:

(1/π) * (cos(0.161π) - cos(1.239π)) ≈ 0.696

Therefore, the total area of the region is:

Area = 2.148 + 0.696

= 2.844 (rounded to three decimal places)

So the area of the region bounded by the curves y = sin(πx), y = 4x - 1, and the x-axis is approximately 2.844 square units.

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If total sales remain the same and assuming a $5 profit margin per order, adding cookie dough ice cream and hot fudge to the menu could result in a profit of $58 if 20 or more orders are sold.

To determine if adding cookie dough ice cream and hot fudge to the menu will result in a profit, we need to consider the cost and potential revenue. If the cost of adding these items is $42, we need to calculate how many orders of cookie dough ice cream with hot fudge we need to sell to cover that cost and make a profit.

Assuming the profit margin on each order of cookie dough ice cream with hot fudge is $5 (for example), we would need to sell at least 9 orders (rounding up from 8.4) to cover the $42 cost and break even. If we sell more than 9 orders, we would make a profit.

Assuming we sell 20 orders of cookie dough ice cream with hot fudge, the total revenue generated would be $100 ($5 profit per order x 20 orders). Subtracting the $42 cost of adding these items, the net profit would be $58.

Therefore, if total sales remain the same and assuming a $5 profit margin per order, adding cookie dough ice cream and hot fudge to the menu could result in a profit of $58 if 20 or more orders are sold.

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The length of the variable x, obtained from the formula for the surface area of a solid about 10.63 inches

What is the surface area of a solid object?

The surface area of a solid object is the area of the outside surface of the  object.

The dimensions of of the open top box with the square corners 2 in by 2 in cut from the corners are;

Length of the box, L = x - 4 inches

Width of the box, W = x - 4 inches

Height of the box, H = 2 inches

The surface area of the box is therefore;

Volume, A = L × W + 2 × L × H + 2 × W × H

Plugging in the expressions for the length, width and height of the box, the surface area = (x - 4) × (x - 4) + 2 × 2 × (x - 4) + 2 × 2 × (x - 4) = 9·x² - 40·x + 16

The surface area of the box, A = 9·x² - 40·x + 16

Second part;

When the surface area = 609 in², we get;

A = 609 = 9·x² - 40·x + 16

9·x² - 40·x + 16 - 609 = 9·x² - 40·x - 593 = 0

x = (20 + √(5737))/9 ≈ 10.63, and x = (20 - √(5737))/9 ≈ -6.19

The length x ≈ 10.63 inches

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Answer:

one

Step-by-step explanation:

The equation 4p + 7 = 3(4p) can be simplified by distributing the 3 on the right-hand side of the equation:

4p + 7 = 12p

Subtracting 4p from both sides of the equation, we get:

7 = 8p

Dividing both sides of the equation by 8, we get:

p = 7/8

Therefore, the equation has only one solution, which is p = 7/8. Answer: one.

a) The percentage of men with a height less than or equal to 55.8 inches is approximately 0.00007 or 0.007%.

b) The door design is inadequate, but because the jet is relatively small and seats only six people, a much higher door would require major changes in the design and cost of the jet, making a larger height not practical.

Option (a) is correct.

a) To determine the percentage of adult men who can fit through the door without bending, we need to find the proportion of men whose height is less than or equal to the doorway height of 55.8 inches. We can use the normal distribution formula and standardize the variable:

Z = (X - μ) / σ

Where X is the doorway height, μ is the mean height of men, and σ is the standard deviation of men's heights.

Z = (55.8 - 67.8) / 3.1 = -3.87

Using a standard normal distribution table, we can find that the percentage of men with a height less than or equal to 55.8 inches is approximately 0.00007 or 0.007%.

Therefore, only a very small percentage of adult men can fit through the door without bending.

b)The door design is inadequate, but because the jet is relatively small and seats only six people, a much higher door would require major changes in the design and cost of the jet, making a larger height not practical.

While it is true that most women will be able to fit through the door without bending, it is not acceptable to design a door that does not accommodate all potential passengers. The door should be designed to allow all passengers to enter without any discomfort or difficulty.

However, in the case of this executive jet, increasing the height of the door to accommodate all potential male passengers would require major redesign and cost implications.

In summary, while the current door design is inadequate, it may not be practical or feasible to make significant changes due to design and cost constraints.

Therefore, the correct option is a.

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The function that can be used to find the number of milligrams of antibiotic in George's bloodstream after h hours is A(h) = 50[tex](0.5)^h[/tex] . This is an exponential function where the initial dose of 50 milligrams is halved every hour.

The problem states that the number of milligrams of the antibiotic in a person's bloodstream is dependent on the number of hours elapsed since taking the antibiotic. We know that George took a 50-milligram dose of the antibiotic and had 25 milligrams of the antibiotic in his bloodstream one hour after taking it.

This means that half of the initial dose remained in his bloodstream after one hour. Similarly, after two hours, he had 12.5 milligrams of the antibiotic in his bloodstream, which means that half of the remaining dose from the first hour remained in his bloodstream.

Therefore, we can conclude that the number of milligrams of the antibiotic in his bloodstream is halved every hour.

Using this information, we can create an exponential function where A(h) represents the number of milligrams of the antibiotic in his bloodstream after h hours. The function is A(h) =  50[tex](0.5)^h[/tex] , where 50 is the initial dose and 0.5 is the halving factor.

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The percentage of body weight the dog lost is 9.2%. Mary would weigh 52.664 kg after losing the same percentage of body weight as her dog.

A) To find the percentage of body weight the dog lost, first, calculate the actual weight loss: 25 kg - 2.3 kg = 22.7 kg. Then, divide the weight loss (2.3 kg) by the original weight (25 kg) and multiply by 100 to get the percentage: (2.3 kg / 25 kg) * 100 = 9.2%.

B) If Mary lost the same percentage of her body weight as the dog did, she would lose 9.2% of her weight. To calculate this, multiply her original weight (58 kg) by the percentage (9.2%): 58 kg * 0.092 = 5.336 kg. Now, subtract this weight loss from her original weight to find her new weight: 58 kg - 5.336 kg = 52.664 kg. So, Mary would weigh 52.664 kg after losing the same percentage of body weight as her dog.

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The probability that during the next three months it has;

Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has included probability to forecast the likelihood of certain events.

The degree to which something is likely to happen is basically what probability means. You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution.

The plant has on the average two breakdowns per month,

so the Poisson distribution is,

[tex]P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}[/tex]

where,

X is the random variable representing the number of events

λ is the average rate at which the events occur

k is the number of events that occur

a)  at least five breakdowns

[tex]P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}[/tex]

P(X =5) = [tex]\frac{e^{-2} 2^5}{5!}[/tex]

= 0.036

Thus, probability that at least five breakdowns in three months is 0.036.

b)  at most eight breakdowns

[tex]P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}[/tex]

[tex]P(X=8) = \frac{e^{-2} 2^8}{8!}[/tex]

= 0.00085.

Therefore, probability of at most eight breakdowns is 0.00085.

c) more than five breakdowns.

[tex]P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}[/tex]

P(X = 6) = [tex]\frac{e^{-2} 2^6}{6!}[/tex]

=0.012

Therefore, probability of more than five breakdowns is 0.012.

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The equation of the plane is -x(tan(r/4)) + z(sin(r/4)) = 1.

Let the equation of the plane be Ax + By + Cz = D. Since the plane contains the y-axis, we know that x = 0 when y = 0. Therefore, the equation becomes:

0A + 0B + Cz = D

=> Cz = D

This means that the plane is perpendicular to the y-axis and intersects the z-axis at z = D/C.

Now, we need to find the values of A, B, and C. Since the plane makes an angle of r/4 with the positive x-axis, we can use the direction cosines to find these values. The direction cosines of a vector are the cosines of the angles it makes with the x, y, and z axes.

Let the direction cosines of the vector perpendicular to the plane be (l, m, n). Then, we have:

cos(r/4) = l/√(l^2 + m^2 + n^2)

=> l = cos(r/4) / √2

cos(π/2) = m/√(l^2 + m^2 + n^2)

=> m = 0

cos(π/2) = n/√(l^2 + m^2 + n^2)

=> n = sin(r/4) / √2

Therefore, the vector perpendicular to the plane is:

(l, m, n) = (cos(r/4) / √2, 0, sin(r/4) / √2)

Since the plane contains the y-axis, we know that it is perpendicular to the vector (0, 1, 0). Therefore, the dot product of the two vectors is zero:

0A + B + 0C = 0

=> B = 0

Finally, we can use the fact that the vector (A, B, C) is perpendicular to the vector (cos(r/4) / √2, 0, sin(r/4) / √2) to find A and C:

A(cos(r/4) / √2) + 0 + C(sin(r/4) / √2) = 0

=> A = -C(tan(r/4) / √2)

Therefore, the equation of the plane is:

-C(tan(r/4) / √2)x + 0y + C(sin(r/4) / √2)z = D

Multiplying through by √2/C and setting D = √2, we get:

-x(tan(r/4)) + z(sin(r/4)) = 1

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The possible amount of time for Dinah to drive 420 miles on the highway is between 6 & 7 hours.

To get the possible time, we need to consider the range of speeds she can drive at.

Because she must drive at least 60 miles per hour and at most 70 miles per hour, we can calculate the possible time ranges using "Time = Distance / Speed"

At 60 miles per hour:

= 420 miles / 60 miles per hour

= 7 hours

At 70 miles per hour:

= 420 miles / 70 miles per hour

= 6 hours.

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