a rectangular container 12 cm long, 8 cm wide, and 36 cm high was one - third full. when some syrup from a bottle was poured into the container, it got half full. find the volume of the syrup poured from the bottle into the container in millimeters

Answer:

꧁. ꧂

Step-by-step explanation:

꧁. ꧂꧁. ꧂

The second urn contains 16 green balls and 46 blue balls.

Let's first find the probability of drawing two green balls, which we'll denote as P(G,G).

We can break it down into two cases:

Case 1: A green ball is drawn from the first urn and a green ball is drawn from the second urn.

The probability of drawing a green ball from the first urn is 4/10, and the probability of drawing a green ball from the second urn is 16/(16+N). Therefore, the probability of this case is (4/10) x (16/(16+N)).

Case 2: A blue ball is drawn from the first urn and a blue ball is drawn from the second urn.

The probability of drawing a blue ball from the first urn is 6/10, and the probability of drawing a blue ball from the second urn is N/(16+N). Therefore, the probability of this case is

(6/10) x (N/(16+N))

The probability of drawing two balls of the same color is the sum of the probabilities of these two cases, which we know to be 0.58:

P(G,G) + P(B,B) = 0.58

We can solve for P(G,G) as follows:

P(G,G) = 0.58 - P(B,B)

= 0.58 - (6/10) x (N/(16+N))

Now, we can set P(G,G) equal to the probability of drawing two green balls from the urns that we found earlier:

(4/10) x (16/(16+N)) = 0.58 - (6/10) x (N/(16+N))

By simplifying this equation, we get:

64 = 58(16+N) - 60N

By solving for N, we get:

N = 46

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Question:

An urn contains 4 green balls and 6 blue balls. A second urn contains 16

green and N blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same colour is 0.58. Find N.

Answer:

Step-by-step explanation:

The lateral area consists of 2 trapezia and 2 rectangles.

= 2 * 1/2 * 4 *(4 + 6) + 2 * 9 * length of the sloping line

= 40 + 18 * length of the sloping line.

Find the length of the sloping lines:

Consider the right triangle whose height is 4 ft.

The trapezium is isosceles so the base of this triangle

= (6 - 4)/ 2 = 1 ft.

By Pythagoras:

Sloping line = √(1^2 + 4^2) = √17

So lateral area

= 40 + 18√17.

= 114.22 ft^2 to nearest hundredth

Total area =

40 + 18√17 + 4*9 + 6*9

= 130 + 18√17

= 204.22 ft^2 to nearest hundredth

Good start.

Surface area:

Area of 2 trapezoids: 2 ((4+6)/2*4) = 40

Area of two side rectangles: [tex]2*9*\sqrt{17}=18\sqrt{17}[/tex] (I got root 17 from constructing a right triangle from the altitude just like last time and then finding the missing side using pythag)

Area of top rectangle: 4*9 = 36

Area of bottom rectangle: 6*9 = 54

Summing these up we get [tex]130+18\sqrt{17}[/tex]

Lateral area is the area excluding the bases (correct me if i'm wrong) so it will be [tex]130+18\sqrt{17}-40=90+18\sqrt{17}[/tex] (we need to subtract the areas of the trapezoids).

There are 4896 ways of arranging 7 pets in a row of 7 chairs such that at least 2 cats are next to each other.

Arrangement of entities can be done using permutation and combination in mathematics.

Consider the distributions where 2 cats are not next to each other as

C D C D C D C.

There are 4! ways to seat the cats, and 3! ways to seat the dogs that gets us N=4! 3! = 144

There are a total of 7!=5040 possible ways to seat both cats and dogs.

At least 2 cats seat together= All possible arrangements- None of the cat seat together.

Thus the number of seatings where at least 2 cats are adjacent is

5040-144 = 4896

So, there are 4896 arrangements can be made so that at least 2 of the cats seat together.

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The regression line is a statistical tool that can be used to analyze the relationship between two variables.

It helps identify trends in the data and can be used to make predictions about future values. It can also be used to assess the accuracy of the model and to determine the strength of the relationship between the two variables.

By examining the slope of the regression line, we can assess the direction and strength of the linear relationship between the two variables.

Additionally, the y-intercept of the regression line can provide insight into the value of the dependent variable when the independent variable is equal to zero.

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In response to the student question about registering multiple accounts on AliExpress for discounts, it is not recommended to create more than one account to apply coupons or discounts. This is because it goes against the platform's terms of service and may result in penalties, such as account suspension or termination.

To enjoy discounts on AliExpress without violating the platform's rules, follow these steps:
Register for an account: Sign up for a single account on AliExpress, providing your personal information and email address.
Subscribe to newsletters and promotions: Opt-in for promotional emails and newsletters from AliExpress. This way, you will be informed of any available discounts, deals, and special offers.
Shop during sales events: AliExpress regularly hosts sales events, such as the annual 11.11 sale, where a wide range of products are offered at discounted prices. Make sure to shop during these events to maximize your savings.
Collect coupons: Browse the website or use the AliExpress mobile app to collect various types of coupons, such as select coupons, store coupons, and AliExpress coupons. These can be applied at checkout to reduce the cost of your purchases.
Use cashback services: Some cashback services partner with AliExpress, offering a percentage of your purchase back as a refund or credit. Register with a cashback service, and make sure to use their links when shopping on
By following these steps, you can enjoy discounts on AliExpress without resorting to creating multiple accounts, which is not suggested and could lead to penalties.

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Answer:

1.   No, the side lengths 3, 5, and 9 cannot make a triangle.

2.  If you are scaling an object by ¹/₃, it makes the object smaller.

     If the real object is 30 inches long, it is 10 inches once it is scaled.

3.  A cube has 6 faces.

     To find the volume, cube the side length:  6³ = 216 in³

Step-by-step explanation:

According to the Triangle Inequality Theorem, the sum of any two sides of a triangle is greater than the length of the third side.

As the sum of the two sides measuring 3 and 5 is 8, and 8 is not greater than 9, the side lengths 3, 5, and 9 cannot make a triangle.

Scaling an object reduces or enlarges its size. When scaling an object:

Therefore, if you are scaling an object by ¹/₃, the object becomes smaller, since ¹/₃ is between zero and 1.

To scale length, simply multiply the length by the scale factor.

Therefore, if a real object is 30 inches long, its length once it is scaled is:

[tex]\implies \sf 30 \times \dfrac{1}{3}=\dfrac{30}{3}=10\;inches[/tex]

A cube is a three-dimensional solid shape with six square faces.

The volume of a cube is the product of its width, length and height.

If the side of a cube measures 6 inches, its volume is:

[tex]\begin{aligned}\implies \sf Volume &=\sf 6 \times 6 \times 6 \\&= \sf 36 \times 6 \\&= \sf 216\;in^3\end{aligned}[/tex]

Answer:

See below, please.

Step-by-step explanation:

Ans 1.

In order to make a triangle, three line segments are needed, and the sum of the lengths of any two sides must be greater than the length of the remaining side. The side lengths 3, 5, and 9 cannot make a triangle because 3 + 5 is less than 9, which violates the triangle inequality.

Ans 2.

Scaling an object by 3 makes the object larger, as each dimension (length, width, and height) is multiplied by 3. If the real object is 30 inches long and is scaled by 3, it will be 90 inches long.

Ans 3.

A cube has 6 faces. To find the volume of a cube when one side measures 6 inches, we use the formula V = s^3, where s is the length of one side. Plugging in s=6, we get V = 6^3 = 216 cubic inches.

Therefore, the area of the polygon is 24 square units.

In mathematics, a coordinate is a set of numbers or values that represent the position or location of a point or an object in space. Typically, coordinates are used to describe points in two-dimensional (2D) or three-dimensional (3D) space.

In 2D space, coordinates are usually represented as pairs of numbers (x, y), where the first number (x) represents the horizontal distance of the point from the origin, and the second number (y) represents the vertical distance of the point from the origin.

by the question.

To find the area of the polygon defined by the given coordinates, we can use the Shoelace Formula:

[tex]Area = 1/2 * |(x1y2 + x2y3 + ... + xny1) - (y1x2 + y2x3 + ... + ynx1)|[/tex]

where?[tex](x_{1} ,y_1), (x_2,y_2), ..., (x_n,y_n)[/tex] are the coordinates of the vertices listed in order.

Plugging in the given coordinates in order, we get:

[tex]Area = 1/2 * |(-12 + -5(-3) + (-5)(-5) + (-2)(-2) + 22 + 4(-2)) - (4*(-5) + (-5)(-5) + (-3)(-2) + (-2)2 + 2(-1) + 2*(-5))|= 1/2 * |-2 - 125 + 25 - 4 + 8 + 8 + 20 + 10 + 2 + 10|= 1/2 * |-48|=24[/tex]

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The value of the angle subtended by the chord WV is: x° = 79°.

The equal angles equal Chords Theorem states that If the angles subtended by the chords of a circle are equal in measure, then it means that the length of the chords is equal.

Now, we are given that the 2 chords are equal and we are given one angle extended by one chord as 79°. Therefore, we can easily say that by using the equal angles equal Chords theorem, that angle x° is equal to 79°.

Thus, we can say that is the value of the angle x°.

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The equation that shows how the amount of money Dalton gets in a week, y, depends on the number of times he walks the dog, x is y = 5 + x

Given that

Weekly allowance = $5

Amount earned for walking the dog = $1

The above means that

Total money = Weekly allowance  + Amount earned for walking the dog  * Number of times

So, we have

y = 5 + 1 * x

Evaluate

y = 5 + x

Hence, the equation is y = 5 + x

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The probability that Todd runs more than 15kms tomorrow is 38.46%.

By simply dividing the favorable number of possibilities by the entire number of possible outcomes, the probability of an occurrence can be determined using the probability formula.

Because the favorable number of outcomes can never exceed the entire number of outcomes, the chance of an event occurring might range from 0 to 1.

Moreover, the proportion of positive outcomes cannot be negative.

Here in the question,

Probability = No.of days with more than 15kms run/total no.of days.

= 5/2+3+3+5

= 5/13

= 0.3846

= 38.46%

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The complete question is:

Todd is training for a marathon. The following histogram summarizes the number of kilometers that he ran each day.

Remember, accurate interpretation of the results is essential for the success of Laura's dissertation.

The sum of squares for the factor in Laura Van Ertia's experiment can be found in the ANOVA display. ANOVA stands for Analysis of Variance, and it is a statistical method used to compare the means of two or more groups. The sum of squares is a measure of the variation within the data, and it helps to determine whether the differences between groups are statistically significant.

In the given question, you have not provided the actual ANOVA display, which is necessary to determine the sum of squares for the factor. The ANOVA table typically consists of columns for sources of variation (such as between groups and within groups), degrees of freedom, sum of squares, mean squares, F-ratio, and p-value. The sum of squares for the factor can be found in the 'sum of squares' column corresponding to the between-group variation row.

Once you have the ANOVA display, locate the sum of squares for the factor and use it in your analysis to understand the results of the experiment and its implications on the unified theory.

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Answer:

The probability of selecting an apple from the overripe fruit is 5/8. Since there are no replacement, the probability of selecting another apple from the remaining overripe fruit is 4/7, and the probability of selecting another apple after that is 3/6, and the probability of selecting another apple after that is 2/5. Therefore, the probability of selecting all apples is:

(5/8) × (4/7) × (3/6) × (2/5) = 0.0429

Rounded to four decimal places, the probability is 0.0429.

The equations without absolute value for the given conditions are: 1. y = -3x + 6 if x > 5; 2. y = -x - 6 if x < -2; 3. y = x - 6 if 3 ≤ x ≤ 5, and y = -x - 6 if x < 3.

1. When x > 5, the expression (x - 3) is positive, (x + 2) is positive, and (x - 5) is positive. Thus, to get absolute value we can rewrite the equation as:

y = (x - 3) + (x + 2) - (x - 5)

Simplifying this, we get:

y = 2x - 4

2. When x < -2, the expression (x - 3) is negative, (x + 2) is negative, and (x - 5) is negative. Thus, we can rewrite the equation as:

y = -(x - 3) - (x + 2) + (x - 5)

Simplifying this, we get:

y = -2x + 6

3. When -2 ≤ x ≤ 3, the expression (x - 3) is negative, (x + 2) is positive, and (x - 5) is negative. Thus, we can rewrite the equation as:

y = -(x - 3) + (x + 2) - (x - 5)

Simplifying this, we get:

y = 10 - x

When x > 3, the expression (x - 3) is negative, (x + 2) is positive, and (x - 5) is positive. Thus, we can rewrite the equation as:

y = -(x - 3) + (x + 2) + (x - 5)

Simplifying this, we get:

y = -2x + 6

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Thus, the function for the total amount paid by the manager is: t = 7 +28s.

a mathematical phrase, rule, or legislation that establishes the link between an independent variable and a dependant variable (the dependent variable).

Given data:

Let the number of bags ordered be 's'.

The total amount paid 't'.

Thus, the function for the total amount paid by the manager is:

t = 7 + 28s.

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the answer is 62x IAM take a guess so

The employees work in the production, service, and sales sections is 3/4 of total number. 1/4 of the employees work in other sections. The total number of employees in the factory is 14,400. The number of cars produced in 2020 is 439,875. The production decreased by 2.25% from 2018 to 2020.

Let the total number of employees in the factory be represented by the variable x. Then, the fraction of employees in the production section is 1/3x, the fraction in the service section is 1/4x, and the fraction in the sales section is 2/5 of the remaining employees, which is (1 - 1/3 - 1/4) x = 7/60x. Therefore, the fraction of employees in the production, service, and sales sections is:

1/3x + 1/4x + 7/60x = 15/20x = 3/4x

The fraction of employees in other sections is the remaining fraction after accounting for those in the production, service, and sales sections:

1 - 3/4x = 1/4x

Let the number of employees in the sales section be represented by the variable y. Then, the number of employees in the production section is y + 6000, and the number of employees in the service section is (1/4)x. Therefore, the total number of employees in the factory is:

y + 6000 + (1/4)x + y + 7/60x + 3/4x = 1x

Simplifying this equation, we get:

5/12x + 2y = 6000

we can find their sum as:

x + y = (12/5)(6000) = 14,400

The number of cars produced in 2019 is:

450,000 x 1.15 = 517,500

The number of cars produced in 2020 is:

517,500 x 0.85 = 439,875

The percentage change in production from 2018 to 2020 is:

[(439,875 - 450,000)/450,000] x 100% = -2.25% (rounded to two decimal places)

Therefore, the production decreased by 2.25% from 2018 to 2020.

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The vertex for the quadratic function and the range of the function are determined in the following answer.The given quadratic function [tex]isy = -32 - 2(x - 1)²[/tex]

It is noted that the quadratic function is in the vertex form, i.e.,[tex]y = a(x - h)² + k[/tex]

where (h, k) is the vertex of the parabola. Comparing the given function with the vertex form of the quadratic equation, we get the value of the vertex, i.e.,(h, k) = (1, -32).This implies that the vertex of the given quadratic function is located at (1, -32).Since the coefficient of the quadratic term is negative in the given function, the parabola opens downwards, and the vertex is the highest point on the graph.Therefore, the vertex (-32, 1) is the highest point on the graph of the given quadratic function. The range of a function is the set of all possible values that the function can attain. The given quadratic function is in the form y = a(x - h)² + k where a, h, and k are constants.

Since the coefficient of the quadratic term is negative in the given function, the parabola opens downwards. Therefore, the range of the given function is (-∞, -32].Hence, the range of the given quadratic function is[tex](-∞, -32].[/tex]

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In order to ensure the client understands the information being taught, the nurse should ask follow-up questions and provide resources, such as written handouts, for the client to review.

No, drinking 2 to 3 liters of fluids every day would not be an indication that additional teaching is required for good bowel habits. Instead, an indication that more teaching is needed could be the client saying that they are constipated, or that they are not having regular bowel movements.

In order to have healthy bowel habits, an older adult should aim for 3 to 4 bowel movements per week, with the stool being soft and easy to pass. Dietary fiber, drinking plenty of fluids, regular physical activity, and timed trips to the bathroom can help.

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The inequality representing the number of players on the court is given by 0 ≤ x ≤ 5.

Let us consider variable 'x' represents the number of players in the basketball team.

As we know,

Number of players is always greater than or equal to zero.

As number of players in the basketball team cannot be negative players.

As per condition,

Number of players in the basketball team = At most 5 players.

This implies,

Number of players in the basketball team is less than or equal to 5

The inequality that best describes the number of players on the basketball court is written as,

⇒ 0 ≤ number of players ≤ 5

⇒ 0 ≤ x ≤ 5

Therefore, the inequality used to represents the number of players in the basketball court is equal to 0 ≤ x ≤ 5.

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The probability of getting exactly two true answers in a quiz that consists of three true or false questions is 3/8 or 0.375. Here's how to get the answer:

There are three possible scenarios where you can get exactly two true answers: T T F, T F T, F T T
To calculate the probability of each scenario, you need to know the probability of getting a true answer (T) or a false answer (F). Since each question is independent, the probability of getting a true or false answer is 0.5.
Scenario 1: T T F,The probability of getting two true answers and one false answer is: P(T T F) = 0.5 x 0.5 x 0.5 = 0.125
Scenario 2: T F T,The probability of getting one true answer and two false answers is:P(T F T) = 0.5 x 0.5 x 0.5 = 0.125
Scenario 3: F T T,The probability of getting one false answer and two true answers is:P(F T T) = 0.5 x 0.5 x 0.5 = 0.125

To get the probability of getting exactly two true answers, you need to add the probabilities of the three scenarios:P(2 True Answers) = P(T T F) + P(T F T) + P(F T T) = 0.125 + 0.125 + 0.125 = 0.375.Therefore, the probability of getting exactly two true answers in a quiz that consists of three true or false questions is 3/8 or 0.375.

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The percentage of the cakes containing nuts using the ratio of different types of cakes is equal to 4.76%.

Let M represents the number of milk chocolate cakes

Let D represents the number of dark chocolate cakes

Let W represents the number of white chocolate cakes

From the given ratios, we have,

Ratio of the number of milk chocolate cakes to dark chocolate cakes

= 10:3

⇒D = M/(10 / 3)

⇒D = 3M/10

Ratio of the number of white chocolate cakes to milk chocolate cakes

= 1:6

⇒ W = M/6

Let N be the total number of cakes containing nuts.

Then the number of cakes not containing nuts is,

For milk chocolate cakes,

Ratio of the number of cakes containing nuts to not containing nuts = 1:8

⇒ 8N for every N containing nuts.

⇒M - N= 8N

⇒M = 9N

For dark chocolate cakes,

Ratio of the number of cakes containing nuts to not containing nuts = 3:2

⇒ 2N for every 3 containing nuts.

⇒ (D - 3N) = 2N

⇒ D = 5N

For white chocolate cakes,

Ratio of the number of cakes containing nuts to not containing nuts = 2:5

⇒5N for every 2 containing nuts

⇒ (W - 2N) = 5N

⇒ W = 7N

The total number of cakes is,

M + D + W

= 9N + 5N + 7N

= 21N

Percentage of cakes containing nuts is equal to,

=N/(21N) × 100

= 100/21

= 4.76%

Therefore, percentage of the cakes containing nuts using given ratio is equal to 4.76%.

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If we see the scatter plot we can conclude that the possible relation between x and y is linear and with a positive correlation since when the values of x increase the values for y increases as well.

[tex]Cov(x,y) =\frac{\sum_1^n(x_i-X')(y_i-Y')}{n-1}[/tex]

 [tex]\sum_1^5(6-16)(6-10)+(11-16)(9-10)....(27-16)(12-10)=106\\\\and\\Cov(x,y)=\frac{106}{4}=26.5\\\\r=0.693[/tex]

For this part we use excel in order to create the scatterplot and we got the result on the figure attached

If we see the scatter plot we can conclude that the possible relation between x and y is linear and with a positive correlation since when the values of x increase the values of y increase as well

The correlation coefficient is a "statistical measure that calculates the strength of the relationship between the relative movements of two variables". It's denoted by r and its always between -1 and 1.

And in order to calculate the correlation coefficient we can use this

[tex]Cov(x,y) =\frac{\sum_1^n(x_i-X')(y_i-Y')}{n-1}[/tex]

:  

[tex]Cov(x,y) =\frac{\sum_1^n(x_i-X')(y_i-Y')}{n-1}[/tex]

 [tex]\sum_1^5(6-16)(6-10)+(11-16)(9-10)....(27-16)(12-10)=106\\\\and\\Cov(x,y)=\frac{106}{4}=26.5\\\\r=0.693[/tex]

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This shows that they read less frequently than 95.7% of their classmates in the same grade level.

In statistics, a percentile is a metric that is used to express where one observation stands in relation to a larger group of data. It displays the proportion of data points that fall under a given value. For instance, if a student achieves a score in the 90th percentile on a standardised test, it signifies that they outperformed 90% of their peers who also took the test. As an alternative, if a child is taller than 25% of children their age and shorter than 75%, their height is in the 25th percentile for their age.

given

We must first compute the z-score using the following formula to determine the percentile of the child's reading proficiency.

z = (x - μ) / σ

where x represents the child's reading rate, represents the grade-level mean reading rate, and represents the standard deviation.

Inputting the values provided yields:

z = (46.2 - 85) / 22

z = -1.727

We can calculate or use a conventional normal distribution table to obtain the value of 0.0428 for the region to the left of z = -1.727. The child's reading rate is at the 4.28th percentile according to this statistic.

As a result, the child's reading proficiency is 4.3 percentile (rounded to one decimal place).

This shows that they read less frequently than 95.7% of their classmates in the same grade level.

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Numeric (continuous) variable: the amount of zantac in milliliters that the nurse will administer.A millilitre is a unit of capacity measurement. It is one thousandth of a litre in size. In other words, a one-liter container may hold  [tex]1,000[/tex] millilitres.

A millilitre is a more manageable unit of measurement for liquid volume or capacity in the metric system. It is used to measure smaller amounts of liquid and is equal to one thousandth of a litre (1 litre = 1000 millilitres).

To calculate how many millilitres of Zantac will the nurse administer to the patient, you need to know the concentration of Zantac in the dosage form and the prescribed dose for the patient.

For example, if the patient is prescribed 150 mg of oral Zantac twice daily and has a bottle of syrup that contains 15 mg/mL of ranitidine, then the nurse will administer [tex]10 mL (150 mg / 15 mg/mL)[/tex]  of syrup per dose.

Therefore, If the patient is prescribed 50 mg of intravenous Zantac every 8 hours and has a vial of solution that contains 25 mg/mL of ranitidine, then the nurse will administer 2 mL (50 mg / 25 mg/mL) of solution per dose.

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In the question given, The inequality will be h₂< h₁<h₃. Thus, Daniel is tallest.

In Mathematics, Inequality defines the relationship between two values that are not equal. When two values are not equal, we use “not equal sign (≠)”. But if there is a comparison between two values, whether it is less than or greater than, different inequalities are used. For example 5<7 , 8>4.

The following problem is to model an inequality situation.

Given that ,

Najid is taller than Emily and shorter than Daniel.

 Let, height of Najid is h₁

         height of Emily is h₂

  and height of Daniel is h₃

By the condition h₁>h₂ and h₁< h₃

Combining both the statement we have the inequality

              h₂< h₁<h₃

From this inequality we have the person having h₃ height is tallest.

Hence, Daniel is tallest.

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Answer:

(- 1, 1 )

Step-by-step explanation:

x = 6y - 7 → (1)

4x + y = - 3 → (2)

substitute x = 6y - 7 into (2)

4(6y - 7) + y = - 3 ← distribute parenthesis and simplify left side

24y - 28 + y = - 3

25y - 28 = - 3 ( add 28 to both sides )

25y = 25 ( divide both sides by 25 )

y = 1

substitute y = 1 into (1)

x = 6(1) - 7 = 6 - 7 = - 1

solution is (- 1, 1 )

Answer:

(-1,1)

Step-by-step explanation:

4(6y-7)+y=-3

24y+y=-3+28 (put them in like terms)

25y/25=25/25 (divide both side by 25)

y=1

x=6×1-7(already we get y and replace it by 1)

x=-1

(-1,1)

Answer: m = 6/5

Step-by-step explanation:

Answer: m=6/5

Step-by-step explanation:

The height of the water is increasing at the rate of 3 meters per minute.

We know that a cylindrical tank with radius r and height h has a volume of V = πr²h.

The rate at which the water is being filled in the tank is given as dV/dt = 2πr²( dh/dt).

On rearranging the above equation we get dh/dt = (dV/dt) ÷ (2πr²)

Given: Radius of the cylindrical tank, r = 3 m

Rate at which the water is being filled,

dV/dt = 54π cubic meters per minute.

we

Height of the cylindrical tank, h = ?

We need to find out how fast is the height of the water increasing.

This can be calculated using the formula.

dh/dt = (dV/dt) ÷ (2πr²)

Substitute the values of dV/dt  and r in the above formula,

we get,

dh/dt = (54π) ÷ (2π x 3²)

dh/dt = 54/18

dh/dt = 3 m/min

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The longest amount of time a patron can spend at the hot springs and still receive the discount is approximately 43.55 minutes.

To determine the longest amount of time a patron can spend at the hot springs and still receive the discount, we need to find the 5th percentile of the time spent by all patrons.

Let's say we have a sample of 1000 patrons and their time spent at the hot springs is normally distributed with a mean of 60 minutes and a standard deviation of 10 minutes. We can use the z-score formula to find the value corresponding to the 5th percentile:

[tex]$z = \frac{x - \mu}{\sigma}$[/tex]

where x is the value at the 5th percentile, [tex]$\mu$[/tex] is the mean, [tex]$\sigma$[/tex] is the standard deviation, and z is the z-score corresponding to the 5th percentile.

Since we want to find the value of x, we can rearrange the formula as:

[tex]$x = z\sigma + \mu$[/tex]

To find the z-score, we can use a standard normal distribution table or a calculator. For the 5th percentile, the z-score is approximately -1.645.

Plugging in the values, we get:

[tex]$x = -1.645 \times 10 + 60 \approx 43.55$[/tex]

Therefore, the longest amount of time a patron can spend at the hot springs and still receive the discount is approximately 43.55 minutes.

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