Three ballet dancers are positioned on stage. Max is 2 feet straight behind Kenji and 4 feet directly left of Lindsey. When the music begins, Max twirls to Lindsey's position, then leaps to Kenji's position, and finally walks back to his original position. How far did Max travel? If necessary, round to the nearest tenth.

At the 5% significance level, we can reject the null hypothesis and conclude that there is sufficient evidence to suggest that the average weight of the vacuums from Nature Abhors is less than 19.64 pounds.

To answer this question, we can use a one-sample t-test, assuming that the population of weights is normally distributed. The null hypothesis is that the true mean weight of the vacuums from Nature Abhors is 19.64 pounds, while the alternative hypothesis is that it is less than 19.64 pounds.

We can calculate the t-statistic using the formula

t = (X - μ) / (s / sqrt(n))

where X is the sample mean (19.33), μ is the hypothesized population mean (19.64), s is the sample standard deviation, and n is the sample size (31).

Since we don't know the population standard deviation, we can estimate it using the sample standard deviation. We calculate the sample standard deviation as follows

s = sqrt(sum((xi - X)^2) / (n - 1))

where xi is the weight of the i-th vacuum in the sample. We assume that the weights are normally distributed, so we can use the t-distribution with (n-1) degrees of freedom to calculate the p-value.

Using a t-table or calculator, we can find the critical t-value for a one-tailed test at the 5% significance level with 30 degrees of freedom to be -1.699.

Plugging in the numbers, we get

t = (19.33 - 19.64) / (s / sqrt(31))

s = sqrt(sum((xi - X)^2) / (n - 1)) = 0.714

t = (-0.31) / (0.714 / sqrt(31)) = -1.84

Since the calculated t-value (-1.84) is less than the critical t-value (-1.699), we can reject the null hypothesis at the 5% significance level. Therefore, we can conclude that there is sufficient evidence to suggest that the average weight of the vacuums from Nature Abhors is less than 19.64 pounds.

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Sophie deposited approximately $16,516.

The digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and so forth are known as whole numbers. They are the figures we use to measure things, such as the quantity of pencils in a box or apples in a basket. In computations like addition, subtraction, multiplication, and division, whole numbers are also employed.

Let's call the amount Sophie deposited "P". After 9 years, the investment would have gone through 18 compounding periods (2 per year for 9 years).

The formula for calculating the future value (FV) of an investment with compound interest is:

[tex]FV = P(1 + r/n)^{(nt)[/tex]

where:

P = the principal (the initial amount invested)

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the number of years

In this case, we have:

P = unknown (what we're trying to find)

r = 4.2% = 0.042

n = 2 (since the interest is compounded semiannually, or twice a year)

t = 9

So the formula becomes:

[tex]24,575.22 = P(1 + 0.042/2)^{(2*9)[/tex]

Simplifying the right side of the equation:

24,575.22 = P(1.021)¹⁸

24,575.22 = 1.485P

Dividing both sides by 1.485:

P ≈ $16,516

Therefore, Sophie deposited approximately $16,516.

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The values of the sides are;

11. SR = 25

12. RU = 25

13. SU = 35.4

14. Perimeter = 100 units

15. Area = 625 square units

It is important to note that a square have 4 equal sides and 4 equal angles.

From the information given, we have that;

Line SR = line RU

But we have that;

SR = 2k + 7

RU = 3k - 2

Equate the sides, we have;

2k + 7 = 3k - 2

collect the like terms

2k - 3k = - 2 - 7

subtract the terms

-k = - 9

k = 9

Then, SR =2(9) + 7 = 25

RU = 3(9) - 2 = 25

The line SU is the hypotenuse side

Using the Pythagorean theorem

SU² = 25² + 25²

SU² = 625 + 625

SU = √1250

SU = 35. 4

15. The perimeter of a square is expressed as;

Perimeter = 4a

Perimeter = 4(25)

Perimeter = 100 units

16. The area of a square is expressed as;

Area = a²

Area = 25²

Area = 625 square units

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Required corresponding sides are side AB corresponds to side A'B', side BC corresponds to side B'C', side AC corresponds to side A'C'

What is corresponding side?

In geometry, when two figures are similar, their corresponding sides are the sides that are in the same position and have the same relative orientation in each figure.

For example, consider two triangles ABC and DEF, where angle A is congruent to angle D, angle B is congruent to angle E, and angle C is congruent to angle F. If the length of AB is proportional to the length of DE, BC is proportional to EF, and AC is proportional to DF, then side AB corresponds to side DE, side BC corresponds to side EF, and side AC corresponds to side DF.

To list all the corresponding sides of two triangles ABC and A'B'C', we need to identify which side of triangle ABC corresponds to which side of triangle A'B'C'. Corresponding sides are those that are in the same position relative to each triangle.

We can identify the corresponding sides by comparing the positions of the vertices. For example, the side opposite vertex A in triangle ABC corresponds to the side opposite vertex A' in triangle A'B'C'. Similarly, the side opposite vertex B in triangle ABC corresponds to the side opposite vertex B' in triangle A'B'C', and so on.

Therefore, the corresponding sides are side AB corresponds to side A'B', side BC corresponds to side B'C', side AC corresponds to side A'C'.

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The correct option regarding which bus has the least spread among the travel times is,

⇒ Bus 14, with an IQR of 6.

Now, First we consider the dot plot, which shows the number of times that each observation appears in the data-set.

Then we consider the interquartile range, which gives the difference between the third quartile and the first quartile of the data-set.

The interquartile range is a better measure of spread compared to the range of a data-set, as it does not consider outliers.

For groups of 15 students, we have that:

The first half is composed by the first seven students, hence the first quartile is the fourth dot, which is the median of the first half.

The second half is composed by the last seven students, hence the first quartile is the eleventh dot, which is the median of the first half.

The quartiles for Bus 14 are given as follows:

Q1 = 12.

Q3 = 18.

Hence the IQR is of:

IQR = Q3 - Q1

= 18 - 12

= 6.

The quartiles for Bus 18 are given as follows:

Q1 = 9.

Q3 = 16.

Hence the IQR is of:

IQR = Q3 - Q1

= 16 - 9

= 7.

Hence, Bus 14 is the more consistent bus, due to the lower IQR.

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For a one-tailed hypothesis test (upper tail), the p-value is computed to be .034. If the test is being conducted at the 5% level of significance, the null hypothesis C. is rejected

A one-tailed hypothesis test refers to a statistical test in which the alternative hypothesis is either greater than or less than the null hypothesis, but not both. Since the problem states that it is a one-tailed hypothesis test (upper tail), the alternative hypothesis, H1, will be a greater than or > sign. Hypothesis testing is the method of determining whether or not an assertion about a population parameter is consistent with sample data. A hypothesis test is conducted to assist in deciding between two competing hypotheses. One hypothesis is the null hypothesis (H0), which is a statement of the population parameter that is being tested.

In a hypothesis test, the null hypothesis is rejected if the test statistic falls in the rejection region. The test statistic, which measures the distance between a sample estimate and the null hypothesis, is used to assess whether or not the null hypothesis should be rejected. Reject the null hypothesis if the p-value is less than or equal to α. Since the p-value of 0.034 is less than the level of significance, α, of 0.05, the null hypothesis will be rejected, according to the question above. Therefore, option C is correct.

The Question was Incomplete, Find the full content below :

For a one-tailed hypothesis test (upper tail), the p-value is computed to be .034. If the test is being conducted at the 5% level of significance, the null hypothesis

a. could be rejected or not rejected depending on the sample size.

b. could be rejected or not rejected depending on the sample mean.

c. is rejected.

d. is not rejected.

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Answer:

10% of $10=$1

Step-by-step explanation

1/10=10% hence 10% of $10 is %1

The sides of the parallelogram are thus congruent. Hence, the parallelogram is similar.

Congruent refers to something that is "absolutely equal" in terms of size and form. The shapes hold true regardless of how we rotate, flip, or turn them. Draw two circles with the same radius, for instance, cut them out, then stack them on top of one another. We will see that they will superimpose, or be positioned entirely on top of, one another. This demonstrates the congruence of the two circles. The circles below may be put perfectly over one another and have the same radius, hence they are considered to be congruent.

For the parallelogram ESPR and ECTF, the scale factor is:

Scale factor = 9/3 = 3

The sides of the parallelogram are thus congruent. Hence, the parallelogram is similar.

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The equation that describes the relationship between the time a zebra runs at a speed and the distance it covers is y = 2/3(x).

A linear equation is an algebraic equation of the form y=mx+b. where m is the slope and b is the y-intercept.

Assuming that the relationship between the time a zebra runs at a speed and the distance it covers is a linear one, we can use the formula for the equation of a straight line:

y = mx + b

where y is the distance covered, x is the time the zebra runs at a speed, m is the slope (i.e., the speed of the zebra), and b is the y-intercept (i.e., the distance covered when the zebra is not running).

From the table two points can be formed as (x, y) that are, (3, 2) and (6, 4)

Using these two points, we can find the slope of the line:

m = (y₂ - y₁)/(x₂ - x₁)

 = (4 - 2)/(6 - 3)

 = 2/3

Now as we have the slope m we can use it in point slope form, i.e.

y - y₁ = m(x - x₁)

Using the point (3, 2),

y - 2 = 2/3(x - 3)

y - 2 = 2/3(x) - 2

y  = 2/3(x) - 2 + 2

y  = 2/3(x)

Hence, the equation that describes the relationship between the time a zebra runs at maximum speed and the distance it covers is:

y  = 2/3(x)

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Complete question:

The relationship between the amount of time a zebra run at maximum speed in the distance it covers is shown.

Write an equation to describe this relationship.

A design that has two conditions with different participants in each condition is independent groups design. Hence option a) independent groups design is correct.

The student question refers to a design where there are two conditions with different participants in each condition.

An independent groups design, also known as a between-subjects design, involves comparing two or more groups of participants who are not the same individuals.

In this type of design, each participant is exposed to only one level of the independent variable.

The groups are formed independently of each other, and any differences between them can be attributed to the manipulation of the independent variable.

An independent groups design: Identify the research question and the independent variable. Form two or more groups of participants, ensuring that they are independent of each other.

Manipulate the independent variable for each group, ensuring that each participant is exposed to only one level of the variable.

Option b is wrong because repeated measures design involves testing the same participants under different conditions or at different times.

Option c is wrong because solomon four-group design is a complex design involving four groups with two groups receiving a pretest, and two groups not receiving a pretest.

Option d is wrong because the pretest-posttest design involves assessing participants before and after a treatment or intervention.

The correct option is a) independent groups design.

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Angle R is right angle (QR perpendicular to PT) then ∆PQR ~ ∆TSR   (both are similar triangle).

Triangles with the same shape but different sizes are called similar triangles. To put it another way, the sides and angles of the same triangle are in proportion to each other. This indicates that the shape of one of the triangles would remain unchanged regardless of how much you scaled it up or down. Similar triangles are important in geometry and have numerous practical uses in architecture and engineering.

From the given figure,

Statement                   Reason

∠P = ∠T                      AA similarity theorem

∠R is right angle        QR ⊥ PT

Then ∆PQR ~ ∆TSR   (both are similar triangle)

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The minimum number of customers that need to be included in the test to determine if the new plan has more than a 10 percentage point effect on customer retention, with 90% confidence is at least 192 customers.

To determine the minimum sample size needed to test if the new plan has more than a 10 percentage point effect on customer retention, we can use the following formula:

n = [(Zα/2 + Zβ)² × (p1 × (1-p1) + p2 × (1-p2))] / (p1 - p2)²

where:

n is the sample size needed

Zα/2 is the z-score corresponding to the desired confidence level (90% confidence corresponds to a z-score of 1.645)

Zβ is the z-score corresponding to the desired power (we'll use a power of 80%, which corresponds to a z-score of 0.84)

p1 is the retention rate under the current plan (55%)

p2 is the retention rate under the new plan (55% + 10% = 65%)

Plugging in the values, we get:

n = [(1.645 + 0.84)² × (0.55 × 0.45 + 0.65 × 0.35)] / (0.10)²

n = 191.34

Rounding up to the nearest whole number, we get a minimum sample size of 192.

Therefore, the company needs to include at least 192 customers in the test to determine if the new plan has more than a 10 percentage point effect on customer retention, with 90% confidence.

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The simple interest rate is therefore 6.25% as the initial principal (P) is 2000 Birr and the final amount (A) is 2500 Birr.

what is interest ?

A creditor assesses interest as a cost to a borrowing in exchange for the use of money; Typically, it is expressed as a portion of the principal amount borrowed. Generally, it is the cost of getting cash.

given

The simple interest formula is

S I = P * r * t,

where I is the interest,

P is the principal (amount of money we start with),

r is the interest rate, and

t is the amount of time in years.

In this instance, we are aware that the initial principal (P) is 2000 Birr and the final amount (A) is 2500 Birr.

The interest rate (r) is constant because we also know that the interest is simple.

We may determine r by using the simple interest formula:

SI = P*r*t= 2500-2000

500 = 2000*r*4

500 = 8000r

r = 500/8000

r = 0.0625

The simple interest rate is therefore 6.25% as the initial principal (P) is 2000 Birr and the final amount (A) is 2500 Birr.

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Answer:

( 2q + 1 ) ( q + 3 )

Step-by-step explanation:

➟ 2q^2+7q+3

➟ ( 2q + 1 ) ( q + 3 )

The probability that the baseball player has exactly 1 hit in his next 7 at-bats is 0.2908, rounded to 4 decimal places.

It is a statistic that measures the performance of baseball players. It is the ratio of the number of hits to the number of at-bats.

Probability is the measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1.

Static means the precise value of something, without approximation or estimation.

It is a process of approximating a number to a specific number of decimal places.

A baseball player has a batting average of 0.365. The probability that he has exactly 1 hit in his next 7 at-bats is to be found.

Step 1: Calculate the probability of getting a hit in one at-bat. The probability of getting a hit in one at-bat is equal to the batting average of the player.

P(hit) = 0.365

Step 2: Calculate the probability of not getting a hit in one at-bat. The probability of not getting a hit in one at-bat is equal to the difference between 1 and the batting average.

P(not hit) = 1 - 0.365 = 0.635

Step 3: Calculate the probability of getting exactly 1 hit in 7 at-bats. The probability of getting exactly 1 hit in 7 at-bats can be calculated using the binomial probability formula:

P(1 hit in 7 at-bats) = 0.2908 (rounded to 4 decimal places)

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After 3 weeks, the dog consumed 3 pounds of food, leaving 7 pounds of food left. Subtracting the amount of food consumed from the initial amount, 10 pounds - 3 pounds = 7 pounds.

After 3 weeks, the dog would have consumed 3 pounds of food (1 pound per week x 3 weeks = 3 pounds).

To find out how many pounds of food are left, we can subtract the amount of food consumed from the initial amount:

10 pounds - 3 pounds = 7 pounds

So there are 7 pounds of food left after 3 weeks.

Expressed in decimal form, it would be 7.0 pounds.

The base-10 numbering system, which employs ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) to represent all conceivable numerical values, is used to express numbers in decimal form. When a number is expressed in decimal form, there can be one or more digits to the left of the decimal point that represent the whole number portion of the value and one or more digits to the right of the decimal point that represent the fractional portion of the value.

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The probability of running out of anchovies before the evening is over is approximately 0.058, or 5.8%. We must presume that the number of pizzas ordered with anchovies follows a binomial distribution with parameters n = 60 (the total number of pizzas) and p = 0.08 in order to answer this issue using the normal approximation for the binomial distribution. (the probability of ordering anchovies on a pizza). 

The standard deviation of this binomial distribution is given by = sqrt(np(1-p)) = sqrt(60 x 0.08 x 0.92) = 2.03, and the mean is given by = np = 60 x 0.08 = 4.8.

Now, we need to determine the likelihood that we will need to prepare more than eight anchovy-topped pies before the evening is through in order to determine the likelihood that we will run out of anchovies. (since we only have enough anchovies for 8 pizzas).

This is equivalent to finding the probability that the number of pizzas with anchovies is greater than 8, or P(X > 8), where X is the number of pizzas with anchovies.

To use the normal approximation for the binomial distribution, we need to standardize the variable X using the standard normal distribution. This gives us:

z = (X - μ) / σ = (8 - 4.8) / 2.03 = 1.57

Using a standard normal table or calculator, we can find the probability that z is greater than 1.57, which is approximately 0.058. Therefore, the probability of running out of anchovies before the evening is over is approximately 0.058, or 5.8%.

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Triangle ABC's sides are not equal in length, hence it's not an equilateral triangle.

It has three equal sides. The three angles are equal. Every edge of an equilateral is 60 degrees since the total of the inner angles equals 180 degrees. It has three regular sides and is a polygon. A vertex's height and median are represented by a single line.

A triangular with equally length sides is known as an equilateral in geometry. The three angles opposing the three equal sides are equal in size because the three are equal. As a result, with each angle measuring 60 degrees, it is sometimes referred to as an equiangular triangle.

Using the distance formula, we can find the lengths of the three sides:

[tex]AB = \sqrt{((5 - 4)^2 + (5 - (-2))^2)} = \sqrt{(2^2 + 7^2)} = \sqrt{(53)}[/tex]

[tex]BC = \sqrt{((-1 - 5)^2 + (3 - 5)^2) } = \sqrt{((-6)^2 + (-2)^2)} = \sqrt{(40)}[/tex]

[tex]CA = \sqrt{((4 - (-1))^2 + (-2 - 3)^2)} = \sqrt{(5^2 + 5^2)} = \sqrt{(50)}[/tex]

Since AB, BC, and CA have different lengths, the triangle is not equilateral.

Therefore, the answer is: Triangle [tex]ABC[/tex] is not an equilateral triangle because its sides have different lengths.

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Since AB and AC have the same length, but BC has a different length, Therefore,  ΔABC is not an equilateral triangle.

Equilateral triangles are those with three sides that are all the same length.

In other words, all three sides of an equilateral triangle are of the same length, and all three angles within the triangle are also equal to 60 degrees.

Equilateral triangles are a type of regular polygon, which means that they have congruent sides and congruent angles.

Triangle ABC is not an equilateral triangle because its side lengths are not all equal.The lengths of the sides can be determined using the distance formula:

AB = √((5-4)² + (5-(-2))²) = √(1 + 49) = √(50)

BC = √((5-(-1))² + (5-3)²) = √(36 + 4) = √(40)

AC = √((4-(-1))² + (-2-3)²) = √(25 + 25) = √(50)

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the altitude of the airplane would decrease by 1628 feet over a period of 37 seconds.

What is Rate?

A rate in arithmetic is a ratio that contrasts two separate values with various unit systems. For instance, if John types 50 words per minute, that means he types 50 words per minute. We are dealing with a rate because the word "per" is there. The symbol "/" can be used in place of the word "per" in issues.

When two or more similar amounts or numbers are being compared using the same units, a ratio is utilized. When referring to the ratio of one quantity "to" the second quantity in spoken language, it is frequently written with a colon.

To find the change in altitude, we need to multiply the rate of decrease by the time period:

change in altitude = rate of decrease × time period

So, in this case:

change in altitude = 44 feet/second × 37 seconds

change in altitude = 1628 feet

Therefore, the altitude of the airplane would decrease by 1628 feet over a period of 37 seconds.

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Answer:

Step-by-step explanation:

Surface area of a cylinder = 2πr(r + h)

radiius = 1/2 diameter so the radius is 1/2 of 2 = 1

Surface are = 2(22/7)(1 )[(1 + 1.25)]

                     44/7 (2.25) = 99/7

This is for ONE muffin - mulitply by 6 to find out how much paper for

6 muffins,

6 × 99/7 = 594/7 = 84 6/7 in²

Answer:

A) -⅕.

Step-by-step explanation:

If lines s and t are perpendicular, then their slopes are negative reciprocals of each other. Therefore, if the slope of line s is 5, the slope of line t is -1/5.

So the answer is A) -⅕.

The slope of line T is -1/5, because perpendicular lines have opposite reciprocal slopes, meaning the opposite sign (positive or negative) and flipped (whole number becomes a fraction).

The point (1, 5) is a solution of the first equation, and is not a solution of the second one (and thus the system)

Here we want to see if the point (1, 5) is a solution of some given equations.

The first equation is:

d + s = 6

let's assume that the notation for the point is (d, s), then we can replace these values in the equation above to get:

1 + 5 = 6

6 = 6

So yes, the point is a solution here.

For the next equation we have:

10d + 28s = 78

Replacing the values:

10*1 + 5*28 = 78

150 = 78

So no, here is not a solution.

And the point is a solution of the system only if it is a solution of both of the equations, then the point is not a solution of the system,

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The 95% confidence interval for the mean difference in the amount absorbed is (0.01, 3.25) ml, with a df of 28. Yes, the interval does not contain zero, suggesting that there is a significant difference between the two brands in the amount absorbed.

To find the 95% confidence interval for the mean difference in the amount absorbed, we can use the formula:

CI = (x1 - x2) ± tα/2 * SE

where x1 - x2 is the difference in means, tα/2 is the critical value from the t-distribution table for a 95% confidence interval with degrees of freedom (df) = 30 - 2 = 28 (assuming equal variances), and SE is the standard error of the difference in means:

SE = sqrt[(s1^2/n1) + (s2^2/n2)]

Plugging in the values, we get:

x1 - x2 = 15.63 - 14 = 1.63

s1 = 3.12, n1 = 16

s2 = 2.53, n2 = 16

SE = sqrt[(3.12^2/16) + (2.53^2/16)] = 0.974

tα/2 for df = 28 and α = 0.05 is 2.048.

CI = (1.63) ± (2.048 * 0.974) = (0.01, 3.25)

Therefore, the 95% confidence interval for the mean difference in the amount absorbed is (0.01, 3.25), with df = 28.

This interval suggests that there is a significant difference between brands in the amount absorbed, since the interval does not include 0. This means that we can reject the null hypothesis that the means are equal at a 5% significance level and conclude that the mean amount of water absorbed for brand A is significantly greater than brand B.

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Therefore , the solution of the given problem of ratio comes out to be John needs to use 36 cans of green beans.

The simple procedure percentage "a / b," once "b" could be a larger than zero, is used to find a group of numbers both "a" and " that appear to be equal. Two ratios are combined to form a ratio. If there were only one man and three ladies, the ratio would be 1:1. In the group, there are 3/4 women and 1/4 males. A division between things, a number, or a particular physical component of a component can all be described as having "parts."

Here,

Let's express the quantity of pumpkin cans as 5x (where x is some multiplier) and the quantity of green bean cans as 2x.

Given that there are 126 cans in total, we can construct the following equation:

=> 5x + 2x = 126

Combining related words gives us:

=>7x = 126

When we multiply both parts by 7, we get:

=> x = 18

John therefore requires 2 cans of green beans and 5 cans of pumpkin:

=> 5x = 5(18) = 90 pumpkin cans

=> 2x = 2(18) = 36 green bean cans.

So, for his exhibit, John needs to use 36 cans of green beans.

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Using the time formula, it will take the lizard  6days to reach the top of the well.

The time formula, using distance and speed, is as follows:

Time = Distance ÷ Speed

The depth of the well and the lizard's position = 12 feet

Daily crawling-up rate = 4 feet

Daily crawling down rate = 2 feet

Difference between crawling up and slipping down rates = 2 foot per day ( 4- 2)

Therefore, the lizard's net average crawling up speed or rate = 2 foot per day

Time (in days) = Distance/Average Rate

12/2 = 6 days

Thus, for the lizard to reach the top of the well, it will take 6 days.

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Answer:

Let (x1, y1) = (-9, 5) and (x2, y2) = (3, 2) be the two points.

The formula for finding the slope between two points is:

m = (y2 - y1) / (x2 - x1)

Substituting the values, we get:

m = (2 - 5) / (3 - (-9))

m = -3 / 12

m = -1/4

Therefore, the slope between the points (-9, 5) and (3, 2) is -1/4.

Answer:

M = -1/4

Step-by-step explanation:

(y2-y1)/(x2-x1) = Slope

we use the x and y components of the given coordinates to find out the answer.

Therefore,

(2-5)/(3-(-9)) = -1/4

The equation of the tangent line to the circle at the point (0, 4) is y = 1/2x + 4

To find the equation of the tangent line to the circle at the point (0,4), we need to first find the slope of the tangent line at that point.

The slope of a tangent line to a circle at a given point is equal to the negative reciprocal of the slope of the radius that passes through that point.

The center of the circle is at the point (2, 0), and the radius passing through the point (0, 4) has a slope of:

m = (0 - 4)/(2 - 0) = -2

Therefore, the slope of the tangent line at (0, 4) is:

m' = -1/m = -1/(-2) = 1/2

Now that we have the slope of the tangent line, we can use the point-slope form of a line to find the equation of the tangent line.

The point-slope form of a line is:

y - y1 = m(x - x1)

where m is the slope of the line, and (x1, y1) is a point on the line.

Using (0,4) as the point on the line and 1/2 as the slope, we get:

y - 4 = 1/2(x - 0)

Simplifying, we get:

y = 1/2x + 4

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The lateral area of the cylinder is 24π square units.

To find the lateral area of the cylinder, we need to use the formula:

Lateral Area = Circumference of circular base × Height.

First, we will find the circumference of the circular base using the area given.
1. We know the area of the circular base is 36π square units.

The formula for the area of a circle is A = πr²,

where A is the area and r is the radius.
2. We are given A = 36π,

so 36π = πr².
3. Divide both sides by π to isolate r²:

36 = r².
4. Take the square root of both sides to find the radius:

r = 6 units.
5. Now that we have the radius, we can find the circumference of the circular base using the formula C = 2πr,

where C is the circumference.
6. Substitute r = 6 into the formula:

C = 2π(6) = 12π.
7. The circumference of the circular base is 12π units.
8. The height of the cylinder is given as 2 units.
9. Finally, we can find the lateral area using the formula Lateral Area = Circumference × Height.
10. Substitute C = 12π and H = 2 into the formula:

Lateral Area = 12π × 2 = 24π.

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