D. A population of rabbits is doubling every 3 months. If there were 2 rabbits to begin
with, how many will there be after 5 years?

1. The number of hours which Sidra did on homework is 12 hours

2. The total number of hours which Sally's friends spend on homework per month is 64 hours.

Sidra does 3 keys worth of homework, and since a key equals 4 hours, she does:

= 3 keys x 4 hours/key

= 12 hours of homework.

To find the total number of hours that Sally's friends spend on homework per month, we need to add up the hours of homework done by each of her friends:

Tim: 5 keys x 4 hours/key = 20 hours

Mary: 2 1/2 keys x 4 hours/key = 10 hours

Keith: 5 1/2 keys x 4 hours/key = 22 hours

Sidra: 3 keys x 4 hours/key = 12 hours

Total hours of homework per month:

= 20 + 10 + 22 + 12

= 64 hours.

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In response to the stated question, we may state that The scatter plot indicates a clustering pattern in the data, and as elevation increases, temperature drops.

"Scatter plots are graphs that show the association of two variables in a data collection. It is a two-dimensional plane or a Cartesian system that represents data points. The X-axis represents the independent variable or characteristic, while the Y-axis represents the dependent variable. These plots are sometimes referred to as scatter graphs or scatter diagrams."

"A scatter plot is also known as a scatter chart, scattergram, or XY graph. The scatter diagram plots numerical data pairings, one variable on each axis, to demonstrate their connection."

Because the graph is a scatter plot, the data displays a clustering pattern.

We may deduce from the figure that as height increases, temperature falls.

As a result, C and E are the proper choices.

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The correct question is -

The scatter plot shows the relationship between elevation and temperature on a certain mountain peak in North America. Which statements are correct?

A. The data shows one potential outlier

B. The data shows a linear association

C. The data shows a clustering pattern

D. The data shows a negative association

E. As elevation increases, temperature decreases

Regardless of the appearance of the normal distribution or the size of the standard deviation, approximately less than 2.98% of observations consistently fall within two deviations types (one high and one low).

The normal distribution, also known as the Gaussian distribution, is a probability distribution symmetrical about the mean, indicating that data near the mean occurs more frequently than data far from the mean. In graphical form, the normal distribution is represented by a "bell curve".

The normal distribution is important in statistics and is often used in the natural and social sciences to represent real-valued random variables whose distribution is unknown. Their importance is partly due to the central limit theorem. It states that in some cases the mean of many samples (observations) of a random variable with finite mean and variance is itself a random variable - whose distribution converges to a normal distribution as the size of the l sample increases. Therefore, physical quantities assumed to be the sum of many independent processes, such as measurement errors, tend to have a near-normal distribution.

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365,000

lots of speed added together with air resistance

Answer:

The bunny ran 68 units and the average speed is 4 units per minute.

Step-by-step explanation:

Since the bunny is running from 34 to -34 you need to add those numbers up to get 68. You then need to divide 68 by 17.

68/17 = 4

So the bunny ran 68 units and the average speed is 4 units per minute

The ramp will rise from the ground at its highest end to a height of approximately 8.56 inches. Rounded to the nearest hundredth of an inch, this is 8.56 inches.

Fiberboard is a type of engineered wood product that is made by combining wood fibers or other plant-based fibers with a binder or adhesive, and then compressing and heating the mixture to create a dense, flat panel

We can use trigonometry to solve this problem. Let's call the height of the ramp "h" and the length of the base "b". We want to find "h".

First, we need to find "b". We know that the fiberboard is 27 inches long, so the base of the ramp will be the length of the hypotenuse of a right triangle. Using trigonometry, we can find:

b = 27 * cos(19°) ≈ 25.68 inches

Next, we can use trigonometry again to find "h". We know that the angle of elevation from the ground is 19°, so the angle of depression from the ramp to the ground is also 19°. This means we have a right triangle with the height "h", the base "b", and an angle of 19°. Using trigonometry, we can find:

tan(19°) = h/b

Solving for "h", we get:

h = b * tan(19°) ≈ 8.56 inches

Therefore, the ramp will rise from the ground at its highest end to a height of approximately 8.56 inches. Rounded to the nearest hundredth of an inch, this is 8.56 inches.

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Answer:

The volume of the sphere 288π cubic meters which is approximately 904.8 cubic meters to the nearest tenth.

Step-by-step explanation:

The given figure is a sphere.

The formula for the volume of a sphere is:

[tex]V=\dfrac{4}{3}\pi r^3[/tex]

where r is the radius.

Given the diameter of the sphere is 12 m, and the radius is half the diameter, r = 6 m.

To calculate the volume of the sphere, substitute r = 6 into the formula:

[tex]\begin{aligned}\implies V&=\dfrac{4}{3}\pi \cdot 6^3\\\\&=\dfrac{4}{3}\pi \cdot 216\\\\&=288\pi \\\\ &\approx 904.8\;\sf m^3\;(nearest\;tenth)\end{aligned}[/tex]

Therefore, the volume of the sphere 288π cubic meters which is approximately 904.8 cubic meters to the nearest tenth.

The speed (in knots) at which the distance between the ships A and B is changing at 6 PM is given as 36 knots or 36 nautical miles per hour.

Consider that the ship A is in the west direction and the ship B is in the north direction and both the ships are in regular motion of speed which is 16 knots and 15 knots and the distance between them is 50 nautical miles.

Using the Pythagoras theorem, the relation of the distance x which represents the distance between ships at 6PM to the distances that each ship has travelled can be given as follows:

x^2 = (50 + 16t)^2 + (15t)^2

where, t is the number of hours that has passed since noon.

Differentiating both sides of the above equation with respect to time, we get:

2x*(dx/dt) = 2(50 + 16t)*(16) + 2*(15t)*(15)

t = 6, at 6 PM, therefore substituting the value and solving, we get:

2x(dx/dt) = 2[(50 + 16(6)]*(16) + 2*[15(6)]*(15)

2x(dx/dt) = 4194

dx/dt = 2097/x

Now substituting the value of x that corresponds to 6 PM:

x^2 = (50 + 16(6))^2 + (15(6))^2

x^2 = 3385

x = √3385 ≅ 58.19

Putting this value in dx/dt, we get:

dx/dt = 2097/58.19 ≅ 36.00 knots

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Approximately 95.44% of the data falls within ±2 standard deviations of the mean in a normal distribution.

To find the percentage of the area under the normal curve that falls between ±2 standard deviations, we need to follow the following steps:

We need to know the mean (μ) and standard deviation (σ) of the normal distribution in question. If we assume a standard normal distribution (i.e., a normal distribution with mean of 0 and standard deviation of 1), then we can use a z-score table to find the percentage of area under the curve.

The z-score formula is:

z = (x - μ) / σ

For ±2 standard deviations, the values of x are μ ± 2σ. Therefore, the z-scores are:

z = (μ + 2σ - μ) / σ = 2

z = (μ - 2σ - μ) / σ = -2

A z-score table gives the percentage of area under the standard normal curve that falls to the left of a given z-score. Since the normal distribution is symmetric, the percentage of area to the right of a negative z-score is the same as the percentage of area to the left of the corresponding positive z-score.

Using a z-score table, we find that the percentage of area under the standard normal curve that falls to the left of z = 2 is 0.9772, or 97.72%. Therefore, the percentage of area under the curve that falls between ±2 standard deviations is:

97.72% - (100% - 97.72%) = 95.44%

This means that approximately 95.44% of the data falls within ±2 standard deviations of the mean in a normal distribution.

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Answer:

146 feet is the correct option

Answer: 146

Step-by-step explanation:

just filling in so you can give other brainliest

As the demand for tobacco is inelastic so the consumers are the group who are less responsive to a higher price as an outcome of it the consumers will have to bear the largest share of the tobacco tax.

This inelasticity of demand will lead to only a small decline in the quantity demanded after the tax have been leived , therefore the deadloss weight will be in really small degree. the percentage increase in price of any amount will overcome a smaller decline in the quantity which eventually would lead to a rise in the tax revenue collection.

Inelasticity of demand refers to the degree to which the quantity demanded of a particular good or service changes in response to a change in its price. When demand is inelastic, a change in price will result in a proportionally smaller change in quantity demanded. This is typically the case for goods or services that are considered necessities or have few substitutes available.

For example, if the price of insulin, a life-saving medication for diabetics, increases by 10%, it is unlikely that the quantity demanded will decrease by 10%. People with diabetes require insulin to manage their condition, and there are few substitutes available, so they are willing to pay a higher price to maintain their health.

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Complete Question: -

Suppose the supply of tobacco is elastic and the demand for tobacco is inelastic. If an excise tax is levied on the suppliers of tobacco, will the incidence fall mostly on consumers or mostly on producers? Will there be a large amount or small amount of deadweight loss? Will tax revenue from the tobacco tax fall or rise?

The answer is 58 days.

How to determine number of days between months?

You must take the start date and finish date and subtract them to find the number of days between the two dates. You should determine how many full years are involved if this spans multiple years.

To determine the number of days between June 27 and August 30, we can count the number of days in each of the months of July and August and add them up.

July has 31 days, and August has 30 days. Therefore, the total number of days between June 27 and August 30 is:

31 days (in July) + 30 days (in August) - 3 days (from June 27 to June 30) = 58 days

So, the answer is 58 days.

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Answer:

During a 5 minute period, the minute hand moves 1 portion out of the 12 portions in a clock. Hence, the minute hand turns π6 radians during a 5 minute period.

For 95% confidence interval, Standard Error is 3, Margin of Error is 6.4, Lower limit is 65.1 and Upper limit is 77.9.

A confidence interval is a range of values that is likely to contain the true value of an unknown population parameter with a certain degree of confidence based on a sample from that population. It is calculated from the sample data and provides a range of values within which the true population parameter is likely to fall.

1) standard error = s / √(n)

standard error = 12 / √(16)

standard error = 3

2) margin of error = t-value x standard error

margin of error = 2.131 x 3

margin of error = 6.393

3) lower limit = sample mean - margin of error

lower limit = 71.5 - 6.393

lower limit = 65.107

4) upper limit = sample mean + margin of error

upper limit = 71.5 + 6.393

upper limit = 77.893

Therefore, the information for a 95% confidence interval is:

Sample mean = 71.5 pounds

Standard Deviation = 12 pounds

Sample size = 16

Z or t = t

Standard Error = 3

Margin of Error = 6.4

Lower limit = 65.1

Upper limit = 77.9

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Answer:

[tex](x + 4)(2x - 5) = 0[/tex]

explanation:

[tex]x \times 2x = 2 {x}^{2} [/tex]

[tex]( - 5 \times x) + (4 \times 2x) = 3x[/tex]

[tex] (- 5) \times 4 = - 20[/tex]

8am : -14F

noon : -14+10  = -4F

increase between noon and 4 is 2*10 =20

4 pm: -4F + 20 = 16F

temperature at 4 pm is 16F

The p-value is greater than the level of significance (0.05). Therefore, we fail to reject the null hypothesis. There is not enough evidence to suggest that the mean time to complete a four-year college degree varies based on gender. The conclusion is that there is no difference between the mean number of years for men and women to complete a four-year college degree.

The null hypothesis in the given data set is there is no difference between the mean number of years for men and women to complete a four-year college degree. The null hypothesis is a statement that describes the population parameter being tested. It is usually given the symbol H0. The null hypothesis is that the population parameter is equal to some value.

The null hypothesis is what is assumed to be true if no evidence is available to contradict it. It can be accepted or rejected based on the results of a statistical test. The degrees of freedom is the number of independent observations in a statistical test. It is usually represented by the symbol df. The degrees of freedom is used to calculate the t-value for a statistical test.

It is calculated as n - 1, where n is the sample size. In this problem, the degrees of freedom is 10 (12 - 2).The t-test is a statistical test used to determine if there is a significant difference between two groups. It is used when the population standard deviation is not known. The t-test uses the t-distribution to determine the p-value. The p-value is the probability of observing the data if the null hypothesis is true.

If the p-value is less than the level of significance, the null hypothesis is rejected. In this problem, the null hypothesis is that there is no difference between the mean number of years for men and women to complete a four-year college degree. To test this hypothesis, we will use a two-sample t-test.

The formula for a two-sample t-test is:t = (x1 - x2) / (s1^2/n1 + s2^2/n2)^(1/2)where x1 is the sample mean for group 1, x2 is the sample mean for group 2, s1 is the sample standard deviation for group 1, s2 is the sample standard deviation for group 2, n1 is the sample size for group 1, n2 is the sample size for group 2, and t is the t-value.

Men: (4 + 5 + 4 + 6 + 4 + 5) / 6 = 28 / 6 = 4.67Women: (6 + 4 + 7 + 5 + 3) / 5 = 25 / 5 = 5The sample mean for men is 4.67 years, and the sample mean for women is 5 years. The pooled variance is a weighted average of the variances of two independent samples. It is used to estimate the population variance when the population variances are assumed to be equal.

The formula for the pooled variance is:s^2 = [(n1 - 1)s1^2 + (n2 - 1)s2^2] / (n1 + n2 - 2)where s1 is the sample standard deviation for group 1, s2 is the sample standard deviation for group 2, n1 is the sample size for group 1, n2 is the sample size for group 2, and s^2 is the pooled variance.s1 = [(4 - 4.67)^2 + (5 - 4.67)^2 + (4 - 4.67)^2 + (6 - 4.67)^2 + (4 - 4.67)^2 + (5 - 4.67)^2] / (6 - 1)^(1/2) = 0.763s2 = [(6 - 5)^2 + (4 - 5)^2 + (7 - 5)^2 + (5 - 5)^2 + (3 - 5)^2] / (5 - 1)^(1/2) = 1.87s^2 = [(6 - 1)0.763^2 + (5 - 1)1.87^2] / (6 + 5 - 2) = 1.243

The t-value is calculated using the formula: t = (x1 - x2) / (s1^2/n1 + s2^2/n2)^(1/2)t = (4.67 - 5) / (1.243/6 + 1.243/5)^(1/2)t = -0.555 The p-value is the probability of observing the data if the null hypothesis is true. The p-value is calculated using a t-table or a t-distribution calculator. The p-value for a two-tailed test with 10 degrees of freedom and a t-value of 0.555 is 0.593.

The p-value is greater than the level of significance (0.05). Therefore, we fail to reject the null hypothesis. There is not enough evidence to suggest that the mean time to complete a four-year college degree varies based on gender. The conclusion is that there is no difference between the mean number of years for men and women to complete a four-year college degree.

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Answer:

Step-by-step explanation:

To solve this problem, we can use the formula for calculating simple interest:

I = P * r * t

where:

I = interest earned

P = principal (initial amount of money)

r = rate of interest

t = time (in years)

We can rearrange the formula to solve for the principal:

P = I / (r * t)

In this case, we know that Gill earned $2306 in interest after one year at a rate of 6%. So:

I = $2306

r = 0.06

t = 1 year

Substituting these values into the formula, we get:

P = $2306 / (0.06 * 1)

P = $38,433.33

Therefore, the initial amount of money that Gill deposited into her account was $38,433.33.

Answer:

[tex]\frac{11}{8}[/tex]

Step-by-step explanation:

3(2k+3)=6-(3k-5)

6k +9=6-3k+5

6k+3k=6+5

8k=11

k=[tex]\frac{11}{8}[/tex]

Answer: I think it is k=2/9

Step-by-step explanation:

Answer:

equal

Step-by-step explanation:

[tex]f(x) = -2(x - 0.5)^2 + 6[/tex] is the equation of the quadratic function that passes through the points (-1, 14), (0, 8), (1, 6), and (2, 8).

What is quadratic function?

f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero, is a quadratic function.

To find the equation of the quadratic function that passes through the points (-1, 14), (0, 8), (1, 6), and (2, 8), we can use the vertex form of the quadratic function, which is:

[tex]f(x) = a(x - h)^2 + k[/tex]

[tex]f(1) = a(1 - h)^2 + k\\\\6 = a(1 - h)^2 + k[/tex]

We can use a second point to find a relationship between h and k. Let's use the point (0, 8):

[tex]f(0) = a(0 - h)^2 + k\\\\8 = a(-h)^2 + k\\\\6 - 8 = a(1 - h)^2 + k - (a(-h)^2 + k)\\\\-2 = a(1 - h)^2 - a(h)^2\\\\-2 = a(1 - 2h + h^2) - a(h^2)\\\\-2 = a - 2ah + ah^2 - ah^2\\\\-2 = a - 2ah\\\\a = -2/(2h - 1)[/tex]

Let's use the second equation:

[tex]8 = a(-h)^2 + k\\\\8 = (-2/(2h - 1))(h^2) + k\\\\8(2h - 1) = -2h^2 + k(2h - 1)\\\\16h - 8 = -2h^2 + k(2h - 1)\\\\-2h^2 + 16h - 8 = k(2h - 1)\\\\k = (-2h^2 + 16h - 8)/(2h - 1)[/tex]

Now we can substitute this value of h into our expressions for a and k to get:

[tex]a = -2/(2(0.5) - 1) = -2\\\\k = (-2(0.5)^2 + 16(0.5) - 8)/(2(0.5) - 1) = 6[/tex]

So the equation of the quadratic function is:

[tex]f(x) = -2(x - 0.5)^2 + 6[/tex]

Therefore, [tex]f(x) = -2(x - 0.5)^2 + 6[/tex] is the equation of the quadratic function that passes through the points (-1, 14), (0, 8), (1, 6), and (2, 8).

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The abiotic factor that can prevent the organism from becoming preserved, AFTER it has been buried is groundwater.

An abiotic factor refers to a non-living component of an ecosystem that influences the survival and growth of living organisms.

Groundwater can cause the dissolution of minerals present in the sediment that encases the buried organism.

This process can lead to the destruction of the fossilized remains, as the minerals provide the framework for the preservation of the organism's hard parts (such as bones or shells).

Groundwater can also cause the erosion of the sediment, which can expose the buried remains and make them vulnerable to biotic factors such as scavengers, predators, and decomposers.

Predators and animals that eat bones, scavengers and plants that use the nutrients of fossils, and other animals of the same species as the organism that died are all biotic factors that can affect the preservation of the organism, but they do so before or during the burial process.

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(a) The probability that the series will last exactly m games is:(1 - (pA + pB))^m(pA (1 - pB) + pB (1 - pA))where pA and pB are the probabilities of teams A and B, respectively. Therefore, the probability that the series will last less than 5 games is the probability that the series will last exactly 3 games or exactly 4 games:1 - (1 - 0.6 * 0.4)^3 - (1 - 0.6 * 0.4)^4 ≈ 0.684.

The probability that the series will last 5 games is the probability that the first four games have two wins for each team, and the last game is won by Team A:0.6^3 * 0.4^3 * 4 ≈ 0.055.The probability that the series will last 6 games is the probability that the first five games have two wins for each team, and the last game is won by Team A:0.6^3 * 0.4^3 * 5 ≈ 0.077The probability that the series will last 7 games is the probability that the first six games have two wins for each team, and the last game is won by Team A:0.6^3 * 0.4^3 * 6 ≈ 0.091.

The expected number of games played is thus approximately:0.684 * 4 + 0.055 * 5 + 0.077 * 6 + 0.091 * 7 + ∑m=8^∞ (1 - (pA + pB))^m(m + 1)(pA (1 - pB) + pB (1 - pA))The sum above can be computed by expressing it as the product of three factors:1 - (pA + pB) is a common factor for all terms, (pA (1 - pB) + pB (1 - pA)) is a sum of two terms that can be replaced by 1 - (1 - pA)(1 - pB), and m + 1 is a sum of m and 1. After replacing the sum of two terms, we obtain:0.684 * 4 + 0.055 * 5 + 0.077 * 6 + 0.091 * 7 + (1 - (0.6 + 0.4))^8(8 + 1)(1 - (1 - 0.6 * 0.4)^2) / (0.6 + 0.4 - 0.6 * 0.4) ≈ 0.684 * 4 + 0.055 * 5 + 0.077 * 6 + 0.091 * 7 + 0.02525 / 0.34 ≈ 5.43.

Therefore, the expected number of games played is approximately 5.43.(b) Given that Team B wins the first game, the series can last 7 or 8 games. The probability that the series will last 8 games is the probability that the first seven games have three wins for each team, and the last game is won by Team A:0.6^3 * 0.4^3 ≈ 0.013824.The probability that the series will last at least 8 games is therefore approximately 0.091 + 0.013824 = 0.104824, or 10.48%.Answer: (a) The expected number of games played is approximately 5.43. (b) The probability that the series will last at least 8 games is approximately 0.104824 or 10.48%.

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Answer:

[tex]{ \rm{s = 12( v_{0} + v_{1} )t}} \\ \\{ \boxed { \rm{t = \frac{s}{12(v_{0} + v_{1})} \: \: }}}[/tex]

The definition of vector space that fails is part A and B  because it does not satisfy the any property of vector space.

The following statement can be determined if at-least one part of vector space definition that fails as:

(a) A is not a vector space because it does not contain the zero vector.

The zero vector is the unique vector that satisfies the property that when it is added to any other vector in the space, the result is the original vector.

However, in this case, the [a1, b1] + [a2, b2] = [a1 - a2, b1 - b2] is 0x² + 1, which is not an element of A. Therefore, A fails to satisfy the requirement of having a zero vector, and it is not a vector space.

(b) B is not a vector space because it does not satisfy the distributive property of scalar multiplication over vector addition.

In general, scalar multiplication must distribute over vector addition, meaning that for any scalar a and any vectors u and v in the space, a(u+v) = au + av.

However, in B, the scalar multiplication is defined as usual for R², but the vector addition is defined differently. In particular,[a1, b1] + [a2, b2] = [a1 - a2, b1 - b2].

The vector space definition fails because the vector addition is not associative, and it is also not commutative, which are the first two conditions for vector spaces. Therefore, the first and second conditions of the definition are not met.

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The 21 square root of 98 divided by 7 square root of 21 = 21√98 / 7√21 = 6.4807407

A square root of a number x is a number y such that y2 = x; in other words, a number y who's square and the result of multiplying the number by itself, or y ⋅ y, is x.

Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by √where the symbol √ is called the radical sign.

Every positive number x has two square roots: √ which is positive, and -√ which is negative. The two roots can be written more concisely using the ± although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root.

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The required ratio of the surface area of the given pyramids is (A) 3:2.

A ratio can be used to show a relationship or to compare two numbers of the same type.

To compare things of the same type, ratios are utilized.

We might use a ratio, for example, to compare the proportion of boys to girls in your class.

If b is not equal to 0, an ordered pair of numbers a and b, denoted as a / b, is a ratio.

A proportion is an equation that equalizes two ratios.

For illustration, the ratio may be expressed as follows: 1: 3 in the case of 1 boy and 3 girls (for every one boy there are 3 girls)

So, the given surface area is:

- 21 in

- 14 in

Now, calculate the ratio as:

= 21/14

= 3/2

= 3:2

Therefore, the required ratio of the surface area of the given pyramids is (A) 3:2.

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all the points that represent the following values of x-1 on number line is -3 -2 -1 0 1 2 3.

What is number line ?

A number line is a visual representation of numbers placed along a straight line, where each point corresponds to a unique number. It helps in visualizing the relative position of numbers and their magnitudes. The numbers to the right of zero on the number line are positive and the numbers to the left of zero are negative. The distance between two consecutive points on a number line is the same and represents the unit distance. Number lines can be used to represent integers, fractions, decimals, and even irrational numbers.

According to the question:
Make a number line and mark all the points that represent the following values of x: -1" is simply to mark the point -1 on the number line.
Here is the number line with -1 marked:
     -3    -2    -1     0     1     2     3

    ---------------------------------------

                  ○

The circle (○) represents the point that corresponds to x = -1.


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Q) Make a number line and mark all the points that represent the following values of x=-1,-2,-3,0,1,2,3 ?

The maximum sum of the products taken two at a time is 180, and this can be achieved by choosing 60, 60, and 60 as the three numbers.

In order to write the number 180 as a sum of three numbers so that the sum of the products taken two at a time is a maximum, one way to do it is to use the formula:

[tex](x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + xz + yz)[/tex]

Let the three numbers be x, y, and z.

Then the product of the numbers taken two at a time is: [tex]xy + xz + yz[/tex]

If we want to maximize the sum of the products taken two at a time, we need to maximize [tex]xy + xz + yz[/tex].

In the formula: [tex](x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + xz + yz)[/tex]

We can see that the first three terms on the right-hand side are fixed since they depend on x, y, and z. Therefore, to maximize the sum of the products taken two at a time, we need to maximize 2(xy + xz + yz). Since we have the number 180, we can let: [tex]x + y + z = 180[/tex]

Then, we need to maximize: 2(xy + xz + yz) Using calculus, we can find that the maximum value of 2(xy + xz + yz) is attained when: [tex]x = y = z = 60[/tex]

Therefore, the three numbers that can be used to write the number 180 as a sum of three numbers so that the sum of the products taken two at a time is a maximum are: 60, 60, 60.

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The required percentage of values that should fall in the interval 56-88 is approximately 74.37%.

Chebyshev’s Theorem:Chebyshev's Theorem states that, for any given data set, the proportion (or percentage) of data points that lie within k standard deviations of the mean must be at least (1 - 1/k2), where k is a positive constant greater than 1.Calculation:Given,Mean (μ) = 72N (Total number of values) = 387Interval (x) = 64-80 and 56-88Minimum values (n) = 344Minimum percentage (p) = (344 / 387) x 100 = 88.85%From the given data we have,1. Calculate the variance of the distribution,Variance = σ2 = [(n × s2 ) / (n-1)]σ2 = [(344 × 42) / 386]σ2 = 18.732. Calculate the standard deviation of the distribution,σ = √(18.73)σ = 4.33. Calculate k = (|x - μ|) / σ for the given interval 56-88,Here, x1 = 56, x2 = 88, k1 = |56-72| / 4.33 = 3.7, k2 = |88-72| / 4.33 = 3.7Thus, k = 3.74. Calculate the minimum percentage of values within the interval 56-88 using Chebyshev's Theorem,p = [1 - (1/k2)] x 100p = [1 - (1/3.7)2] x 100p = 74.37% (approximately)Therefore, the required percentage of values that should fall in the interval 56-88 is approximately 74.37%.

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By answering the presented question, we may conclude that If you use a equation different fertiliser with a different dilution rate, you must modify the amount of fertiliser to attain the appropriate concentration.

In mathematics, an equation is an assertion that affirms the equivalence of two factors. An algebraic equation (=) separates two sides of an equation. For instance, the assertion  [tex]"2x + 3 = 9"[/tex] states that the word  [tex]"2x + 3"[/tex] corresponds to the number "9".

The goal of solution solving is to figure out which variable(s) must still be adjusted for the equations to be true. It is possible to have simple or intricate equations, recurring or complex equations, and equations with one or more components.

For example, in the equations  [tex]"x^2 + 2x - 3 = 0,"[/tex] the variable x is lifted to the powercell. Lines are utilized in many areas of mathematics, include algebra, arithmetic, and geometry.

The amount of fertiliser required for three applications of a 3-gallon watering can is determined by the desired fertiliser concentration in the water.

For example, if the suggested dilution rate for a fertiliser is 1 tablespoon per gallon of water, you would need to add 3 tablespoons of fertiliser to the 3-gallon watering can for each application. So, for 3 applications, you would need a total of 9 tablespoons of fertiliser.

Therefore, If you use a different fertiliser with a different dilution rate, you must modify the amount of fertiliser to attain the appropriate concentration.

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