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The area under the curve [tex]y = 37x^3[/tex] from x = 1 to x = t is [tex]37/4(t^4 - 1)[/tex] square units.

To find the area under the curve [tex]y = 37x^3[/tex] from x = 1 to x = t, we need to integrate the function with respect to x over the given interval:

[tex]∫[1, t] 37x^3 dx[/tex]

Using the power rule of integration, we can evaluate the integral as:

[tex][37/4 x^4][/tex] from 1 to t

= [tex]37/4(t^4 - 1)[/tex]

Therefore, the area under the curve [tex]y = 37x^3[/tex] from x = 1 to x = t is [tex]37/4(t^4 - 1)[/tex] square units.

Integration is a mathematical concept that involves finding the integral of a function. It is the reverse process of differentiation and allows us to determine the antiderivative of a given function. The integral of a function represents the area under the curve of that function over a given interval.

The symbol used to denote integration is ∫ (integral symbol), and the process of finding an integral is often referred to as integration. Integration is used in various branches of mathematics, including calculus, physics, engineering, and economics.

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The upper fences (inner and outer) are 400 and 550. The correct option is C. 400 and 550.

To calculate the upper fences, we need to use the formula:

Upper inner fence = Q3 + 1.5(Q3-Q1)
Upper outer fence = Q3 + 3(Q3-Q1)

Plugging in the given values, we get:

Upper inner fence = 250 + 1.5(250-150) = 400
Upper outer fence = 250 + 3(250-150) = 550

Therefore, the correct answer is C. The upper inner fence is 400 and the upper outer fence is 550. It is important to note that these fences are used in outlier detection to determine if there are any extreme values in the data set. Any values beyond the outer fence are considered potential outliers and should be further investigated.

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Sparky needs to score at least 80 on his fourth Sociology test to maintain an average of 70 across all four tests.

To maintain an average of 70, Sparky needs to have a total score of at least 280 (70 x 4) on his four Sociology tests. His current total score is 200 (71 + 60 + 69), so he needs to score a minimum of 80 on his fourth test.

Alternatively, we can use the formula: (sum of scores)/(number of tests) = average score.

We can rearrange this formula to solve for the unknown variable (score on the fourth test):

(score on fourth test) = (average score) x (number of tests) - (sum of scores)

Substituting the values given, we get:

(score on fourth test) = 70 x 4 - (71 + 60 + 69) = 280 - 200 = 80

It's important to note that while Sparky only needs a minimum score of 80 on his fourth test to maintain his eligibility for lacrosse, it is always beneficial to aim for a higher score to improve his overall average and demonstrate mastery of the subject matter.

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The total number of milligrams of sodium recommended for a 2000-calorie diet is 2429 mg.

In mathematics, an expression is a combination of numbers, variables, and mathematical operations (such as addition, subtraction, multiplication, and division) that can be evaluated to obtain a value.

If one serving of toasted corn kernels contains 170 milligrams of sodium and it represents 7% of the daily value, we can calculate the recommended daily value as follows:

Let X be the total number of milligrams of sodium recommended for a 2000-calorie diet.

7% of X = 170 mg

0.07X = 170 mg

X = 170 mg / 0.07

X = 2428.57 mg

The total number of milligrams of sodium recommended for a 2000-calorie diet is 2429 mg.

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The value of the given integral is π(e - 1).

We can solve this double integral by using polar coordinates. First, we convert the limits of integration to polar coordinates:

0 ≤ r ≤ 1 (since [tex]x^2 + y^2[/tex] ≤ 1 for all points within the unit circle)

0 ≤ θ ≤ 2π

Next, we convert the integrand, emax([tex]x^2, y^2[/tex]), to polar coordinates. Since [tex]e^x[/tex] is an increasing function, emax([tex]x^2, y^2[/tex]) = [tex]e^{(max(x^2, y^2)}[/tex] = [tex]e^{(r^2)[/tex], where r is the distance from the origin. Therefore, we can write:

emax([tex]x^2, y^2[/tex]) = [tex]e^{(max{x^2, y^2)} = e^{(r^2)[/tex]

Now we can write the integral in polar coordinates:

∫10∫10emax(x2,y2)dxdy = ∫[tex]_0^1[/tex]∫[tex]_0^{2\pi[/tex] [tex]e^{(r^2)[/tex] r dθ dr

To solve this integral, we can use the substitution u = [tex]r^2[/tex], du = 2r dr:

∫[tex]_0^1[/tex]∫[tex]_0^{2\pi[/tex] [tex]e^{(r^2)[/tex] r dθ dr = (1/2) ∫[tex]_0^1 e^u[/tex] du ∫[tex]_0^{2\pi[/tex] dθ

= (1/2) [e - 1] [2π]

= π(e - 1)

Therefore, the value of the given integral is π(e - 1).

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An airline has a requirement that the total outside dimensions (length, width, and height) of a checked bag not exceed 73 inches. You want to check a bag with a height equal to its width. To find the largest volume bag of this shape that you can check, we will use the constraint provided and optimize the volume.

Let L, W, and H represent the length, width, and height of the bag, respectively. According to the constraint, L + W + H ≤ 73 inches. Since H = W, we can rewrite the constraint as L + 2W ≤ 73.

The volume (V) of the bag can be represented as V = L × W × H. Substituting H = W, we get V = L × W². To maximize the volume, we need to rewrite this equation in terms of one variable. Using the constraint, we can express L as L = 73 - 2W. Now, substitute this into the volume equation: V = (73 - 2W) × W².

To find the maximum volume, we can use calculus or simply observe that the function V(W) is a downward-opening parabolic function. The maximum volume occurs at the vertex, which is found at the W-coordinate W = -b/2a in the general quadratic equation f(x) = ax^2 + bx + c. In our case, a = -2, b = 73, so W = 73/(2×-2) = 18.25 inches.

an airline requires that the total outside dimensions (length width height) of a checked bag not exceed 73 inches. suppose you want to check a bag whose height equals its width, Now that we have W, we can find H (which is equal to W) and L. H = 18.25 inches, and L = 73 - 2(18.25) = 36.5 inches. Thus, the largest volume bag you can check is V = 36.5 × 18.25² ≈ 12,104.16 cubic inches (rounded to two decimal places).

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The sampling method described in this example is called random sampling with replacement. This means that every member of the population has an equal chance of being selected for the sample, and after each selection, the individual is returned to the population, so they could be selected again.

This method is commonly used when the population size is small, and it is not possible to use other sampling methods. However, it is important to note that this method may not provide a representative sample, as certain individuals may be selected multiple times, while others may not be selected at all. Therefore, the results obtained from this type of sampling should be interpreted with caution.

In this example, Dr. Anderson is using a sampling method called "simple random sampling with replacement." This method involves placing all individuals' names in a hat, selecting a name, and then replacing it back into the hat before making the next selection. By doing this, each individual has an equal chance of being selected during each draw, and the same individual can be selected more than once. This sampling method is useful for obtaining a representative sample of a small population and ensures unbiased results, as long as the process remains random and independent for each selection.

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Out of the given options, verifying a large prime is not computationally difficult. This is because verifying a prime can be done using basic mathematical operations such as division and multiplication, and does not require any complex algorithms.

However, factoring a large number, computing a primitive root of a large number, and computing the discrete logarithm of a large number are computationally difficult tasks that require advanced algorithms and high computational power. Factoring involves breaking down a number into its prime factors, which can be a challenging task for large numbers with many digits. Computing a primitive root involves finding a number that generates all the possible residues of a given modulus, which can be a computationally intensive task for large numbers. Computing the discrete logarithm involves finding the exponent to which a given number must be raised to obtain another number, which is a difficult task for large numbers.

In summary, out of the given options, verifying a large prime is the only task that is not computationally difficult.

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A benchmark fraction is a commonly used fraction that is easy to remember and can be used to estimate the value of other fractions or percentages. The most commonly used benchmark fractions are 1/2, 1/3, 1/4, 1/5, 1/6, 1/8, and 1/10.

To use a benchmark fraction to tell if a percent is reasonable, you can first convert the percent to a fraction by dividing it by 100. Then, you can compare this fraction to the benchmark fractions to see which one it is closest to.

For example, if you wanted to estimate whether 75% is a reasonable amount, you could first convert it to a fraction by dividing 75 by 100, which gives you 0.75. Then, you could compare this fraction to the benchmark fractions to see which one it is closest to. 0.75 is closest to 3/4, which is one of the benchmark fractions. This suggests that 75% is a reasonable amount, since it is close to 3/4.

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The correct answer is option C. At the point (2,1) on the curve x² - xy + y²= 3, the derivative dy/dx exists, and the tangent line at that point is neither horizontal nor vertical.

To determine the correct option, we need to analyze the properties of the curve x² - xy + y² = 3 at the point (2,1). First, let's find the derivative dy/dx. Taking the derivative of the given equation implicitly with respect to x, we get:

2x - y - x(dy/dx) + 2y(dy/dx) = 0

Rearranging the terms, we have:

dy/dx = (2x - y) / (x - 2y)

Now, substituting the values x = 2 and y = 1 into the expression for dy/dx, we can determine its value at the point (2,1):

dy/dx = (2(2) - 1) / (2 - 2(1)) = 3 / 0

Since the denominator is zero, dy/dx is undefined at (2,1). Therefore, option D, stating that dy/dx does not exist at (2,1) and the tangent line at that point is vertical, is incorrect.

However, we can still determine the nature of the tangent line at (2,1). Although the derivative is undefined, it is still possible for the tangent line to exist and have a defined slope. In this case, the tangent line would be neither horizontal nor vertical. Therefore, option C, which states that dy/dx exists at (2,1) and the tangent line at that point is neither horizontal nor vertical, is the correct answer.

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Answer:

[tex](2x + 8)(6x + 2)[/tex]

[tex] = 12 {x}^{2} + 52x + 16[/tex]

C is the correct answer.

In a normal distribution, approximately 68% of the population is within one standard deviation of the mean. This can be explained by the empirical rule, which states that for a normal distribution, about 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations.

The standard deviation is a measure of how spread out the data is from the mean, so if the data is normally distributed, we can use the empirical rule to estimate the proportion of the population that falls within a certain number of standard deviations from the mean.

Therefore, we can say with confidence that approximately 68% of the population falls within one standard deviation of the mean in a normal distribution.

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The best sentence that describes the sample the owner could use to make inferences about the population is: "Randomly select a sample of shopping malls from the chain throughout the nation and survey a random sample of the people who shop at the mall each day of the week."

In statistics, a population is the entire group of individuals or objects that we are interested in studying, and from which we want to draw conclusions or make inferences. This group can be as large or as small as necessary, and it can be defined in different ways depending on the research question. For example, the population could be all the students in a school, all the customers of a business, or all the trees in a forest. The important thing is to clearly define the population and to ensure that it is representative of the group we want to study

The best sample for making inferences about the population would be option 3: randomly select a sample of shopping malls from the chain throughout the nation and survey a random sample of the people who shop at the mall on Friday.

This option involves selecting a random sample of shopping malls, which helps to ensure that the sample is representative of the entire population of shopping malls in the chain. Surveying a random sample of people who shop on Friday further helps to ensure that the sample is representative of the entire population of shoppers who visit the malls.

Option 1 only surveys one shopping mall, which may not be representative of the entire population of shopping malls in the chain. Option 2 only surveys the managers of the three largest stores, which may not provide a representative sample of the shoppers. Option 4 surveys a random sample of people who shop each day, which may not be practical or necessary for the owner's purposes.

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Based on the given information, we can assume that the table provides us with the number of hours that a group of students spent studying for the SAT during their first week of preparation. The mean study time for the second week is 10.8 hours, the median study time is 11 hours, there is no mode, and the range of study times is 13 hours.

Mean: To find the mean, we need to add up all the study times in the second week and divide by the number of students. Let's say the table shows that there were 10 students, and their study times in the first week were 2, 4, 5, 6, 7, 8, 9, 10, 12, and 15 hours. If each student added 4 hours in the second week, their new study times would be 6, 8, 9, 10, 11, 12, 13, 14, 16, and 19 hours. Adding up these new study times gives us a total of 108 hours. Dividing by the number of students (10) gives us a mean of 10.8 hours.

Median: To find the median, we need to put the new study times in order from lowest to highest and find the middle value. In this case, the new study times in order are: 6, 8, 9, 10, 11, 12, 13, 14, 16, 19. The middle value is 11, which is the median.

Mode: To find the mode, we need to look for the value that appears most frequently in the new study times. In this case, there is no mode as each value appears only once.

Range: To find the range, we need to subtract the lowest value from the highest value. In this case, the lowest value is 6, and the highest value is 19. Therefore, the range is 19 - 6 = 13.

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The probability of not getting a red jack when selecting one card at random from a standard deck of 52 cards is (50/52) or approximately 0.962.

This is because there are 50 cards that are not red jacks out of a total of 52 cards in the deck.

The probability of not getting a red jack when selecting one card at random from a standard deck of 52 cards can be calculated using these terms:

There are 2 red jacks in the deck, and 52 cards in total. Therefore, there are 50 cards that are not red jacks (52 - 2).

To find the probability, divide the number of favorable outcomes (not getting a red jack) by the total number of possible outcomes (total cards in the deck).

Probability = (Number of favorable outcomes) / (Total possible outcomes) = 50/52 = 25/26 ≈ 0.96 or 96%.

So, the probability of not getting a red jack when selecting one card at random is approximately 96%.

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The median of the box-and-whisker plot is the value that lies exactly in the middle of the data set. the median is a useful measure of central tendency for the data set, as it provides a representative value that summarizes the central location of the data.

In this case, the box-and-whisker plot represents the average speed of a certain car for several laps at a race track. To find the median, we need to locate the middle value of the data set, which is the point where half of the data is below it and half is above it. The median is represented by the line in the middle of the box on the plot. It indicates that half of the laps recorded had an average speed below this value and half had an average speed above it. Therefore, the median is a useful measure of central tendency for the data set, as it provides a representative value that summarizes the central location of the data.

In the context of a race track and a box-and-whisker plot, the median represents the middle value of the average speeds recorded for a certain car during several laps. The plot organizes the data into four quartiles, with the median being the boundary between the second and third quartiles. To find the median, you would look at the box-and-whisker plot and identify the line within the box that separates the lower and upper halves of the data. This value indicates the median average speed for the car over the measured laps.

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Since we need to round the number of scoops to the nearest scoop, we can round 67.32 scoops to 67 scoops. Therefore, you'll need to add 67 scoops of drink mix to make a full batch of sports drink for the campers on a hot day.

To determine the number of scoops of drink mix needed for a full batch, we'll first need to calculate the volume of the container and convert it to gallons. The container's dimensions are 1 foot by 1.5 feet by 3 feet.
Step 1: Calculate the volume of the container.
Volume = length × width × height
Volume = 1 ft × 1.5 ft × 3 ft
Volume = 4.5 cubic feet
Step 2: Convert the volume to gallons.
1 cubic foot is equivalent to 7.48 gallons, so we'll multiply the volume in cubic feet by the conversion factor.
4.5 cubic feet × 7.48 gallons/cubic foot ≈ 33.66 gallons
Step 3: Calculate the number of scoops of drink mix needed.
You must add two scoops of drink mix for each gallon of water.
33.66 gallons × 2 scoops/gallon ≈ 67.32 scoops

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Answer:(x−3)^2−2

Step-by-step explanation:

Angela has been making greeting cards for 12 days by cutting one piece of construction paper into 3 pieces each day. Therefore, she has used a total of 12 pieces of paper.

To determine how many greeting cards she has made, we need to multiply 12 by 3 since each paper is cut into 3 pieces. Therefore, Angela has made 36 greeting cards in total. It's important to note that this assumes that Angela uses all three pieces of paper from each cut to make greeting cards. If she only uses one or two pieces from each cut, the total number of greeting cards would be different.

Angela makes 3 greeting cards each day using a piece of construction paper. Over 12 days, she has made a total of 12 days x 3 cards per day = 36 greeting cards. This calculation shows the number of cards Angela has created by cutting one piece of construction paper into three parts daily for her friends during the 12-day period.

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T he joint moment generating function of x and y is M(t1, t2) = (1/36)Σx=1^6Σz=1^6 e^(t1x + t2z)

Let X be the value of the first die, which takes on values {1, 2, 3, 4, 5, 6} with equal probability of 1/6 each. Let Y be the sum of the values of two dice, so Y takes on values {2, 3, ..., 12}.

The joint moment generating function of X and Y is given by:

M(t1, t2) = E[e^(t1X + t2Y)]

To compute this, we can use the law of total probability and conditioning on the value of X:

M(t1, t2) = E[e^(t1X + t2Y)]

= Σx P(X=x) E[e^(t1X + t2Y) | X=x]

= (1/6)Σx=1^6 E[e^(t1x + t2Y) | X=x]

Now we need to compute E[e^(t1x + t2Y) | X=x]. We can use the fact that the sum of two dice is the sum of two independent uniform random variables on {1, 2, 3, 4, 5, 6}:

E[e^(t1x + t2Y) | X=x] = E[e^(t1x + t2(x+Z))]

= E[e^(tx) e^(t2Z)]

= MZ(t2) e^(tx)

where Z is a uniform random variable on {1, 2, 3, 4, 5, 6} and MZ(t2) is its moment generating function, which is:

MZ(t2) = E[e^(t2Z)]

= (1/6)Σz=1^6 e^(t2z)

Substituting this back into the expression for M(t1, t2), we get:

M(t1, t2) = (1/6)Σx=1^6 E[e^(t1x + t2Y) | X=x]

= (1/6)Σx=1^6 MZ(t2) e^(tx)

= (1/6)Σx=1^6 [(1/6)Σz=1^6 e^(t2z)] e^(tx)

Simplifying the expression, we get:

M(t1, t2) = (1/36)Σx=1^6Σz=1^6 e^(t1x + t2z)

This is the joint moment generating function of X and Y.

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Answer:

The center is ( -2, 2.5)

Step-by-step explanation:

The center would be the middle of the diameter.

The x coordinate would be

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

The y coordinate would be

(3+2)/2 = 5/2 = 2.5

The center is ( -2, 2.5)

The vector function r(s) is t = s/a, the direction of motion T(s) explicitly is r'(t) = a and the length of the  curve r(t) = ecos(t)i + et sin(t)j + e'k  is given by curvature  r = [tex]\frac{||T'(t)||}{||r'(t)||}[/tex] = 1/a.

Curvature is any of many closely related geometric notions in mathematics. Curvature is intuitively defined as the amount a curve deviates from being a straight line or a surface deviates from being a plane. The most common example of a curve is a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, resulting in more curvature.

The curvature of a differentiable curve at a location is the curvature of the circle that best approximates the curve near this point. A straight line has no curvature. In contrast to the tangent, which is a vector quantity, the curvature at a point is normally a scalar quantity, defined by a single real integer.

Circle with center the origin and having radius a

a) r(t) = (acost, asint)

s = [tex]\int\limits^a_b {\sqrt{a^2sin^t+a^2cos^t} } \, dt[/tex]

= at

t = s/a.

b) r(s) = (a cos(s/a), asin(s/a))

= (-a sint, a cost)

r'(t) = a.

c) Graphing the vector include the co-ordinates

T(t) = (-sint, cost)

T(s) = (-sin(s/a), cos(s/a))

Curvature r = [tex]\frac{||T'(t)||}{||r'(t)||}[/tex]

= 1/a.

Therefore, the length of the  curvature is 1/a.

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The requried, company can expect to lose about $55.20.

If the bottles pass inspection 98/100 of the time, then the probability that a bottle does not pass inspection is 2/100 or 0.02.

For each bottle that does not pass inspection, the company loses the unit cost of $3.45.

To find the expected amount of money the company can expect to lose, we can multiply the total number of bottles produced by the probability that a bottle does not pass inspection and by the unit cost of $3.45:

Expected loss = 800 bottles * 0.02 * $3.45 = $55.20

Therefore, the company can expect to lose about $55.20.

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The expression "arcsin(1/2)" represents the inverse sine function, which returns the angle whose sine is equal to 1/2.

To find the exact value of arcsin(1/2), we need to determine the angle whose sine is 1/2. In other words, we are looking for the angle θ such that sin(θ) = 1/2.

By recalling the trigonometric values for common angles, we know that sin(30°) = 1/2. Therefore, the exact value of arcsin(1/2) is 30 degrees or π/6 radians.

In summary, arcsin(1/2) = 30° = π/6 radians.

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The correct answer is option d) 60 cm^2.

The area of the triangle with side lengths 8 cm, 17 cm, and 15 cm can be calculated using Heron's formula. The correct answer can be found by substituting the side lengths into the formula and evaluating the expression.

To find the area of a triangle when the lengths of all three sides are known, we can use Heron's formula. Heron's formula states that the area of a triangle with side lengths a, b, and c is given by:

Area = sqrt(s(s - a)(s - b)(s - c))

where s is the semi-perimeter of the triangle, calculated as:

s = (a + b + c) / 2

In this case, the side lengths are given as 8 cm, 17 cm, and 15 cm. Plugging these values into the formula, we have:

s = (8 cm + 17 cm + 15 cm) / 2 = 20 cm

Area = sqrt(20 cm(20 cm - 8 cm)(20 cm - 17 cm)(20 cm - 15 cm))

     = sqrt(20 cm * 12 cm * 3 cm * 5 cm)

     = sqrt(3600 cm^2)

     = 60 cm^2

Therefore, the correct answer is option d) 60 cm^2.

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the complete question is:

A triangle has sides 8 cm, 17 cm and 15 cm The area of the triangle is

a) 50 cm^2

b) 68  cm^2

c) 40 cm^2

d) 60 cm^2

The FDA regulates that a fish consumed should contain at most 1 mg/kg of mercury. In Florida, bass fish from 52 different lakes were collected to determine if the mercury levels in the fish exceed the allowable amount.

To assess if there's enough evidence to show that fish in all Florida lakes have a higher mercury level than permitted, we will conduct a hypothesis test using the data collected.
The random variable (X) represents the average mercury level of bass fish in a sampled lake. The population parameter (μ) stands for the true mean mercury level of bass fish in all Florida lakes.
The null hypothesis (H₀) states that the average mercury level in the fish does not exceed the FDA's allowable amount, which means μ ≤ 1 mg/kg. The alternative hypothesis (H₁) claims that the average mercury level in the fish is higher than the FDA's allowable amount, which means μ > 1 mg/kg.
To determine if there is enough evidence to support H₁, we will perform a hypothesis test using the appropriate statistical test and significance level, while considering the sample size and mercury levels collected from the 52 lakes. If the test results lead to the rejection of H₀, we can conclude that there is evidence suggesting the fish in all Florida lakes have a mercury level higher than the allowable amount.

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The values of x (to the nearest hundredth) for which the functions f(X) = g(x) are

To find the values of x where f(x) = g(x), we need to equate the two given functions and solve for x.

f(x) = g(x)

sin(2x) = -(1/2x) - 1

sin(2x) + (1/2x) + 1 = 0

Using graphing calculator to plot the values we have the solution as

-2.91 to the nearest hundredth

-1.42 to the nearest hundredth

-0.45 to the nearest hundredth

The graph is attached

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The constant of proportionality is 0.5. 9 points are awarded for 18 minutes of play.

Part A) To find the constant of proportionality, we can set up a proportion using the information given:

1 minute of play = 0.5 points

5 minutes of play = 2.5 points

0.5/1 = 2.5/5

Simplifying the proportion:

0.5 = 0.5

Therefore, the constant of proportionality is 0.5.

Part B) Using the constant of proportionality, we can write the equation that represents the relationship between the time played (in minutes) and the points awarded:

points = 0.5 x time played

Part C) To graph the relationship, we can plot the time played (in minutes) on the x-axis and the points awarded on the y-axis. We would then plot two points: (1, 0.5) and (5, 2.5). These points represent the proportionality of 1 minute of play to 0.5 points, and 5 minutes of play to 2.5 points, respectively. We can then draw a straight line through these two points, which represents the relationship between time played and points awarded.

Part D) Using the equation we found in Part B, we can calculate the number of points awarded for 18 minutes of play:

points = 0.5 x time played

points = 0.5 x 18

points = 9

Therefore, 9 points are awarded for 18 minutes of play.

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https://brainly.com/question/29126727

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