Find the global minimum and maximum of the continuous F(x) = ×2 - 8 In(x) on [1, 4].
Global minimum value = ______
Global maximum value =______

To find the velocity and acceleration 8 seconds after the rocket is launched, we need to find the first and second derivatives of the height function with respect to time.

The first derivative of h(t) gives the velocity function v(t):
v(t) = h'(t) = 3t^2 + 1.2t

Substituting t = 8 into this equation gives us the velocity of the rocket at 8 seconds after launch:
v(8) = 3(8)^2 + 1.2(8) = 204.8 feet per second

So the velocity of the rocket 8 seconds after launch is 204.8 feet per second.

The second derivative of h(t) gives the acceleration function a(t):
a(t) = h''(t) = 6t + 1.2

Substituting t = 8 into this equation gives us the acceleration of the rocket at 8 seconds after launch:
a(8) = 6(8) + 1.2 = 49.2 feet per second squared

So the acceleration of the rocket 8 seconds after launch is 49.2 feet per second squared.
To find the velocity and acceleration of the rocket 8 seconds after it is launched, we need to determine the first and second derivatives of the height function h(t) with respect to time t.

Given h(t) = t^3 + 0.6t^2, let's find its first and second derivatives:

1. Velocity (first derivative of h(t)):
v(t) = dh/dt = 3t^2 + 1.2t

2. Acceleration (second derivative of h(t)):
a(t) = d^2h/dt^2 = d(v(t))/dt = 6t + 1.2

Now, let's evaluate the velocity and acceleration at t = 8 seconds:

Velocity at t=8:
v(8) = 3(8^2) + 1.2(8) = 192 + 9.6 = 201.6 ft/s

Acceleration at t=8:
a(8) = 6(8) + 1.2 = 48 + 1.2 = 49.2 ft/s^2

So, 8 seconds after the rocket is launched, its velocity is 201.6 ft/s and its acceleration is 49.2 ft/s^2.

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Mr. Robins earns a commission of 15% on the airfares he books, as he earned $31.29 on $208.60 worth of airfare bookings.

Let x be the amount of commission earned by Mr. Robins on the airfares he booked. Then, we can write the equation:

x = 15% of $208.60

Simplifying this equation, we get:

x = 0.15 x $208.60

x = $31.29

Therefore, Mr. Robins earned a commission of $31.29 on $208.60 worth of airfare bookings. To verify this, we can calculate his commission rate as:

Commission rate = Commission earned / Airfare bookings

Commission rate = $31.29 / $208.60

Commission rate = 0.15 or 15%

Hence, Mr. Robins earns a commission of 15% on the airfares he books.

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With probability 1, both machines are in the repair facility, and we move to state 2 (both machines working) two days later.

Let the state of the system be the number of machines working at the beginning of the day. We have two machines, so the state space is {0, 1, 2}.

Let the probability of transitioning from state i to state j be P(i,j).

To fill in the entries of the transition probability matrix, we need to consider the possible transitions between states.

If both machines are working at the beginning of the day (state 2):

If one machine is working at the beginning of the day (state 1):

If neither machine is working at the beginning of the day (state 0):

Putting this all together, we get the following transition probability matrix:

|   | 0        | 1        | 2        |

|---|----------|----------|----------|

| 0 | 0        | 0        | 1        |

| 1 | 1/3      | 2/3      | 0        |

| 2 | 1/9      | 4/9      | 4/9      |

For example, the entry in row 1 and column 2 represents the probability of transitioning from state 1 (one machine working) to state 2 (both machines working) and is 4/9.

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Answer:

24π cm²

Concepts Applied:

SA (TSA) of a cone = π · r · ( l+r )

Relation between l, h, and r i.e. l²=h²+r²

(h: cone height, r: base radius, l: slant height)

Step-by-step explanation:

Calculating the Slant height:

l²=h²+r²

l = sqrt(h²+r²)

l = sqrt(16+9)

l = sqrt(25)

l = +5 cm (distance is a scalar quantity)

Calculating the TSA:

= π · 3 · (5+3)

= 24π cm²

Answer:

34π cm^2 is the correct answer

If one line has a slope of 0. 5 and another distinct line has a slope of those two lines are Parallel. The correct answer is A.

Two lines are parallel if and only if they have the same slope. If two distinct lines have different slopes, then they cannot be parallel. In this case, one line has a slope of 0.5 and the other line's slope is unknown, so we cannot determine whether they are parallel or not just by looking at their slopes.

However, if the other line's slope is perpendicular to 0.5, then the lines would be perpendicular. Two lines are perpendicular if and only if the product of their slopes is -1. Therefore, if the other line's slope is -2, then the lines would be perpendicular (0.5 * -2 = -1).

If the other line's slope is undefined (i.e., the line is vertical), then the lines would not be parallel or perpendicular, but rather they would be skew lines.

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The number of coffee drinks sold on a day that 32 ice cream treats were sold using the equation of line of best fit is y = 32m + b, where y represents the number of coffee drinks sold and ice cream treats that  were sold = 32.

To estimate the number of coffee drinks sold on a day that 32 ice cream treats were sold, we would need to use the equation of the line of best fit. This equation represents the trend of the data collected and can be used to make predictions based on that trend.

Assuming that the data collected shows a positive correlation between the number of ice cream treats sold and the number of coffee drinks sold, we can use the equation of the line of best fit to estimate the number of coffee drinks sold on a day that 32 ice cream treats were sold.

Let's say that the equation of the line of best fit is y = mx + b, where y represents the number of coffee drinks sold and x represents the number of ice cream treats sold. Using the data collected, we can find the values of m and b that best fit the trend.

Once we have the equation, we can substitute x = 32 into the equation and solve for y. This will give us an estimate of the number of coffee drinks sold on a day that 32 ice cream treats were sold.

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Applying the compound interest formula, rounding to the nearest whole number, we get that Sophie deposited approximately $11,200.

We can use the formula for compound interest to solve this problem:

A = P * (1 + r/n)^(nt)

where A is the ending balance, P is the principal (the amount Sophie deposited), r is the annual interest rate (3.3%), n is the number of times the interest is compounded per year (2 for semiannual), and t is the number of years.

Substituting the given values, we get:

19786.19 = P * (1 + 0.033/2)^(2*11)

Simplifying and solving for P, we get:

P = 19786.19 / (1 + 0.033/2)^(2*11)

P ≈ 11200

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The negative slope indicates a decline in population over the 18-year period. The most accurate statement based on this information is that the city's population experienced a decline of approximately 50 people per year on average between 1953 and 1971.

To compute the slope of the population growth or decline, we need to use the formula:

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

where y2 is the final population, y1 is the initial population, x2 is the final year, and x1 is the initial year.

Plugging in the values we have:

slope = (2,694,850 - 2,695,750) / (1971 - 1953)

slope = -900 / 18

slope = -50

The negative slope indicates a decline in population over the 18-year period.

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The difference between the amount of potassium called for in the experiment (0.227 grams) and 1 gram is 0.773 grams.

The amount of potassium called for in the experiment is 227 milligrams. To convert milligrams to grams, we divide by 1000: 227/1000 = 0.227 grams.

The amount of 1 gram is larger than 0.227 grams. To find the difference between the two amounts, we subtract the smaller amount from the larger amount:

1 gram - 0.227 grams = 0.773 grams

Therefore, the difference between the amount of potassium called for in the experiment (0.227 grams) and 1 gram is 0.773 grams.

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The minimum value of S is 10 when both x and y are equal to 5.

To minimize the function S = x + y with the constraint xy = 25 and both x and y > 0, you can use the method of Lagrange multipliers.

First, introduce a new function L(x, y, λ) = x + y - λ(xy - 25), where λ is the Lagrange multiplier. Now find the partial derivatives with respect to x, y, and λ:

∂L/∂x = 1 - λy = 0
∂L/∂y = 1 - λx = 0
∂L/∂λ = xy - 25 = 0

Solve the first two equations for λ:

λ = 1/y and λ = 1/x

Now, set these two equations equal:

1/y = 1/x

Since x and y are positive, you can safely cross-multiply:

x = y

Now, use the constraint equation (xy = 25):

x(x) = 25
x^2 = 25
x = ±5 (but x > 0, so x = 5)

Since x = y, we also have y = 5. The minimum value of S = x + y is:

S = 5 + 5 = 10

So, the minimum value of S is 10 when both x and y are equal to 5.

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She will need 4 quarts of water.

Given Hannah decided to make finger gelatin for a huge children’s party. She had to make 8 packs of gelatin. Each pack needed 2 cups of water.

Since each pack of gelatin requires 2 cups of water, Hannah will need a total of:

8 packs x 2 cups/pack = 16 cups of water

To convert cups to quarts, we need to divide the number of cups by 4 (since there are 4 cups in a quart):

16 cups ÷ 4 cups/quart = 4 quarts

Therefore, Hannah will need 4 quarts of water to make 8 packs of finger gelatin for the children’s party.

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We know that the P(Marching Band and Team Sport) ≠ P(Marching Band) * P(Team Sport), the two events are dependent and associated.

A) The conditional probability of being on the marching band given that the student plays a team sport can be calculated using the formula:

P(Marching Band | Team Sport) = P(Marching Band and Team Sport) / P(Team Sport)

where P(Marching Band and Team Sport) is the probability of being on the marching band and playing a team sport, and P(Team Sport) is the probability of playing a team sport.

Let's say that out of a total of 500 students, 100 students play a team sport and 50 of them are also on the marching band. Then,

P(Marching Band and Team Sport) = 50/500 = 0.1
P(Team Sport) = 100/500 = 0.2

Plugging these values into the formula, we get:

P(Marching Band | Team Sport) = 0.1 / 0.2 = 0.5

Therefore, the conditional probability of being on the marching band given that the student plays a team sport is 0.5 or 50%.

b. The probability of being on the marching band can be calculated as:

P(Marching Band) = (Number of students on the marching band) / (Total number of students)

Let's say that out of the same 500 students, 75 students are on the marching band. Then,

P(Marching Band) = 75/500 = 0.15 or 15%

The difference between part (a) and part (b) is that in part (a), we are given additional information (the student plays a team sport) and we want to find the probability of being on the marching band. In part (b), we are simply asked for the probability of being on the marching band without any other information.

c. The two events, {on the marching band} and {on a team sport}, may or may not be associated. We can use probabilities to determine whether they are associated or not.

If the probability of being on the marching band and playing a team sport is different from the product of the probabilities of being on the marching band and playing a team sport separately, then the events are dependent and associated. If they are the same, then the events are independent and not associated.

Let's calculate the probabilities:

P(Marching Band and Team Sport) = 50/500 = 0.1
P(Marching Band) = 75/500 = 0.15
P(Team Sport) = 100/500 = 0.2

Product of the probabilities:

P(Marching Band) * P(Team Sport) = 0.15 * 0.2 = 0.03

Since P(Marching Band and Team Sport) ≠ P(Marching Band) * P(Team Sport), the two events are dependent and associated. This means that knowing whether a student is on the marching band affects the probability of them playing a team sport, and vice versa.

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Using the triple integral to find the volume of the solid bounded by the parabolic cylinder is 32/15 cubic units.

The given solid is bounded by the parabolic cylinder y = 2x², the plane z = 0, the plane z = 2, and the plane y = 4.

To find the volume of the solid using a triple integral, we can set up the integral as follows:

∫∫∫E dV

where E is the region of integration in three dimensions.

Region E can be described as:

0 ≤ z ≤ 2

0 ≤ y ≤ 4

0 ≤ x ≤ √(y/2)

Therefore, the triple integral can be written as:

∫0² ∫[tex]0^4[/tex] ∫[tex]0^{\sqrt(y/2)}[/tex] dx dy dz

Evaluating the integral gives us the volume of the solid:

V = ∫0² ∫[tex]0^4[/tex] ∫[tex]0^{\sqrt(y/2)}[/tex] dx dy dz = 32/15

Hence, the volume of the solid is 32/15 cubic units.

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For approximately 64% of the shows, the theater was half full or less than half full.

Since the seat capacity of the theater is 300, half full would be 150 seats or less. Looking at the histogram, we can see that there are 3 bars representing showings with 150 or less viewers.

The first bar represents showings with 0-50 viewers. From the histogram, it looks like there were about 5 showings with this number of viewers.

The second bar represents showings with 50-100 viewers. From the histogram, it looks like there were about 8 showings with this number of viewers.

The third bar represents showings with 100-150 viewers. From the histogram, it looks like there were about 3 showings with this number of viewers.

So the total number of showings with 150 or less viewers is 5+8+3 = 16.

Since the histogram shows a total of 25 showings, the fraction of shows that was half full or less than half is:

16/25 = 0.64 or 64%

Therefore, for approximately 64% of the shows, the theater was half full or less than half full.

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The exact value for each trigonometric function are -12/13, -5/13 and 12/5

To find the reference angle, we can use the properties of right triangles. We can draw a line from the point (-5, 12) to the x-axis to form a right triangle. The hypotenuse of the triangle is the distance from the point (-5, 12) to the origin, which is the square root of the sum of the squares of the x and y coordinates:

√((-5)² + 12²) = 13

The reference angle is the acute angle between the x-axis and the adjacent side of the triangle, which is the x-coordinate of the point (-5, 12) divided by the hypotenuse:

cosθ = -5/13

θ = arccos(-5/13)

θ ≈ 2.214 radians

The angle's standard form is given by the equation:

θ = n(2π) ± α

where n is an integer, and α is the angle's reference angle. Since the point (-5, 12) is in the second quadrant, the angle's terminal side intersects the unit circle at an angle of θ = π + α. Therefore, the standard form of the angle is:

θ = (2n + 1)π - arccos(-5/13)

To find the exact value of the trigonometric functions of this angle, we can use the properties of the unit circle. Since the sine function is positive in the second quadrant, we have:

sinθ = sin(π + α) = -sinα = -12/13

Similarly, since the cosine function is negative in the second quadrant, we have:

cosθ = cos(π + α) = -cosα = -5/13

Finally, since the tangent function is the ratio of the sine and cosine functions, we have:

tanθ = tan(π + α) = -tanα = 12/5

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1. The value of d is 10

2. measure of angle BSR is 120°

3. measure of angle RSM is 60°

The sum of angles on a straight line is 180°. This angles are adjascent angles.

angle BSR and RSM are on a straight line, therefore;

10d+20+6d = 180

16d = 180-20

16d = 160

d = 160/16

d = 10

therefore the value of d is 10

angle BSR = 10d+20 = 10×10+20

= 100+20 = 120°

angle RSM = 6d = 6 × 10

= 60°

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The critical value for this hypothesis test is 1.676.

To test the claim that manufacturing workers in Racine and Kenosha had the same salary as workers across the state, we can conduct a hypothesis test with the following null and alternative hypotheses:

We can use a t-test for the sample mean to test this hypothesis. At a 90% confidence level, we have a significance level of alpha = 0.10. Since this is a two-tailed test (we are testing for a difference in either direction), we will split the significance level evenly between the two tails, so alpha/2 = 0.05.

We need to calculate the critical value of the t-distribution with n-1 degrees of freedom, where n is the sample size. In this case, n = 54, so the degrees of freedom is 53. We can use a t-distribution table or a calculator to find the critical value. For a two-tailed test with alpha/2 = 0.05 and 53 degrees of freedom, the critical value is approximately 1.676.

Therefore, the critical value for this hypothesis test is 1.676.

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The basketball coach can order approximately 34 shooting shirts for $300.

To determine the number of shooting shirts the basketball coach can order for $300, we need to solve the equation C = 0.2x^2 + 1.6x + 15, where C represents the cost and x represents the number of shooting shirts.

The equation is given as C = 0.2x^2 + 1.6x + 15.

To find the number of shooting shirts for $300, we set the cost C equal to 300 and solve for x:

0.2x^2 + 1.6x + 15 = 300

0.2x^2 + 1.6x + 15 - 300 = 0

0.2x^2 + 1.6x - 285 = 0

Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

For this equation, a = 0.2, b = 1.6, and c = -285. Plugging in these values into the quadratic formula:

x = (-1.6 ± sqrt(1.6^2 - 4 * 0.2 * -285)) / (2 * 0.2)

Simplifying the equation further:

x = (-1.6 ± sqrt(2.56 + 228)) / 0.4

x = (-1.6 ± sqrt(230.56)) / 0.4

x = (-1.6 ± 15.18) / 0.4

Now we have two solutions:

x1 = (-1.6 + 15.18) / 0.4 = 33.95

x2 = (-1.6 - 15.18) / 0.4 = -44.95

Since the number of shooting shirts cannot be negative, we discard the negative solution.

Therefore, the basketball coach can order approximately 34 shooting shirts for $300.

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To find the new salary after a 2.6% increase, we need to add 2.6% of the original salary to the original salary.

2.6% of $29,000 can be calculated as:

(2.6/100) x $29,000 = $754

Therefore, the new salary will be:

$29,000 + $754 = $29,754

So the woman's new salary will be $29,754.

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The true statements about the rectangular prism are Prism B has a greater volume than Prism C, Prism B has the greatest volume, and Prism A has the least volume.

To understand, the differences between the prisms and therefore to verify the statements about them, let's calculate the volume of each by using the formula length x width x height.

Prism A: 2 x 2 x  3= 12 cubic units

Prism B: 2 x 3 x 4 = 24 cubic units

Prism C: 3 x 2 x 3 = 18cubic units

Based on this, the statements A, D, and E are correct.

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With regard to the similar triangles,

In ΔACD

BE ∥ CD

In ΔACD and ΔABE

BE ∥ CD

∠ACD =∠ABE           (corresponding angles)

∠ADC = ∠AEB          (corresponding angles)

∠A = ∠A                    (common angle)

ACD ∼ ΔABE

So,  The corresponding sides are in proportion.

Now, find ED

AB/BC = AE/ED
ED = AE (BC/AB)
ED = 26(5/20)
ED = 6.5cm


For BE

AB/AC = BE/CD
BE = CD (AB/AC)
BE = 18 (20/25)
BE  = 14.4cm

Now, find BE

Therefore, the length of ED is 6.5cm and the length of BE is 14.4 cm.

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An equation that express each relationship for the joint variation is X = kyz. Solving for y will give the equation y = X/(kz)

Joint variation is a mathematical concept that describes the relationship between two or more variables. 

If X varies jointly as y and z, we can express this relationship mathematically using the formula:

X = k × y × z

X = kyz

where k is a constant of proportionality.

We can solve for y by dividing both sides by kz follows:

X/kz = kyz/kz

X/(kz) = y

Therefore, the equation that express each relationship for the joint variation is X = kyz. And the equation solved for y is y = X/(kz).

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Answer:

An integer to describe 6 meters above the sea level would be  meters.

As per the question statement, We are supposed to use an integer to describe the following situation "6 meters above sea level".

We assume that sea level is the datum line and anything above that would be positive and below that would be negative.

So 6 meters above the sea level can be described as .

Integers: Set of whole number containing both positive and negative values of it.

PLS MARK BRAINLIEST

Step-by-step explanation:

The calculated value of the circumference of the circle is 31.4 units

From the question, we have the following parameters that can be used in our computation:

Radius, r = 5

Using the above as a guide, we have the following:

Circumference = 2 * π * r

Substitute the known values in the above equation, so, we have the following representation

Circumference = 2 * 5 * 3.14

Evaluate the products

Circumference = 31.4

HEnce, the value of the circumference is 31.4 units

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If ad and bd are tangent to circle c, and ad= 13 and bd= 4x-3, then the value of x is 4 or 4.5

To solve this problem, we'll need to use some geometry and algebra.

Using the Pythagorean theorem, we can set up two equations:

OA² + AD² = OD² (for right triangle OAD)

OB² + BD² = OD² (for right triangle OBD)

In these equations, "OD" is the radius of the circle. We don't know this value yet, but we can express it in terms of "x" using the fact that "BD" = 4x-3.

Now, we can simplify these equations by substituting in the values we know. We get:

OA² + 13² = OD²

OB² + (4x-3)² = OD²

This means we can set up the equation:

OA = OB

Now we can substitute in the expressions we found for "OA" and "OB" earlier:

√(OD² - 13²) = √(OD² - (4x-3)²)

We can then square both sides to eliminate the square roots:

OD² - 13² = OD² - (4x-3)²

Simplifying this equation, we get:

169 = (4x-3)²

Taking the square root of both sides (and remembering to include the positive and negative solutions), we get:

4x-3 = ±13

Solving for "x," we get two possible values:

x = 4

x = 4.5

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The percentage of the data values that are greater than 16, as shown in the box plot is: 75%.

A box plot shows how the data points of a data set are distributed, in such a way that, 25% of the data points lie below the lower quartile, % lie below the median, and 75% lie below the upper quartile.

In the box plot given, the values that are greater than 16 lie above the upper quartile, which equals about 75% of the data values.

Therefore, the percentage of the data values that are greater than 65, as shown in the box plot is: 75%.

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The maximum sum of two positive numbers in the range [1, 450] with a product of 450 is approximately 42.42.

1. Let the two numbers be x and y.
2. Given that their product is 450, we have the equation xy = 450.
3. To find the maximum sum, we will use the fact that the sum of two numbers is maximum when they are equal. So, x = y.
4. From the product equation, we get x * x = 450, which implies x^2 = 450.
5. Taking the square root of both sides, we have x = √450 ≈ 21.21 (approximately).
6. Since x = y, the maximum sum is x + y = 21.21 + 21.21 ≈ 42.42.

Therefore, the maximum sum of two positive numbers in the range [1, 450] with a product of 450 is approximately 42.42.

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The expectation of the product of two discrete random variables x and y is given by E(xy) = ∑(x∑(yP(x,y))) where P(x,y) is the joint probability distribution of x and y.

To find the expectation of the product of two random variables, we need to use the formula:

E(XY) = ΣΣ(xy)p(x,y)

where p(x,y) is the joint probability mass function of X and Y.

So, for the given joint distribution of X and Y, we have:

E(XY) = ΣΣ(xy)p(x,y)

We need to sum this over all possible values of X and Y. If the joint distribution is given in a table or a function form, we can simply plug in the values of X and Y and calculate the sum.

However, without any specific information about the joint distribution of X and Y, it is impossible to calculate the expectation of X times Y. We would need to know either the joint probability mass function or the joint probability density function of X and Y.

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the correct ratio to represent the relationship between Hallie's donation of jeans and Mattie's donation of jeans is 2:3. Thus, option B is correct.

To write a ratio to represent the relationship between Hallie's donation of jeans and Mattie's donation of jeans, we need to find the common factor between the number of pairs of blue jeans each person donated.

Hallie donated 4 pairs of blue jeans, and Mattie donated 6 pairs of blue jeans.

The common factor between 4 and 6 is 2. We can divide both 4 and 6 by 2 to get:

Hallie donated 2 pairs of blue jeans.

Mattie donated 3 pairs of blue jeans.

Now we can write the ratio of Hallie's donation to Mattie's donation as:

2:3

Therefore, the correct ratio to represent the relationship between Hallie's donation of jeans and Mattie's donation of jeans is 2:3.

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The function f(x) = (¹/₂)x⁴ +2x³ is concave up when f''(x) > 0, which is true when x > 0 or x < -2.

The concavity of a function is determined by taking the second derivative.

f'(x) = 2x³ + 6x²

f''(x) = 6x² + 12x

To find out when f(x) is concave up, we need to determine when f''(x) is positive;

f''(x) > 0

6x² + 12x > 0

6x(x + 2) > 0

When x > 0, both factors are positive, and the inequality is true.

When x < -2, both factors are negative, and the inequality is true.

Learn more about concave of function here: https://brainly.com/question/29121586

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