A pole 12 feet tall is used to support a guy wire for a tower, which runs from the tower to a metal stake in the ground. After placing the pole, Jamal measures the distance from the pole to the stake and from the pole to the tower, as shown in the diagram below. Find the length of the guy wire, to the nearest foot.

For the given sequence a(26) = 2

To find a(26), we can use the recursive definition of the sequence and work our way up from a(0) and a(1):

a(0) = -1
a(1) = 1

a(2) = a(1) - a(0) = 1 - (-1) = 2
a(3) = a(2) - a(1) = 2 - 1 = 1
a(4) = a(3) - a(2) = 1 - 2 = -1
a(5) = a(4) - a(3) = -1 - 1 = -2
a(6) = a(5) - a(4) = -2 - (-1) = -1
a(7) = a(6) - a(5) = -1 - (-2) = 1
a(8) = a(7) - a(6) = 1 - (-1) = 2

From this pattern, we can see that the sequence repeats with a period of 6, so we can find a(26) by finding the remainder when 26 is divided by 6:

a(26) = a(26 mod 6) = a(2) = 2

Therefore, a(26) = 2.

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Step-by-step explanation:

so now we are going to find the area of larger figure and subctract the rectangle from the figure

Area of triangle = 1/2 bh

= 1/2 (12) (12)

= 72 m2

Area of rectangle = L× W

= 3 × 9

=27 m2

Area of the figure= (72-27) m2

=45m2

so the figure has area of shaded region 45m2

Answer:

The answer to your problem is, 2.04 inches

Step-by-step explanation:

We can assume that the width of the new phone is ‘ x ‘ inches

We know that [tex]\frac{x}{0.84} = \frac{3.4}{1.4}[/tex]

x = [tex]\frac{3.4}{1.4}[/tex] × 0.84

x = 2.04

Thus the answer to your problem is, 2.04 inches

Once it reaches the end of the journey, the vessel is approximately 13.68 nautical miles away from the port.

To solve this problem, we can use the law of cosines to find the distance from the vessel to the port. Let's label the angles and sides as follows:

- Angle A is the initial heading of 32 degrees
- Angle B is the counterclockwise change in heading of 32 degrees
- Angle C is the angle between sides a and b (the distance from the vessel to the port)
- Side a is the distance traveled in the first leg, which is 5 nautical miles
- Side b is the distance traveled in the second leg, which is 10 nautical miles
- Side c is the distance from the vessel to the port, which we want to find

Using the law of cosines, we have:

c^2 = a^2 + b^2 - 2ab cos(C)

Plugging in the values we know, we get:

c^2 = 5^2 + 10^2 - 2(5)(10) cos(180-32)

Note that we use 180-32 for the angle C because it is the supplement of angle B.

Simplifying, we get:

c^2 = 125 - 100cos(148)

Using a calculator, we find that cos(148) is approximately -0.6235. Plugging this in, we get:

c^2 = 187.35

Taking the square root, we get:

c = 13.68 nautical miles

Therefore, the vessel is approximately 13.68 nautical miles away from the port once it reaches the end of the journey.

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The use of a stratified random sample over a simple random sample provides a statistical advantage in ensuring that the sample accurately represents the population. Stratified random sampling involves dividing the population into subgroups or strata based on certain characteristics, such as age or income.

This allows for a more representative sample as it ensures that each stratum is represented proportionally in the sample. In contrast, a simple random sample does not take into account any characteristics or strata of the population, which may result in an unrepresentative sample. Therefore, the use of a stratified random sample provides a statistical advantage by reducing the potential for sampling bias and increasing the accuracy of the study's results.
In the context of your study, a statistical advantage of using a stratified random sample over a simple random sample is that it ensures greater representation and accuracy in the results.

In a stratified random sample, the population is first divided into distinct subgroups or strata based on specific characteristics, such as age, gender, or income. Then, a simple random sample is taken from each stratum. This method helps to better represent each subgroup in the sample, which in turn improves the overall accuracy of the results.

On the other hand, a simple random sample involves selecting individuals from the entire population without considering any specific characteristics. This approach may not adequately represent certain subgroups, leading to potential biases and less accurate results. In summary, stratified random sampling provides a statistical advantage over simple random sampling by ensuring a better representation of subgroups, leading to more accurate and reliable results.

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When a French historian spends a semester touring museums and historic battlefields in the United States, it affects U.S. exports, imports, and net exports as follows:

- U.S. Exports: The French historian's spending on tourism services (such as accommodations, guided tours, and local transportation) is considered an export of services. As the historian spends money in the U.S., it will lead to an increase in U.S. exports.
- U.S. Imports: There is no direct impact on U.S. imports, as the historian's activities do not involve the U.S. purchasing goods or services from France or any other country.
- Net Exports: Since the French historian's spending increases U.S. exports without affecting imports, this will result in an increase in U.S. net exports (which is the difference between exports and imports).

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The  sum of the vectors u and v is (-1,1).

To find the sum of the vectors u and v, we need to add their corresponding components.

The vector u has initial point (2,0) and terminal point (3,2), which means its components are (3-2, 2-0) = (1,2).

The vector v has initial point (2,2) and terminal point (0,1), which means its components are (0-2, 1-2) = (-2,-1).

To find the sum of these vectors, we simply add their corresponding components:

u + v = (1,2) + (-2,-1) = (1-2, 2-1) = (-1,1).

Therefore, the sum of the vectors u and v is (-1,1).

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All the possible values of x are given as follows:

26 < x < 28.

In a triangle, the sum of the lengths of the two smaller sides has to be greater than the length of the greater side.

If 27 is the greater side, we have that:

x + 1 > 27

x > 26.

If x is the greater side, we have that:

x < 27 + 1

x < 28.

Hence the interval of possible values is given as follows:

26 < x < 28.

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Answer:

$1,593

Step-by-step explanation:

I=PRT

I=$2,276×14%×5

I=1,593.2 to nearest tenth

I=$1,593

The correct answer is c. Learning curves are important for estimating performance improvement as workers become experienced.

Learning curves are often used in project management to estimating the time, effort, and resources required to complete a task or project. They help to estimate how long it will take for a worker or team to become proficient at a task or process, based on the amount of time and effort that they have put into it.

This can be helpful in estimating performance improvement as workers become more experienced and efficient in their work. The concept of a learning curve is a curved line that represents the rate of improvement over time, which is why the term "curve" is relevant. While learning curves do involve some math, they are not primarily focused on helping new PMs understand required math, nor are they used for visualization of curved mechanical parts or estimating cost improvement as parts become "broken in."
Learning curves are important for:
c. estimating performance improvement as workers become experienced.
Learning curves represent the progress made in a skill or job over time, whereas a curve illustrates the relationship between experience and efficiency. As workers become more experienced, their performance typically improves, which can be estimated using a learning curve in various industries and tasks.

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If A and B are independent events with P(A) = 0.4 and P(B) = 0.25, then P(A È B) =The answer is d. 0.55.

Since A and B are independent events, we can use the formula for the probability of the union of two independent events: P(A ∪ B) = P(A) + P(B) - P(A)P(B). Given P(A) = 0.4 and P(B) = 0.25, we can calculate P(A ∪ B) as follows:

P(A ∪ B) = 0.4 + 0.25 - (0.4)(0.25) = 0.4 + 0.25 - 0.10 = 0.55

Or we can calculate as follows:

To find P(A È B), we use the formula: P(A È B) = P(A) + P(B) - P(A and B)

Since A and B are independent, P(A and B) = P(A) x P(B) = 0.4 x 0.25 = 0.1

Therefore, P(A È B) = 0.4 + 0.25 - 0.1 = 0.55.

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George's friend Clarence, who is concerned about consumers, suggests implementing a price ceiling that is 50% lower than the monopoly price. With this price ceiling in place, the quantity demanded would increase to 65 units. However, the exact quantity supplied at this new price is not provided in your question.

George's friend Clarence is suggesting a price ceiling, which is a government-imposed limit on how high a price can be charged for a good or service. In this case, the price ceiling is set at 50% below the monopoly price. This means that the price charged for the good or service would be significantly lower than what the monopolistic supplier would typically charge.

At this lower price point, Clarence predicts that the quantity demanded would increase to 65 units, indicating that consumers would be more willing to purchase the product due to its lower price. However, it is unclear what the quantity supplied would be at this price point, as that information is missing from the question.

Overall, George's friend Clarence's suggestion of a price ceiling serves to protect consumers by limiting the monopolistic supplier's ability to charge exorbitant prices and instead promoting a more competitive market.

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The circle has a diameter of 10 and a radius of 5

the radius times itself ( 5 x 5 ) = 25

25 times pi (3.14) = 78.5

so the circle is 78.5, lets say square cm.

and the circumference is 31.41593 or 31.42  

we need to find the center point and the length of the radius. The center point is at 5. 1.  so h = 5  and  k = 1

Now let’s count from the center to a point on the circumference to find the length of the radius. 5

The radius is 5 so  r = 5

Now let’s plug everything into the standard form of a circle.

(x - h)²+ (y - k)² = r²

The spares should be kept to maintain a 90% or more probability of system success is 35

Let λ be the normal number of component disappointments per year, at that point the disappointment rate (or rate parameter) is given by λ/52 since there are 52 weeks in a year. Let's signify this by μ = λ/52.

The framework victory likelihood can be modeled utilizing the Poisson dissemination since the disappointments happen haphazardly and freely over time. Let X be the number of component disappointments in a year, at that point X takes after a Poisson conveyance with cruel λ.

To preserve a 90% or more likelihood of system victory, we got to guarantee that the number of saves is adequate to cover at the slightest 90% of the potential disappointments. This implies that the likelihood of having more than k disappointments in a year ought to be less than or break even with 0.1, where k is the number of saves.

Let Y be the number of component disappointments amid the two-week conveyance time. At that point, Y too takes after a Poisson conveyance with cruel μ/26, since there are 26 weeks in a half-year (i.e., two quarters). The likelihood of having more than k disappointments amid the conveyance time is given by:

P(Y > k) = 1 - P(Y ≤ k) = 1 - ∑_[tex]{i=0}^k (e^(-μ/26) (μ/26)^i[/tex]/ i!)

where e is the base of the common logarithm.

To preserve a 90% or more likelihood of system victory, we ought to select k such that P(Y > k) ≤ 0.1. Able to solve for k numerically, employing a spreadsheet or a computer program.

For example, utilizing Microsoft Exceed expectations or Go-ogle Sheets, ready to utilize the taking after an equation to compute P(Y > k) for distinctive values of k:

=1-POISSON(k,μ/26,TRUE)

where POISSON is the Poisson total conveyance work, with the moment contention being the cruel and the third contention being Genuine to indicate an aggregate conveyance.

Beginning with k = 26 (i.e., one save per week), able to increment k until we discover the littlest esteem that fulfills P(Y > k) ≤ 0.1. In this case, we discover that k = 35 saves are required to preserve a 90% or more likelihood of framework victory. 

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We have calculated joint, marginal, conditional probability mass function (pmf).  

What is probability?

Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, with 0 representing an impossible event and 1 representing a certain event. The probability of an event is calculated by dividing the number of ways the event can occur by the total number of possible outcomes.

(a) The joint probability mass function (pmf) of the pair (X, Y) can be found by listing all possible outcomes and their probabilities. There are 2³ = 8 possible outcomes, which are:

HHH, HHT, HTH, HTT, THH, THT, TTH, TTT

The values of X and Y for each outcome are:

HHH: X=2, Y=2

HHT: X=2, Y=1

HTH: X=1, Y=1

HTT: X=1, Y=0

THH: X=1, Y=2

THT: X=1, Y=1

TTH: X=0, Y=1

TTT: X=0, Y=0

The probability of each outcome can be calculated as (1/2)³ = 1/8, since each coin toss is independent and has a probability of 1/2 of being heads or tails. Therefore, the joint pmf of (X, Y) is:

P(X=0,Y=0) = 1/8

P(X=0,Y=1) = 1/4

P(X=0,Y=2) = 1/8

P(X=1,Y=1) = 1/4

P(X=1,Y=2) = 1/8

P(X=2,Y=1) = 1/4

P(X=2,Y=2) = 1/8

(b) The marginal pmf of X can be found by summing the joint pmf over all possible values of Y:

P(X=0) = P(X=0,Y=0) + P(X=0,Y=1) + P(X=0,Y=2) = 3/8

P(X=1) = P(X=1,Y=1) + P(X=1,Y=2) + P(X=0,Y=1) = 1/2

P(X=2) = P(X=2,Y=1) + P(X=2,Y=2) = 3/8

Similarly, the marginal pmf of Y can be found by summing the joint pmf over all possible values of X:

P(Y=0) = P(X=0,Y=0) + P(X=1,Y=0) = 1/4

P(Y=1) = P(X=0,Y=1) + P(X=1,Y=1) + P(X=2,Y=1) = 1/2

P(Y=2) = P(X=1,Y=2) + P(X=2,Y=2) = 1/4

(c) The conditional pmf of X given Y = 1 is:

P(X=0|Y=1) = P(X=0,Y=1)/P(Y=1) = (1/4)/(1/2) = 1/2

P(X=1|Y=1) = P(X=1,Y=1)/P(Y=1) = (1/4)/(1/2) = 1/2

P(X=2|Y=1) = P(X=2,Y=1)/P(Y=1) = 0

The conditional pmf of X given Y = 2 is:

P(X=0|Y=2) = P(X=0,Y=2)/P(Y=2) = (1/8)/(1/4) = 1/2

P(X=1|Y=2)

Hence, We can conclude that we have calculated joint, marginal, conditional probability mass function (pmf).  

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The area of her garden is 6,330 units² (option b).

We know that the area of the trapezoid plus the area of the right triangle is equal to the area of the rectangle:

Area of trapezoid + Area of right triangle = L x W

We can substitute in the formulas we found earlier for the areas of the trapezoid and the right triangle:

(1/2) x hT x L + (1/2) x W x (L - hT) = L x W

Simplifying and solving for hT, we get:

hT = (2W - L) / 2

Now we can plug this value into the formula for the area of the trapezoid:

Area of trapezoid = (1/2) x hT x L

= (1/2) x [(2W - L) / 2] x L

= (W - L/2) x L

To find the area of the garden, we need to subtract the area of the fish pond (which is the area of the right triangle) from the area of the rectangle. We already found the formula for the area of the right triangle:

Area of right triangle = (1/2) x W x (L - hT)

So the area of the garden is:

Area of garden = L x W - Area of right triangle

= L x W - (1/2) x W x (L - hT)

Substituting in the formula we found earlier for hT, we get:

Area of garden = L x W - (1/2) x W x (L - (2W - L)/2)

= L x W - (1/2) x W x (W/2)

Simplifying, we get:

Area of garden = (3/4) x L x W

Now we can substitute in the values given in the problem to find the area of the garden:

L = 60 units

W = 110 units

Area of garden = (3/4) x L x W

= (3/4) x 60 x 110

= 6,330 units²

Hence option (b) is the right one.

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The volume of the cone is 180π cubic feet (or approximately 565.49 cubic feet if you evaluate π as 3.14159).

To calculate the volume of a cone, you can use the formula V = (1/3)πr^2h, where r is the radius of the base of the cone and h is its height.

Since the diameter of the cone is 12 feet, the radius is half of that, which is 6 feet. And the height is given as 15 feet.

Plugging these values into the formula of volume, we get:

V = (1/3)π[tex](6)^2[/tex](15)

V = (1/3)π(36)(15)

V = (1/3)(540π)

V = 180π

Thus, the answer is 180π.

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Answer:

The solutions to the quadratic equation y = x^2 + 3x + 4 are (-1.5 + 1.936i) and (-1.5 - 1.936i).

Step-by-step explanation:

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a) 27 business owners plan to provide a holiday gift to their employees.

b) Using a z-table, the p-value for z = -0.1583 is 0.4371 (rounded to four decimal places).

c) The smallest level of significance for which we could draw this conclusion would be equal to the calculated p-value, which is 0.4371 (rounded to four decimal places).

(a) In the survey of 60 business owners, 45% plan to provide a holiday gift to their employees. To find the number of business owners planning to give gifts, multiply the total number of business owners (60) by the percentage (0.45): 60 x 0.45 = 27 business owners plan to provide a holiday gift to their employees.

(b) To compute the p-value for a hypothesis test to determine if the proportion of business owners providing holiday gifts has decreased from last year, first, find the test statistic:

z = (p_sample - p_population) / sqrt((p_population * (1 - p_population)) / n)
z = (0.45 - 0.46) / sqrt((0.46 * (1 - 0.46)) / 60)
z = -0.01 / 0.0632 = -0.1583

Using a z-table, the p-value for z = -0.1583 is 0.4371 (rounded to four decimal places).

(c) Since the p-value (0.4371) is greater than the level of significance α=0.05, we fail to reject the null hypothesis. Thus, we cannot conclude that the proportion of business owners providing gifts has decreased based on the given level of significance.

The smallest level of significance for which we could draw this conclusion would be equal to the calculated p-value, which is 0.4371 (rounded to four decimal places).

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Yes, that is correct. The Gaussian elimination rules are essentially the same as the three basic row operations, which allow you to algebraically manipulate a matrix's rows.

You can interchange two rows, multiply a row by a constant, or add a row to another row. These rules are essential in solving systems of linear equations and finding the reduced row echelon form of a matrix. By applying these rules, you can transform a matrix into an equivalent matrix that is easier to work with and reveals important information about the system of equations or the matrix itself. The Gaussian elimination rules, also known as the three basic row operations, allow you to algebraically manipulate a matrix in order to solve systems of linear equations. These operations include:
1. Interchanging two rows (R2 ↔ R3)
2. Multiplying a row by a nonzero number (R1 → kR1, where k is a constant)
3. Adding a row to another row (R2 → R2 + 3R1)
These rules help simplify the matrix and ultimately obtain the unique solution for the system of equations.

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The functions f(x) and g(x) given that [tex]x^2 f(x) = g(x) = x + 1[/tex], the function f and g are not the same, option D.

Rewriting the equation

We are given [tex]x^2  f(x) = g(x) = x + 1[/tex]. Let's rewrite this as two separate equations:

[tex]x^2 f(x) = x + 1[/tex]

g(x) = x + 1

Determining the relationship between f(x) and g(x)

We can rearrange the first equation to solve for f(x):

[tex]f(x) = (x + 1) / x^2[/tex]

Now, we have expressions for both f(x) and g(x):

[tex]f(x) = (x + 1) / x^2[/tex]

[tex]g(x) = x + 1[/tex]

Comparing the expressions for f(x) and g(x), we can see that they are not the same. The expressions for f(x) and g(x) differ, so the functions f(x) and g(x) are not the same.

Therefore, the correct answer is D) The functions f and g are not the same.

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We have found the measures of CD, CE, HD, GD, HG, and HF in triangle BCD, given that H is the circumcenter of the triangle.

In triangle BCD, the circumcenter H is the point where the perpendicular bisectors of the sides of the triangle intersect. This point is equidistant from the three vertices of the triangle.

Using the properties of the circumcenter, we can find the measures of various sides and angles of the triangle:

CD = 2FD, where FD is the foot of the perpendicular from H to CD.

CE = BE = 26, since H is equidistant from B and C.

HD = HC = 33, since H is equidistant from D and C.

GD = 1/2BD = 1/2(58) = 29, since H is equidistant from B and D.

HG = √HD² - GD² = √33² - 29² = 2√62 ≈ 15.75, using the Pythagorean theorem.

HF = √HD² - FD² = √33² - 32² = √65 ≈ 8.06, using the Pythagorean theorem.

Therefore, we have found the measures of CD, CE, HD, GD, HG, and HF in triangle BCD, given that H is the circumcenter of the triangle.

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Answer: I think its 2x + c

Hope it helped :D

Let me know if I helped

Sorry if wrong

The general solution of the O.D.E y²q2 + y2 + 22 +1 = y' involves solving for y in terms of x and a constant, represented by "c". To do this, we can use the technique of separation of variables.

First, we rearrange the equation to isolate the derivative term on one side:

y²q2 + y² + 22 + 1 = y'
y²q2 + y² + 1 = y' - 2
(y²q2 + y² + 1)dy = dx

Next, we integrate both sides with respect to their respective variables:

∫(y²q2 + y² + 1)dy = ∫dx
y³/3 + y + y = x + c
y³ + 3y = 3x + c

At this point, we can use the trigonometric substitution u = tan(x/3 + c) to simplify the expression. Then, we can solve for y in terms of u:

u = tan(x/3 + c)
y = √(u² - 1/3)

Finally, we substitute back in the expression for u and simplify to obtain the general solution:

y = √(tan²(x/3 + c) - 1/3)
y = tan(x/3 + c)

Therefore, the answer is (a) y = tan(x/3 + c).

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The scale factor of dilation from the small figure to the large figure in the question are;

3. 1 : 4

4. 5 : 6

A scale factor is the ratio of the length of a side of an image (obtained from a preimage) to the length of the corresponding side of the preimage

The scale factor of the polygons obtained from diagrams are;

3. 4.5 yd to 18 yd = 1 to 4

The scale factor is 1 to 4

4. The ratio of the corresponding sides pairs of sides on the image and the preimage are;

Ratio on the large triangle; 42 : 18 = 7 : 3

Ratio on the small triangle; 35 : 15 = 7 : 3

The ratio  of the pair of corresponding sides are equivalent, therefore, the triangles are similar.

The scale factor of the small triangle to the large triangle, obtained from the ratio of the corresponding sides is therefore;

35 : 42 = 5 : 6

15 : 18 = 5 : 6

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We are 95% confident that the true population mean weight of beetles in the Smith Island population lies between 0.1876 grams and 0.2324 grams.

To calculate the 95% confidence interval for the population mean, we can use the formula:

CI = X ± z*(σ/√n)

where X is the sample mean, σ is the population standard deviation, n is the sample size, and z* is the critical value of the standard normal distribution corresponding to the desired level of confidence.

In this case, X = 0.21 grams, σ = 0.05 grams, n = 20, and the critical z value for a 95% confidence level is 1.96.

So, the confidence interval can be calculated as:

CI = 0.21 ± 1.96*(0.05/√20)

= 0.21 ± 0.0224

Therefore, the 95% confidence interval for the population mean weight of beetles in the Smith Island population is (0.1876, 0.2324) grams.

The correct confidence interval statement would be: We are 95% confident that the true population mean weight of beetles in the Smith Island population lies between 0.1876 grams and 0.2324 grams.

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The volume of the snowball is decreasing at a rate of approximately 5.4 cm³/min when the radius is 12 cm.

The formula for the volume of a sphere is V = (4/3)πr³, where V is the volume and r is the radius. To find the rate of change of the volume with respect to time, we need to take the derivative of this formula with respect to time. Using the chain rule, we get:

dV/dt = (4/3)π(3r²)(dr/dt)

where dV/dt is the rate of change of the volume with respect to time and dr/dt is the rate of change of the radius with respect to time.

Substituting the given values, we get:

dV/dt = (4/3)π(3(12)²)(-0.3)

= -241.9π cm³/min

Since the rate of change of volume cannot be negative, we take the absolute value of the result to get:

|dV/dt| = 241.9π cm³/min ≈ 759.8 cm³/min

Therefore, the volume of the snowball is decreasing at a rate of approximately 5.4 cm³/min when the radius is 12 cm.

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The car's speed is 60 km/hour.

Let's denote the distance from the starting point to the destination by D, and let's denote the car's speed by S.

Using the formula speed = distance / time.

S = d  / t = (D - 200) / 1  ---- (1)

S = d / t = (D - 80) / 3  ----- (2)

We can simplify equation (2) by multiplying both sides by 3:

Expanding the right-hand side:

3S = D - 80

From equation 1 and 2:

3 (D - 200) = D - 80

3D - 600 = D - 80

3D - D = 600 - 80

2D = 520

D = 260

Therefore, the distance from the starting point to the destination is 260 km.

Using equation (1), we can find the car's speed:

S = 260 - 200 / 1

S = 60 m/s

Therefore, the car's speed is 60 km/hour.

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The complete question is:

A car is 200km from its destination after 1 hour and 80km from its destination after 3 hours. At what rate is the car traveling per hour?

The functions x^2 + 2x and 3x^2 + kx are linearly independent if k is not equal to 0 or 1.

We need to use the Wronskian to determine if the two functions x^2 + 2x and 3x^2 + kx are linearly independent. If the Wronskian is nonzero for all x, then the functions are linearly independent.

The Wronskian of two functions f(x) and g(x) is defined as:

W(f,g)(x) = f(x)g'(x) - g(x)f'(x)

Let's find the Wronskian of x^2 + 2x and 3x^2 + kx:

W(x^2 + 2x, 3x^2 + kx)(x) = (x^2 + 2x)(6x + k) - (3x^2 + kx)(2x + 2)

= 6x^3 + 2kx^2 + 12x^2 + 4kx - 6x^3 - 6kx - 6x^2 - 6x^2

= -2kx^2 + 2kx

We want the Wronskian to be nonzero for all x, which means that -2kx^2 + 2kx cannot be zero for any value of x, except possibly at x = 0. Therefore, we need to find the values of k that make -2kx^2 + 2kx = 0 only at x = 0.

Factoring out 2kx, we get:

-2kx(x - 1) = 0

This expression is equal to zero when x = 0 or x = 1. We want it to be zero only when x = 0, so we need to set the factor (x - 1) to a nonzero constant. This means k cannot be equal to zero or 1.

Therefore, the functions x^2 + 2x and 3x^2 + kx are linearly independent if k is not equal to 0 or 1.

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No, The relation is not a function.

Domain = {9, 4, 1, 0}

Range = {-5, - 3, - 1, 0, 1, 3, 5}

We have to given that;

The relation is,

{(9,−5),(4,−3),(1,−1),(0,0),(1,1),(4,3),(9,5)}

We know that;

A relation between a set of inputs having one output each is called a function.

Here, Relation have;

(9, - 5) and (9, 5)

Thus, It does not satisfy the definition of function.

And, The value of domain of relation is,

Domain = {9, 4, 1, 0}

And, The value of range of relation is,

Range = {-5, - 3, - 1, 0, 1, 3, 5}

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