The random variable x is known to be uniformly distributed between 10 and 20. Show the graph of the probability density function: Compute P(x 15). Compute P(12 =x= 18). St Compute E(x). Compute Var(x).

Answer:

Tank 1: 6 litres

Tank 2: 6.75 litres

Tank 3: 4.5 litres

Step-by-step explanation:

Initially, James had 18 litres of water, and he poured three-quarters of it from tank 1 into tank 2. This means that the volume of water left in tank 1 is:

18 - (3/4) * 18 = 4.5 litres

The volume of water in tank 2 is:

(3/4) * 18 = 13.5 litres

Next, he poured half of the water in tank 2 (which is now 13.5 litres) into tank 3. The volume of water left in tank 2 is:

(1/2) * 13.5 = 6.75 litres

The volume of water in tank 3 is:

13.5 * (1/2) = 6.75 litres

Finally, he poured one-third of the water in tank 3 (which is now 6.75 litres) into tank 1. The volume of water in tank 1 after this is:

4.5 + (1/3) * 6.75 = 6 litres

The volume of water in tank 3 after this is:

6.75 * (2/3) = 4.5 litres

So the final volume of water in each tank is:

Tank 1: 6 litres

Tank 2: 6.75 litres

Tank 3: 4.5 litres

Answer:best thing for you is drink milk

Step-by-step explanation:

Option A is the correct option for determining the total number of possible outcomes when selecting a day of the week and a month of the year, which is 84.

To find the total number of outcomes for picking a day of the week and a month of the year, you need to multiply the number of options for each category. There are 7 days in a week, so there are 7 options for picking a day. There are 12 months in a year, so there are 12 options for picking a month. To find the total number of outcomes, you multiply the number of options for each category: 7 x 12 = 84. Therefore, the total number of outcomes for picking a day of the week and a month of the year is 84. Option A is the correct answer.

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The largest possible whole-number length for the third side is 18, which satisfies all three inequalities.

The triangle inequality theorem explains the relationship between the three sides of a triangle. This theorem states that for any triangle, the sum of the lengths of the first two sides is always larger than the length of the third side.

Let x be the length of the third side. By the triangle inequality, we have:

3 + 16 > x and 16 + x > 3 and 3 + x > 16

Simplifying, we get:

19 > x and x > 13 and x < 19

The largest possible whole-number length for the third side is 18, which satisfies all three inequalities.

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From the  X coordinate is -0.45 centimeter and Y coordinate is -2.0 cm of the center of mass for the uniform plate.

According to the Question:

Since the plate is uniform, we can divide it into three rectangles, each with a mass proportional to its area and with its barycenter at its geometric center. We will refer to the large piece of 35cm x 10cm; it occupies 63.6% of the total surface and its barycenter is at

(x₁, y₁) = (−5.0cm,−2.5cm).

Now,

The top 20cm× 5cm piece has 18.2 % of the total area; its center of mass is at (x₂, y₂) = (10cm,12.5cm).  

Now,

The bottom 10cm× 10cm piece (section 3) also has 18.2 % of the total area; its center of mass is at (x₃, y₃)​ = (5cm,−15cm).

(a) The x coordinate of the center of mass for the plate is

   [tex]X_{com} = (0.636)X_1 + (0.182)X_2 + (0.182)X_3[/tex]

            = -0.45 cm

(b) The y coordinate of the center of mass for the plate is

    [tex]Y_{com} = (0.636)Y_1 + (0.182)Y_2 + (0.182)Y_3[/tex]

             = -2.0 cm.

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In quadratic functiοn -

          g(x)                                    h(x)                                     d(x)

vertical shift dοwn 3     reflectiοn acrοss the x-axis   vertical shift dοwn 3

hοrizοntal shift left 3     vertical strecht οf 3               hοrizοntal shift right 3

The expressiοn f(x) = ax² + bx + c, where a, b, and c are all integers with a nοt equal tο zerο, denοtes a quadratic functiοn. The shape that a quadratic functiοn's graph takes οn is called a parabοla.

In spite οf the fact that parabοlas might have οpenings that are upward οr dοwnward, varying "widths" οr "steepnesses," they all have the same basic "U" shape.

Quadratic parent: f(x)=x²

The graph is a parabοla with vertex V=(0,0) at the οrigin and οpens up.

When x=1→f(1)=1²→f(1)=1

1) g(x)

The graph οpens up, then there is nοt a reflectiοn acrοss the x-axis.

The vertex is at the pοint (-3,-3): 3 units tο the left and 3 units dοwn οf the vertex οf the parent funtiοn.

When x is 1 unit tο the right frοm the vertex g(x)=1

Then the transfοrmatiοns were applied tο the cuadratic parent functiοn are:

1.1) vertical shift dοwn 3.

1.2) hοrizοntal shift left 3.

2) h(x)

The graph οpens dοwn, then there is a reflectiοn acrοss the x-axis.

The vertex is at the οrigin (0,0).

When x is 1 unit tο the right frοm the vertex h(x)=-3

Then the transfοrmatiοns were applied tο the quadratic parent functiοn are:

2.1) reflectiοn acrοss the x-axis.

2.2) vertical strecht οf 3.

3) d(x)

The graph οpens up, then there is nοt a reflectiοn acrοss the x-axis.

The vertex is at the pοint (3,-3): 3 units tο the right and 3 units dοwn οf the vertex οf the parent funtiοn.

When x is 1 unit tο the right frοm the vertex d(x)=1

Then the transfοrmatiοns were applied tο the cuadratic parent functiοn are:

3.1) vertical shift dοwn 3.

3.2) hοrizοntal shift right 3.

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The complete question is -

The given graphs show functions which have been transformed from the quadratic parent, f(x) = x2. Determine which transformations were applied to the quadratic parent function to result in each graph.

Step-by-step explanation:

The change fro 225 to 320 is 95

95 is what percentage of 225?

95/225 x 100% = 42.2% increase

The probability that a student completed the homework given that they failed the test is 3/8.

Probability is a branch of mathematics that deals with the study of randomness and uncertainty in events. It involves the measurement and analysis of the likelihood or chance of an event occurring. Probability is expressed as a number between 0 and 1, with 0 indicating an impossible event and 1 indicating a certain event.

The concept of probability is used in many fields, such as statistics, physics, finance, and engineering, to make predictions and decisions based on incomplete information. It provides a framework for analyzing situations in which multiple outcomes are possible and assigns a quantitative value to the likelihood of each outcome.

To find the probability that a student completed the homework given that they failed the test, we need to use conditional probability.

Let A be the event that a student completed the homework and B be the event that a student failed the test.

We want to find P(A | B), the probability that a student completed the homework given that they failed the test.

Using the data from the table, we can see that the total number of students who failed the test is 8 (3 who completed the homework and 5 who did not).

Therefore, P(B) = 8/18 = 4/9 (the probability that a student failed the test)

Out of the 8 students who passed the test, 3 also completed the homework, so P(A and not B) = 3/18 = 1/6.

So, P(A | B) = P(A and B) / P(B) = (1/6) / (4/9) = 3/8.

Therefore, the probability that a student completed the homework given that they failed the test is 3/8.

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Based on the above figures and the formula for calculating the debt-assets ratio, the company's debt-assets ratio (where debt is defined to include both short-term and long-term debt) is 0.413 or 41.3%.

The given balance sheet data,

Balance Sheet Data

Cash on Hand 2,000

Total Current Assets 78,000

Total Assets 315,000

Overdraft Loan Payable 4,000

1- Year Bank Loan Payable 8,000

Current Portion of Long-Term Loans 13,000

Total Current Liabilities 48,000

Long-Term Bank Loans Outstanding 105,000

Shareholder Equity Year11 Balance Year 12 Change Common Stock 10,000 0 10,000

Additional Capital 100,000 0 100,000

Retained Earning 30,000 22,000 52,000

Total Shareholder Equity 140,000 +22,000 162,000

          The debt-assets ratio is calculated by dividing the total debt (current liabilities + long-term loans) by the total assets.

           In this case, the total debt is 48,000 + 105,000 = 153,000

           and the total assets are 315,000.

   Thus, the debt-assets ratio is 153,000/315,000 = 0.413.

           Also, the debt-assets ratio is 0.413 * 100 = 41.3%

           The correct answer is b) 0.413.

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(E)The interval would have been wider and would have less precision than the original  .

A beef rancher randomly sampled 42 cattle from her large herd to obtain a 95% confidence interval to estimate the mean weight of the cows in the herd. The interval obtained was (1010, 1321). If the rancher had used a 90% confidence interval instead, the interval would have been narrower and would have less precision than the original estimate.

If the sample size is constant, then we can say that as the level of confidence is increased, the width of the interval will also increase. This is because a higher level of confidence will require a larger interval to include the population mean. Similarly, if the level of confidence is decreased, then the width of the interval will also decrease as a smaller interval is needed to include the population mean.

This directly implies that if the confidence interval is wider, then it has more precision and vice versa. Hence, Option (E) is the correct answer.

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The 90% confidence interval for the proportion of district residents in favor of starting the school day 15 minutes later is (0.392, 0.548). The true proportion is estimated to lie within this interval with 90% confidence.

To calculate the 90% confidence interval for the true proportion of district residents who are in favor of starting the school day 15 minutes later, we can use the following formula:

CI = p ± z*(√(p*(1-p)/n))

where:

CI: confidence interval

p: proportion of residents in favor of starting the day later

z: z- score based on the confidence level (90% in this case)

n: sample size

First, we need to calculate the sample proportion:

p = 94/200 = 0.47

Next, we need to find the z- score corresponding to the 90% confidence level. Since we want a two-tailed test, we need to find the z- score that cuts off 5% of the area in each tail of the standard normal distribution. Using a z-table, we find that the z- score is 1.645.

Substituting the values into the formula, we get:

CI = 0.47 ± 1.645*(√(0.47*(1-0.47)/200))

Simplifying this expression gives:

CI = 0.47 ± 0.078

Therefore, the 90% confidence interval for the true proportion of district residents who are in favor of starting the school day 15 minutes later is (0.392, 0.548). We can be 90% confident that the true proportion lies within this interval.

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Answer:

Question 4

a) 0.667

b) 0.833

c) 0.833

d) 0

Question 5

a) 32 yards

b) $312

Step-by-step explanation:

Probability question 4

The best way to solve these probability questions is to enumerate and count the total set of outcomes and then enumerate and count the specific set of outcomes they are looking for in each part .

Since the experiment consists of rolling a fair single 6-sided die  we can enumerate the set of  possible outcomes as
{1, 2, 3, 4, 5, 6}. This is also referred to in probability as sample space.

There are 6 possible outcomes

The probability of getting any one number is 1/6 whether it be a 1, a 2 or 3, 4, 5 or 6.

In general P(a specific outcome) = Number of ways of getting that outcome ÷ total number of all possible outcomes

part a)
Rolling an even==>  Set of outcomes  = {2, 4, 6}
Rolling a 3 => {3}

Rolling a 3 or even = {2, 3, 4, 6}; 4 elements
P(even or 3 ) = 4/6 = 2/3 = 0.667

Part b)
Number > 2 ==> { 3, 4, 5, 6}
Even: {2, 4, 6}

Even or greater than 2 = {2, 3, 4, 5, 6}; 5 elements  - we do not duplicate

P(Even or greater than 2 ) = 5/6 = 0.833

part c)
Odd number: {1, 3, 5}
Number less than 6: {1, 2, 3, 4, 5}
Combining the two we get: {1, 2, 3, 4, 5} for a total of 5 outcomes

P(odd or less-than 6 ) = 5/6 = 0.833

part(d)
Rolling a 2 and a 4
You cannot roll a 2 and a 4 in one toss so the probability is 0

Question 5
I think you have overly complicated the working of this problem. There is no need to use right triangle, hypotenuse etc

If you have a rectangle with length = L and width = W, the perimeter is just

P = 2( L + W)

Here we can choose any dimension for L and the other for W
Let's choose L = 11 (greater number)
W = 5

Perimeter P = 2 (11 + 5) = 2(16) = 32 yards

b) It costs $3.25 per foot for the fencing..
Careful! The  only tricky part here is the units. It is given in $ per square foot whereas the dimensions of the plot are given in yards and the perimeter is also in yards. Watch out for these tricks in other questions

Convert perimeter from yards to feet

32 yards = 32 x 3 = 96 feet  (there are 3 feet to a yard)

Cost of 96 feet of fencing at $3.25 per foot
= 96 x 3.25 = $312

Answer: One possible solution is to work 12.3 hours landscaping and 3.7 hours clearing tables, earning approximately $167.1.

Step-by-step explanation:

Let x be the number of hours landscaping and y be the number of hours clearing tables.

The system of inequalities can be written as:

x ≤ 13 (maximum of 13 hours landscaping)

x + y ≤ 16 (maximum of 16 hours total)

13x + 8y ≥ 160 (minimum earnings of $160)

To graph this system of inequalities, we can start by graphing the boundary lines of each inequality as follows:

x = 13 (vertical line at x = 13)

x + y = 16 (line with intercepts at (0, 16) and (16, 0))

13x + 8y = 160 (line with intercepts at (0, 20) and (12.3, 0))

Note that we only need to graph the portion of the lines that are in the feasible region (i.e. where x and y are non-negative).

The feasible region is the triangle formed by the intersection of the three boundary lines, as shown below:

    |

20 --|--------------------------

    |              /|

    |           /   |

    |        /      |

    |     /         |

16 --|------------------

    |  /              |

    |/                 |

    -------------------------

           13         12.3

Let x be the number of hours landscaping and y be the number of hours clearing tables.

The system of inequalities can be written as:

x ≤ 13 (maximum of 13 hours landscaping)

x + y ≤ 16 (maximum of 16 hours total)

13x + 8y ≥ 160 (minimum earnings of $160)

To graph this system of inequalities, we can start by graphing the boundary lines of each inequality as follows:

x = 13 (vertical line at x = 13)

x + y = 16 (line with intercepts at (0, 16) and (16, 0))

13x + 8y = 160 (line with intercepts at (0, 20) and (12.3, 0))

Note that we only need to graph the portion of the lines that are in the feasible region (i.e. where x and y are non-negative).

The feasible region is the triangle formed by the intersection of the three boundary lines, as shown below:

lua

Copy code

    |

20 --|--------------------------

    |              /|

    |           /   |

    |        /      |

    |     /         |

16 --|------------------

    |  /              |

    |/                 |

    -------------------------

           13         12.3

The vertices of the feasible region are (0, 16), (12.3, 3.7), and (13, 0).

To determine one possible solution, we can evaluate the objective function (total earnings) at each vertex:

(0, 16): 13(0) + 8(16) = $128

(12.3, 3.7): 13(12.3) + 8(3.7) ≈ $167.1

(13, 0): 13(13) + 8(0) = $169

Therefore, one possible solution is to work 12.3 hours landscaping and 3.7 hours clearing tables, earning approximately $167.1.

The expectation for a $1 bet on a U.S. roulette wheel for a 0 & 00 split is -$2. This means that you are expected to lose $2 for every $1 you bet.

The expected payout for a $1 bet on the 0 & 00 split in a U.S. roulette wheel is $0.45.

On a U.S. roulette wheel, there are 38 possible outcomes, consisting of the numbers 1 through 36, 0, and 00.

The 0 and 00 numbers are green, while all other numbers are either red or black.

Placing a $1 bet on the 0 & 00 split means that the bettor is betting that either the 0 or the 00 number will hit. If either of these numbers hits, the payout is 17 to 1, meaning the bettor will receive $17 for every $1 bet.

To calculate the expected payout for a $1 bet on the 0 & 00 split, we multiply the probability of winning (1/38) by the payout ($17), resulting in an expected payout of $0.447.

Therefore, the expected payout for a $1 bet on the 0 & 00 split is:

(1/38) × 17 × $1 = $0.447

Rounding this to the nearest cent, the expected payout for a $1 bet on the 0 & 00 split is $0.45.

Hence, the answer is $0.45.

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Answer:

D) he did not multiply -3 with -2 to get +6 instead of -6

Step-by-step explanation:

-3(x - 2)

-3x + 6

Compare the two

-3x + 6 ≠ -3x - 6

They are not equal! Looking at the options, D is the answer.

Answer:

The correct answer is D) he did not multiply -3 with -2 to get +6 instead of -6.

To expand -3(x-2), we need to distribute -3 to both terms inside the parentheses, which gives us:

-3(x-2) = -3x + (-3(-2)) = -3x + 6

Therefore, the correct equivalent expression is -3x + 6, not -3x - 6 as Rob wrote. He made a mistake in multiplying -3 and -2, which should have given him +6 instead of -6.

(please mark my answer as brainliest)

Answer:

$109.99

Step-by-step explanation:

The original price of each tire is [tex]\[/tex][tex]109.99[/tex]

Take the amount saved and divide by 4 to find the amount saved on each tire

[tex]96.16\div4 =24.04[/tex]

Add that to the sale price of each tire to find the original price

[tex]85.95+24.04 =109.99[/tex]

Therefore, The original price is $109.99.

A) The value of  x = 45 degrees

B) Lines AD and BC are not parallel when ABCD is drawn to scale.

To solve this problem, we can use the fact that the sum of the angles in a quadrilateral is 360 degrees.

A) angle A + angle B + angle C + angle D = 360

2x + 90 + x + 3x = 360

6x + 90 = 360

6x = 270

x = 45

Therefore, x = 45 degrees.

B) To determine if lines AD and BC are parallel, we can look at the opposite angles of the quadrilateral. If they are supplementary (add up to 180 degrees), then the lines are parallel.

angle A + angle C = 2x + x = 3x = 135 degrees

angle B + angle D = 90 + 3x = 90 + 135 = 225 degrees

Since angle A + angle C and angle B + angle D do not add up to 180 degrees, the opposite angles are not supplementary, and therefore, lines AD and BC are not parallel.

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The given question is incomplete, the complete question is:

ABCD is a quadrilateral A) Calculate the value of x. B) When ABCD is drawn to scale, would the lines AD and BC be parallel or not?

Geographic area D has an area of /8 square units.

Area is a phrase used to describe how much room a 2D form or surface occupies. Area is measured in square units, such as cm2 or m2. The area of a form is computed by dividing its length by its breadth.

Area is the entire amount of space occupied by a plain (2-D) area or an object's form. The area of a planar figure is the area that its perimeter encloses. The quantity of unit square that cover a closed figure's surface is its area.

0 = √θ

θ = 0

Similarly, the curve intersects the [tex]x-axis[/tex]  again when [tex]x = r cos[/tex] θ = √θ cos θ = 0, which implies θ = π/2.

Thus, the limits of integration for θ are [tex]0[/tex] and π/2. For r, the curve is given by [tex]r =[/tex] √θ, so the limits of integration are [tex]0[/tex] and √(π/2).

Evaluating this integral gives:

∫(θ=0 to π/2) [[tex]1/2 r^2[/tex]] (r=0 to √(π/2)) dθ

= ∫(θ=0 to π/2) [[tex]1/2[/tex] (π/2)] dθ

= π/8

Therefore, the area of region D is π/8 square units.

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The triangles within kite with corresponding angles and sides are similar, so options A, B, D, and E are true statements while options C and F are wrong.

Similar triangles are two triangles that have the same shape, but not necessarily the same size. This means that corresponding angles of the two triangles are equal, and corresponding sides are in proportion.

The line FH bisects the kite so:

triangle ∆FGK is similar to ∆FJK, ∆FGK ≅ ∆FJK

angle m∠GKH corresponds to m∠JKH, m∠GKH ≅ m∠JKH

angle m∠GFH corresponds to m∠JFH, m∠GFH ≅ m∠JFH

triangle ∆GKH is similar to ∆JKH, ∆GKH ≅ ∆JKH.

The line FG is not shown to be same with KG, likewise lines FG and JK, so lines FG ≅ KG and

FG ≅ JK are not true statements.

Therefore, options A, B, D, and E are true statements for the triangles within the kite with corresponding angles and sides that are similar, while options C and F are wrong.

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According to the given information value of X =16.06

The volume of a cone defines the space or the capacity of the cone. A cone is a three-dimensional geometric shape having a circular base that tapers from a flat base to a point called apex or vertex. A cone is formed by a set of line segments, half-lines or lines connecting a common point, the apex, to all the points on a base that is in a plane that does not contain the apex.

A cone can be seen as a set of non-congruent circular disks that are stacked on one another such that the ratio of the radius of adjacent disks remains constant.

Cone volume formula

A cone is a solid that has a circular base and a single vertex. To calculate its volume, you need to multiply the base area (area of a circle: π × r²) by height and by 1/3:

volume = (1/3) × π × r² × h

5677 = (1/3) x 22/7 x r² x 21

x = 16.06

Therefore, according to the given information value of X =16.06

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The two-period moving average model and the two-period weighted moving average model are both common forecasting methods used to predict future demand. and we understand that the model with the lower MAD and MSE values will have the most accurate forecast.

To determine which model is better for this particular data set, we need to compare the accuracy of each model. To do this, we will calculate the Mean Absolute Deviation (MAD) and the Mean Squared Error (MSE) for each model.
For the two-period moving average model, we can calculate the forecast for period 7 by taking the average of p5 and 6:

Period 7 forecast = (Gallons in Period 5 + Gallons in Period 6)/2

For the two-period weighted moving average model, we can calculate the forecast for period 7 by using the weights of 0.6 and 0.4:

Period 7 forecast = (0.6 x Gallons in Period 5) + (0.4 x Gallons in Period 6)

We can then compare the accuracy of each model by calculating the MAD and MSE. To calculate MAD, we need to subtract the actual demand in each period from the forecasted demand and take the absolute value:

MAD = |Actual demand – Forecasted demand|

To calculate MSE, we need to square the differences between the actual demand and the forecasted demand:

MSE = (Actual demand – Forecasted demand)^2

After calculating the MAD and MSE for each model, we can compare the results to determine which model is better for this data set. The model with the lower MAD and MSE values will have the most accurate forecast.

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The cost of the notebook is $4.50

Let's call the cost of the necklace "N" and the cost of the scarf "S". We know that Elise spent a total of $184 on the scarf, necklace, and notebook, so we can write:

S + N + NB = 184

We also know that after making her purchases, Elise had $89.50 left, so we can write:

89.5 = 184 - (S + N + NB)

Simplifying this equation, we get:

S + N + NB = 94.5

We're also given that the scarf cost three-fifths the cost of the necklace, so we can write

S = (3/5)N

Finally, we're told that the notebook cost one-sixth as much as the scarf, so we can write:

NB = (1/6)S

Now we can the substitution method here

(3/5)N + N + (1/6)(3/5)N = 184 - 89.5

Simplifying the right-hand side

(3/5)N + N + (1/6)(3/5)N = 94.5

Combining like terms on the left-hand side

(21/10)N = 94.5

Multiplying both sides by 10/21

N = 45

So the necklace cost $45. We can use this to find the cost of the scarf

S = (3/5)N = (3/5)($45) = $27

And we can use the cost of the scarf to find the cost of the notebook

NB = (1/6)S = (1/6)($27) = $4.50

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m∠R is therefore 76 degrees as a triangle's total sides equal 180 degrees.

The degree of rotation between two lines or two planes around a central point is measured by an angle. Typically, it is expressed in radians or degrees. Angles are used in many mathematical and scientific uses, including trigonometry, physics, and engineering, where they are crucial in determining the shape and characteristics of geometric figures. Angles come in four different varieties: acute (less than 90 degrees), right (exactly 90 degrees), oblique (more than 90 degrees), and straight (exactly 180 degrees).

given

Because a triangle's total sides equal 180 degrees, we have:

R, S, and T add up to 180.

Inputting the numbers provided yields:

m∠R + 72 + 32 = 180

Simplifying the equation:

m∠R = 76

m∠R is therefore 76 degrees as a triangle's total sides equal 180 degrees.

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The points scored by the remaining players on the team is 31 points.

A component or fraction of a whole can be expressed as a percentage by using a number out of 100. Mathematicians frequently use comparison to represent changes in quantities over time or to compare two numbers. For instance, if a student receives an 80 percent on a test, they properly answered 80 of the 100 questions.

For performing a range of mathematical operations, such as determining the percentage rise or decrease of a quantity or computing interest rates on loans or investments, percentages can be expressed as fractions or decimals.

From the given percentages the total percent is:

Total percentage = 15% + 10% + 13% = 38%

Now, from 100 the remaining percentage is: 100% - 38% = 62%

For the remaining players the score is given as:

Remaining points = 50 * (62/100) = 31

Hence, the points scored by the remaining players on the team is 31 points.

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The equivalent set of first-order equations in matrix form is [tex]w' = Cw + [0,[/tex] [tex]5sin(3t)^{T}[/tex] and the initial value of the vector-valued solution is w(1) = [tex][3,4.5]^{T}[/tex].

The given second-order differential equation as a set of first-order equations, we can define a new variable v(t) to represent the derivative of u(t), i.e., v(t) = u'(t). Then, we can rewrite the second-order differential equation as two first-order differential equations:

[tex]u' = v[/tex]

[tex]v' = 5sin(3t) + 5u - 2v[/tex]

where u' and v' represent the derivatives of u and v with respect to t, respectively.

To write this first-order system in matrix form, we can define the vector-valued function w(t) = [u(t), v(t)]^T and write the system as:

[tex]w' = [v, 5sin(3t) + 5u - 2v]^T[/tex]

where the superscript T denotes the transpose of a vector or matrix.

Using this notation, the initial value of the solution vector is given by w(1) =[tex][u(1), v(1)]^T = [3, 4.5]^T.[/tex]

In matrix form, the first-order system can be written as:

[tex][ u' ] [ v ]\\[ ] = [ ]\\[ v' ] [ 5sin(3t) + 5u - 2v ][/tex]

or

[tex]w' =\\[ u' ]\\[ v' ]\\[ v ]\\[ 5sin(3t) + 5u - 2v ][/tex]

where

w =

[tex][ u ]\\[ v ][/tex]

and the coefficient matrix C is given by:

[tex]C =\\[ 0 1 ]\\[ 5 -2 ][/tex]

A differential equation is a mathematical equation that relates a function to its derivatives. It is a powerful tool used in many fields of science and engineering to describe how physical systems change over time. The equation typically includes the independent variable (such as time) and one or more derivatives of the dependent variable (such as position, velocity, or temperature).

Differential equations can be classified based on their order, which refers to the highest derivative present in the equation, and their linearity, which determines whether the equation is a linear combination of the dependent variable and its derivatives. Solving a differential equation involves finding a function that satisfies the equation. This can be done analytically or numerically, depending on the complexity of the equation and the available tools.

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The probability that both events occur simultaneously is 0.1913.

We can model the waiting time for the first claim from a good driver and the waiting time for the first claim from a bad driver as independent exponential distributions with mean 6 years and 3 years, respectively.

Let X be the waiting time for the first claim from a good driver and Y be the waiting time for the first claim from a bad driver.

Then, P(X ≤ 3) = 1 - e^(-3/6) = 0.3935 and P(Y ≤ 2) = 1 - e^(-2/3) = 0.4866.

Since X and Y are independent, the probability that the first claim from a good driver will be filed within 3 years and the first claim from a bad driver will be filed within 2 years is

(X ≤ 3) * P(Y ≤ 2) = 0.3935 * 0.4866 = 0.1913(rounded to four decimal places).

(rounded to four decimal places).

Therefore, the probability that both events occur simultaneously is 0.1913.

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There ratio are [tex]28[/tex] starfish inside an aquarium, with a dolphin to pufferfish ratio of 1.

The ratio may be defined as the number of ways you can represent one item as a percentage of another. Only when the two quantities in a ratio share the same unit can they be compared. Ratios are employed to compare two items.

Ratios contrast two figures by typically dividing them. A/B would be your formula if you were comparing each data point (A) to that other data point (B). This indicates that you are multiplying info A by info B. If A is five and B is ten, for instance, your ratio will indeed be 5/10.

Dolphins to pufferfish to starfish: [tex]2:3:7[/tex]

So for every [tex]2[/tex] dolphins, there are [tex]7[/tex] starfish. We know there are 8 dolphins, so we can set up a proportion:

[tex]2[/tex] dolphins / [tex]7[/tex] starfish = 8 dolphins / x starfish

where [tex]x[/tex] is the number of starfish.

Solving for x, we get:

[tex]x[/tex] starfish = (7 starfish * [tex]8[/tex] dolphins) / [tex]2[/tex] dolphins

[tex]x[/tex] starfish = [tex]28[/tex] starfish

Therefore, there are [tex]28[/tex] starfish in the aquarium.

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The angle between the two planes is approximately 64.63°.

The two planes x + 4 y − 3 z = 1 and − 3 x + 6 y + 7 z = 0 are neither parallel nor perpendicular. The angle between them can be determined by the following formula:

[tex]\theta = \cos^{-1}\left(\frac{A\cdot B}{\left\lvert A\right\rvert\left\lvert B\right\rvert}\right)[/tex]
where A and B are the normal vectors of the two planes, and \theta is the angle between them.

A normal vector for the plane x + 4 y − 3 z = 1 can be calculated as:

A = \begin{bmatrix} 1 \\ 4 \\ -3 \end{bmatrix}

A normal vector for the plane − 3 x + 6 y + 7 z = 0 can be calculated as:

B = \begin{bmatrix} -3 \\ 6 \\ 7 \end{bmatrix}

The angle between the two planes can be found by plugging the vectors into the formula above:

[tex]\theta

= \cos^{-1}\left(\frac{A\cdot B}{\left\lvert A\right\rvert\left\lvert B\right\rvert}\right)

= \cos^{-1}\left(\frac{(-3)\cdot(-3) + 6 \cdot 4 + 7 \cdot (-3)}{\sqrt{1 + 16 + 9}\sqrt{9 + 36 + 49}}\right)

= \cos^{-1}\left(\frac{-37}{\sqrt{26}\sqrt{94}}\right)\approx 64.63^\circ[/tex]

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Answer:

the cross is the blue color