Let (Y1, Y2) denote the coordinates of a point chosen at random inside a unit circle whose center is at the origin. That is, Y1 and Y2 have a joint density function given by
f (y1, y2) = 1/ π , y^2 1 + y^2 2 ≤ 1, 0, elsewhere.
Find P(Y1 ≤ Y2).
Solve using polar coordinates.

a) For the ratio 8 : 5 : 7, if Maurice has 960 stickers then Elle collects 600 stickers and Justin collects 840 stickers.

b) For the ratio 8 : 5 : 7, if the total number of stickers is 1500 then Maurice collects 600 stickers, Elle collects 375 stickers and Justin collects 525 stickers.

What is ratio?

Comparing two amounts of the same units and determining the ratio tells us how much of one quantity is in the other. Two categories can be used to categorise ratios. Part to whole ratio is one, and part to part ratio is the other. The part-to-part ratio shows the relationship between two separate entities or groupings.

a) The ratio of stickers Maurice collects = 8

The number of stickers Maurice collects = 960

The ratio of stickers Elle collects = 5

The ratio of stickers Justin collects = 7

Let the number of stickers be x.

So, the equation for Maurice's collection will be -

8x = 960

x = 960/8

x = 120

So, Elle collects 5x stickers which is -

5 × 120 = 600 (substitute the value of x)

Justin collects 7x stickers which is -

7 × 120 = 840 (substitute the value of x)

Therefore, Elle and Justin have 600 and 840 stickers.

b) The total number of stickers is = 1500.

The ratio of stickers Maurice collects = 8x

The ratio of stickers Elle collects = 5x

The ratio of stickers Justin collects = 7x

The equation formed is -

8x + 5x + 7x = 1500

20x = 1500

x = 1500 / 20

x = 75

Substitute the value of x to find number of stickers.

So, Maurice collects = 8 × 75 = 600 stickers

So, Elle collects = 5 × 75 = 375 stickers

So, Justin collects = 7 × 75 = 525 stickers

Therefore, the number of stickers  Maurice, Elle and Justin have is 600, 375 and 525 respectively.

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The decision to rent or purchase a home for the Bainters will depend on several factors, including their financial situation, personal preferences, and long-term plans.

If the Bainters only plan to live in the house for 1 year, it may make more sense for them to rent. Renting allows for greater flexibility and mobility, as well as reduced responsibility for maintenance and repairs. The upfront costs of purchasing a home, such as closing costs and a down payment, may not be worth it for such a short period of time. Additionally, the monthly costs of renting may be lower than the monthly mortgage payments and related expenses for owning a home.

If the Bainters plan to live in the house for 5 years, it may make more sense for them to purchase a home. Over a 5-year period, the costs of renting can add up, and may end up being higher than the cost of owning. Additionally, a portion of a homeowner's monthly mortgage payment can go towards building equity in the property, which can be beneficial if the home's value appreciates over time.

If the Bainters plan to live in the house for 10 years or more, purchasing a home may be the more cost-effective option. Over a longer period of time, the costs of renting can significantly exceed the costs of owning, as renters do not build equity in the property. Additionally, if the Bainters plan to live in the same location for a long time, purchasing a home can provide stability and a sense of community.

To support their decision, the Bainters should calculate the total costs of renting and owning over each respective time period, taking into consideration all expenses, such as rent or mortgage payments, insurance, taxes, maintenance, and repairs. They should also consider factors such as their ability to afford a down payment, their credit score, and their employment stability. It may also be helpful for the Bainters to consult with a financial advisor or real estate professional.

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Let the width be [tex]w[/tex] inches and the length be [tex]w+15[/tex] inches.

So, the dimensions of the box are [tex]w-6[/tex], [tex]w+9[/tex], and [tex]3[/tex] inches.

[tex]3(w-6)(w+9)=1350 \\ \\ (w-6)(w+9)=450 \\ \\ w^2+3w-54=450 \\ \\ w^2+3w-504=0 \\ \\ (w+24)(w-21)=0 \\ \\ w=-24,21[/tex]

Rejecting the negative case, [tex]w=21[/tex].

So, the dimensions are 21 by 36 inches.

Answer: Let's call the number of species N. The total length of the beetles walking up the gangplank will be N * (1 cm + 7 cm) = N * 8 cm.

The total time it will take for all the beetles to walk up the gangplank is the total length of the beetles divided by the walking speed of each beetle, which is (N * 8 cm) / (10 cm/s) = 0.8 * N seconds.

Finally, to convert this to hours, we divide by 3600 seconds per hour: (0.8 * N) / 3600 s/hr = 0.0002 * N hours.

So, for 1/4 million species, N = 1/4 * 10^6, it will take 0.0002 * N = 0.0002 * 1/4 * 10^6 = 25 hours.

Step-by-step explanation:

Based on the information, it can be inferred that the total number of students surveyed is: 2,498.

To find the number of students surveyed we can perform the following operation.

1. Add all the results of each category

854 + 644 + 199 + 407 + 109 + 108 + 84 + 93 = 2,498

According to the above, the total number of students surveyed would be 2,498. Additionally, this information can be classified in a Venn diagram to identify students who belong to two or three categories at the same time.

Note: This question is incomplete because there is some information missing. Here is the complete information

tasks

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Answer:

A

Step-by-step explanation:

As you should know, in a log function f(x) = a * log[k(x-d)] + c, the variable d would be your asymptote. Since the a value of the function is 1, nothing has been done to the graph and thus we can assume that it goes in the same direction of the parent function f(x) = log x. Therefore, as the x value gets closer to -3, the y value will continue to negative infinity.

The difference between the maximum and the minimum values of the trigonometric function is 6.

We have to determine the difference between the maximum and the minimum values of the trigonometric function.

The function is y = -3cos(2x) + 8

The maximum and the minimum value of cos can be defined at the value of 0 and 1.

As we know that, the value of cos0 = 1 and the value of cos90 = 0

Now the value of the function at x = 0

y = -3cos(2×0) + 8

y = -3cos(0) + 8

y = -3×1 + 8

y = -3 + 8

y = 5

Now the value of the function at x = 90

y = -3cos(2×90) + 8

y = -3(-1) + 8

y = 3 + 8

y = 11

The difference between the maximum and the minimum values of the trigonometric function = 11 - 5

The difference between the maximum and the minimum values of the trigonometric function = 6

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For the following triangle Joseph is incorrect and for rectangle Chad is incorrect.

Transform the shapes on a coordinate plane by rotating, reflecting, or translating them. In geometry, object transformations constitute the foundation for the majority of proofs. The transformation, or f: X X, is the name given to a function, f, that maps to itself. After the transformation, the pre-image X becomes the picture X. Any operation, or a combination of operations, such as translation, rotation, reflection, and dilation, can be used in this transformation.

The coordinate of X is (x , y ) is (-1, 7), hence the value of X' when the coordinates are transformed would be:

(x,y) → (y, -x)

(-1, 7) → (7, -1)

Hence, Tara is correct in the first question.

The transformation corresponding to (x,y) → (-x, -y) is rotation of 180 degrees, Hence, Darla is correct.

Hence, for the following questions Joseph and Chad are incorrect.

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Answer:

Equation: 55D+40F+25L

Solution:$1595

Step-by-step explanation:

Equation: 55D+40F+25L

                55(6)+40(11)+25(33)

                330+440+825

Solution: $1595

Answer:

156

Step-by-step explanation:

[tex]\frac{1}{3}[/tex] of 36 = 36 ÷ 3 = 12

difference of 15 and 2 = 15 - 2 = 13

then

12 × 13 = 156

In word problem , the Total bill including tax = $34.78+$2.08 = $36.86

and  you have to give server one $20 bill , one $10 bill, one $5 bill and two $1 bill, so yo don't get any pennies back.

What is word problem?

Word problems are often described verbally as instances where a problem exists and one or more questions are posed, the solutions to which can be found by applying mathematical operations to the numerical information provided in the problem statement. Determining whether two provided statements are equal with respect to a collection of rewritings is known as a word problem in computational mathematics.

Here the bill = $34.78

Sales tax= 6% of Bill

=> Sales tax = 6% of 34.78

=> Sales tax = [tex]\frac{6}{100}\times 34.78[/tex]

=> Sales tax = $2.08

Now the Total bill including tax = $34.78+$2.08 = $36.86

Hence you have to give server one $20 bill , one $10 bill, one $5 bill and two $1 bill, so yo don't get any pennies back.

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The slope of the tangent will be 1280 and the equation of the tangent is y = 1280x - 3846.

A coordinate plane is a 2D plane that is formed by the intersection of two perpendicular lines known as the x-axis and y-axis. A coordinate system in geometry is a method for determining the positions of the points by using one or more numbers or coordinates.

Given that the equation is,

f(x) = 5x⁴ - 6

y = 5 x (4)⁴ - 6 = 1274

The slope of the tangent will be calculated as:-

f'(x) = 20x³ = 20 x ( 4 )³ = 1280

The equation of the line is,

y = mx + c

y = 1280x + c

1274 = 1280(4) + c

c = -3846

The equation is written as:-

y = 1280x - 3846

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Answer:

4/3 miles

Step-by-step explanation:

First, let's turn the statement given to us into an equation:

1/4 of the distance between Cameron's house and the park is 1/3 miles

1/4 * x = 1/3 mile

Now, we need to solve for x.

[tex]\frac{1}{4}x=\frac{1}{3}[/tex]   [Multiply by 4 to get x by itself]

x = 4/3.

So, if x is the distance between Cameron's house and the park and x=4/3, Cameron's house must be 4/3 of a mile away from the park.

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(i) The average number of machinists being served or waiting to be served at any given time can be calculated as the average of the arrival rate and the service rate, which is (2 + 3) / 2 = 2.5.

What is an example of a unitary method?

We can use the unitary approach to calculate 100% of an amount. For instance, if 10% of the marbles in a pot are 35, then the total number of marbles in the pot will be 35 x 10%, or 350.

(ii) The average time a machinist spends waiting for service can be calculated as the difference between the average time between arrivals and the average service time, which is 1 / (2 - 3) = -0.5 hours. However, this result is not physically meaningful as service rate is higher than the arrival rate, which means that all customers can be served without waiting.

(iii) The total cost of operating the system for an eight-hour day can be calculated as the sum of the attendant's wage and the cost of the machinist away from his work, which is 8 * 21.50 + 8 * 4 = 172.

(iv) The cost of the system if there were two attendants working together as a team can be calculated as the sum of the wage of the two attendants and the cost of the machinist away from his work, which is 8 * (2 * 11.50) + 8 * 4 = 216.

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The value of an F'(1) is 114.

If F(x) = f(x f(x f(x))),

where f(1) = 4, f(4) = 6, f '(1) = 4, f '(4) = 5, and f '(6) = 6.

We will differentiate F and wade our way slowly and methodically through finding the derivative of the right. First with the outermost function and chain rule:

[tex]$$F^{\prime}(x)=f^{\prime}(x f(x f(x)))\+x\left(\frac{d}{d x} f(x f(x))\right)\right]$$[/tex]

Now applying the product rule:

[tex]$$F^{\prime}(x)=f^{\prime}(x f(x f(x)))\left[f(x f(x))+x\left(\frac{d}{d x} f(x f(x))\right)\right]$$[/tex]

Reapplying the chain rule:

[tex]$$\begin{aligned}& F^{\prime}(x)=f^{\prime}(x f(x f(x)))\left[f(x f(x))+x\left(f^{\prime}(x f(x))+\left(\frac{d}{d x} x f(x)\right)\right)\right] \\& F^{\prime}(x)=f^{\prime}(x f(x f(x)))\left[f(x f(x))+x f^{\prime}(x f(x))+x\left(\frac{d}{d x} x f(x)\right)\right]\end{aligned}$$[/tex]

Product rule once more:

[tex]$$F^{\prime}(x)=f^{\prime}(x f(x f(x)))\left[f(x f(x))+x f^{\prime}(x f(x))+x\left(f(x)+x f^{\prime}(x)\right)\right]$$[/tex]

We could simplify this a little more.

Evaluating at x=1 :

[tex]$$& F^{\prime}(1)=f^{\prime}(f(f(1)))\left[f(f(1))+f^{\prime}(f(1))+1\left(f(1)+f^{\prime}(1)\right)\right] \\$$[/tex]

[tex]F'(1)=f'(f(4))) [f(4))+f'(4)+1(4+4))]\\[/tex]

[tex]F'(1)=f'(6)[6+5+8]\\[/tex]

[tex]F'(1)=6[19]=114[/tex]

[tex]F'(1)=114[/tex]

Therefore, the value of F'(1)=114.

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The mode of the data set of temperatures in various cities in San Diego County is 78°.

Mode is a measure of central tendency that represents the value that occurs most frequently in a data set. It gives an idea about which value is the most likely to appear in the data set. In a set of data, there can be one mode, more than one mode (referred to as bimodal or multimodal), or no mode at all (referred to as being non-modal).

In this case, the number 78 appears twice, while all the others appear only once.

Therefore, the mode is 78.

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[tex]f(x) = 2x is linear.f(x) = 4^x is exponential.[/tex] This is where at each level, each branch extends into 4 branches.

In the first prompt, the number of boxes in each row is increasing by a constant amount, so the function is linear. In the second prompt, the number of branches is increasing by a factor of 4 with each level, so the function is exponential.

A linear function is defined as one where the output increases by a constant amount for each increase in the input. An exponential function is defined as one where the output increases by a constant factor for each increase in the input. The first prompt corresponds to a linear function, while the second prompt corresponds to an exponential function.

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The complete question is-

Is the function f(x) = 4^x, where x is the number of branches at level x in a tree diagram, linear or exponential? Explain your answer.

The p-value of 0.0761 indicates that there is not sufficient evidence to reject the null hypothesis, which states that the survival rates for the two treatments are equal.

Therefore, based on this study, the use of CC alone does not produce a significantly higher survival rate than the use of both CC and standard MMR.

The test of significance used a significance level of α = 0.05. This means that in order for the null hypothesis to be rejected, the p-value must be less than 0.05. However, the p-value in this study is 0.0761, which is greater than 0.05.

Therefore, there is not sufficient evidence to reject the null hypothesis and we must accept it, meaning that the survival rates for the two treatments are not significantly different. In this case, a Type I error could have been made, where the null hypothesis is rejected when it should have been accepted.

The consequence of this would be that the treatment using CC alone would have been incorrectly identified as providing significantly higher survival rates, leading to a potential misallocation of resources.

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One possible solution from the graph(attached at the end of the solution) is selling 9 bags of popcorn and 5 bags of candies.

As per the given data:

The number of popcorn bags is x

The number of candies is y

The cost of each popcorn bag is $7

The cost of each candy is $4

She has $80 to spend on them

Inequality is the nonequal comparison of two or more variables and numbers.

So the first inequality is

7 x + 4 y ≤ 80

Since she must buy no less than 16 bags of popcorn and candies

The second inequality is

x + y ≥ 16

The red area represents the 1st inequality

The blue area represents the 2nd inequality

The solutions lie in the common part of red and blue colors

One possible solution is ordered pair (9,5)

(Graph attached at the end of solution)

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The statement that is not​ true is option B. A compound linear inequality involving the word​ "and" must always have a solution. In other​ words, the solution is never the empty set.

Option B is not true because a compound linear inequality involving the word "and" does not necessarily have a solution. The solution to a compound linear inequality involving the word "and" is the intersection of the solutions to each inequality.

This means that the solution is the set of values that satisfies both inequalities at the same time. However, if the solution to each inequality does not have any common values, then the solution to the compound inequality will be the empty set, which means there are no values that satisfy both inequalities.

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Answer:

59

Step-by-step explanation:

writing down a problem is a problem halved

472/8

The answer is 59

The actual distance between the two cities is 70 kilometers.

The scale factor states the scale or measurement by which a figure is bigger or smaller than the other figure. The size by which the figure would be reduced or enlarged is called its scale factor.

Given that;

The distance between 2 cities is 3.5 cemtimetes

the map uses a scale in which 1 centimeter represents 20 kilometers

Since, One centimeter represents 20 kilometers

3.5 x 20 = 70

Theefore, the actual distance between the two cities;

70 kilometers.

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Thelma has to save by at least 4.8 weeks, according to the information we get. The proper course of action is B).

According to economists, there are four basic sorts of money: commercial money, fiduciary funds, fiat money, and commodity money. Commodity money is money that derives its worth from the commodities from which it is created.

B. 12 + 10w ≥ 60

10w ≥ 48

w ≥ 4.8

Thelma must so set aside money over the past 4.8 weeks. Keep in mind this is the absolute minimum, so if she not save the entire $10 each week, she may require saving for longer.

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The Complete Question :

Thelma wants to buy a jacket that costs $60. She has $12 in her

savings account and plans to save an additional $10 per week. Which

of these shows a correct procedure Thelma could use to determine w,

the number of weeks she needs to save to have enough money to buy

the jacket?

A. 12+10w≥60

12+wz50

w≤4.17

B.  12 + 10w ≥ 60

10w ≥ 48

w ≥ 4.8

C. 12+10w≥60

12+w≥50

w24.17

D. 12+10w≥60

10wz48

w≤4.8

Answer:

Each cup cost 6.5 cents or 6 1/2 cents

Step-by-step explanation:

13/200- .065

Answer:

15 cents a cup

Step-by-step explanation:

divide 13.00 by 200 and round (*ゝω･*)

Therefore , the solution of the given problem of percentage comes out to be for the tax to be $13154.00, the income has to be around $84991.67 (around because taxes are rounded).

Any quantity that is expressed as a ratio of 100 is referred to as a percentage in mathematics. There are also sporadic uses of the acronyms "pct.," "pct," and "pc." But it is commonly indicated by the "%" symbol. The percentage amount is flat with no dimensions. Percentages actually function as fractions because their numerator is 100. The percent symbol (%) should be used in front of a number to indicate that it is a percentage. If you correctly answered 75 out of total test questions, for example, you would score 75%.

Here,

Given :

The required range for income is $84200 to $160700 because the tax is over $12962 but under $31322.

The tax on the first $84202 is $12962, and the surplus, which we'll refer to as x, is subject to a 24% tax.

=> 24%*x = $13154 - $12962 = $190

=> x = $190/(24%) = $791.(6) (6)

Therefore, the total income is approximately $84,200 plus $791.

=> $84991.

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The required step is adding 14 both sides of the equation to make it a whole square.

A quadratic equation can be written in the standard form as ax² + bx + c = 0, where a, b, c are constants and x is the variable. The values of x that satisfy the equation are called solutions of the equation, and a quadratic equation has at most two solutions.

Given here: The expression as x²-27x+31=-13x-14

Evaluating the expression we get

x²-27x+31=-13x-14

Adding both sides by 13x+14 we get

x²-27x+31+13x+14=-13x-14+13x+14

x²-14x+35=0

Again adding 14 both sides we get

x²-14x+35+14=14

(x-7)²=14

x-7=√14

x=√14+7

Hence, The required step is adding 14 both sides of the equation to make it a whole square.

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a. T(n) = O(n^4) and Ω(n^4): For the upper bound, we can use the substitution method to prove that T(n) is O(n^4). Let S(n) = cn^4, where c is a constant. We can then substitute S(n) into the recurrence to prove that T(n) is O(n^4):

T(n) = 2T(n/2) + n^4 <= 2S(n/2) + n^4 <= 2cn^4/2 + n^4 <= cn^4 + n^4 = (c + 1)n^4

Since (c + 1) is a constant, this expression is equal to O(n^4).

For the lower bound, we can use the recurrence tree method to prove that T(n) is Ω(n^4). Let k be the number of levels in the tree. We can then calculate the total cost of the tree:

Total Cost = 2^k * n^4 = n^4 * (2^k/n^4) = n^4 * (2/n)^k

Since 2/n is a constant and k is a function of n, this expression is equal to Ω(n^4).

b. For the upper bound, we can use the substitution method to prove that T(n) is O(n log n). Let S(n) = cn log n, where c is a constant. We can then substitute S(n) into the recurrence to prove that T(n) is O(n log n):

T(n) = T(7n/10) + n <= S(7n/10) + n <= c(7n/10) log (7n/10) + n = c(7/10)n log n + n = (c(7/10) + 1)n log n

Since (c(7/10) + 1) is a constant, this expression is equal to O(n log n).

For the lower bound, we can use the recurrence tree method to prove that T(n) is Ω(n). Let k be the number of levels in the tree. We can then calculate the total cost of the tree:

Total Cost = 7^k * n = n * (7^k/n) = n * (7/n)^k

Since 7/n is a constant and k is a function of n, this expression is equal to Ω(n).

c. For the upper bound, we can use the substitution method to prove that T(n) is O(n^2). Let S(n) = cn^2, where c is a constant. We can then substitute S(n) into the recurrence to prove that T(n) is O(n^2):

T(n) = 16T(n/4) + n^2 <= 16S(n/4) + n^2 <= 16cn^2/4 + n^2 <= 4cn^2 + n^2 = (4c + 1)n^2

Since (4c + 1) is a constant, this expression is equal to O(n^2).

For the lower bound, we can use the recurrence tree method to prove that T(n) is Ω(n^2). Let k be the number of levels in the tree. We can then calculate the total cost of the tree:

Total Cost = 16^k * n^2 = n^2 * (16^k/n^2) = n^2 * (16/n^2)^k

Since 16/n^2 is a constant and k is a function of n, this expression is equal to Ω(n^2).

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f(7) = -1.906 (rounded to the nearest hundredth.

To find f(7), we first need to substitute 7 for x in the expression for f(x). So, we have:

f(7) = (3/(7+2)) - √(7-2)

f(7) = (3/(9)) - √(5)

f(7) = (1/3) - 2.236

f(7) = 0.33 - 2.236

f(7) = -1.906

So, f(7) = -1.906 (rounded to the nearest hundredth).

An expression is a mathematical phrase that can contain numbers, variables, and operators. Expressions are used to represent values or computations in mathematics. For example, 2x + 3 is an expression that represents the sum of two times x and three. Expressions can be evaluated to obtain a numerical result by substituting numerical values for variables and performing the operations. For instance, if we substitute x=5 into the expression 2x + 3, we get 2(5) + 3 = 10 + 3 = 13.

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Answer:

here

Step-by-step explanation:

Call the two integers a and b . We are given that a+b=−3, and ab=−10. Simple plug and chug yields the answers −5 and 2.

Answer:

Step-by-step explanation:

x=0 and xyz=4 so that would equal 40 and 46.