9 + 22 = x + y

x = 5

y = ?

To find the value of n, we can use the concept of combinations. By setting up and solving an equation using the combination formula, we find that the value of n is 6.

To find the value of n, we can use the concept of combinations. Since Columba has 2 dozen (24) each of n different colored beads, the total number of beads she has is 24n. If she can select 20 beads with repetitions allowed in 230,230 ways, we can set up the equation:

24n choose 20 = 230,230

To solve this equation, we need to use the concept of combinations. The formula for combinations is given by nCr = n! / (r!(n-r)!), where n is the total number of items, r is the number of items being selected, and ! represents the factorial function.

Plugging in the values, we have:

24n! / (20!(24n-20)!) = 230,230

Simplifying the equation, we get:

n! / (20!(n-20)!) = 10

To find the value of n, we can try different values of n and calculate the factorial on both sides of the equation. Starting with n = 2, we have:

2! / (20!(2-20)!) = 1 / (20!(18)!) = 1 / (20!(18!)) = 1 / (20 * 19) = 1 / 380 = 0.00263

Since this value is smaller than 10, we need to try a larger value of n. By trying different values, we find that when n = 6, the equation holds:

6! / (20!(6-20)!) = 6! / (20!(14)!) = 720 / (20 * 19 * 18 * 17 * 16 * 15 * 14!) = 720 / (20 * 19 * 9 * 17 * 16 * 15) = 720 / 9909000 = 0.00007

Therefore, the value of n is 6.

Shown in the explanation

A Rhombus is a quadrilateral having four sides of equal length each. Here, we know that the vertices of this shape are:

[tex]W(7,2) \\ \\ X(5,-1) \\ \\ Y(3,2) \\ \\ Z(5,5)[/tex]

So the rhombus is named as WXYZ. To find its perimeter (P), we just need to find the length of one side and multiply that value by 4. By using the distance formula, we know that:

[tex]\overline{WX}=\sqrt{(x_{1}-x_{2})^2+(y_{1}-y_{2})^2} \\ \\ W(7,2)=W(x_{1},y_{1}) \\ \\ X(5,-1)=X(x_{2},y_{2}) \\ \\ \\ \overline{WX}=\sqrt{(7-5)^2+(2-(-1))^2} \\ \\  \overline{WX}=\sqrt{(2)^2+(3)^2} \\ \\ \overline{WX}=\sqrt{4+9} \\ \\ \overline{WX}=\sqrt{13}[/tex]

Finally, the Perimeter (P) is:

[tex]P=4(\sqrt{13}) \\ \\ \boxed{P=4\sqrt{13}\ units}[/tex]

Answer:

4 13

Step-by-step explanation:

Answer:

There were 49 people that chose chocolate and vanilla twist.

Step-by-step explanation:

This problem can be solved by building a Venn diagram of this set, where:

-A is the number of the people that tasted the vanilla

-B is the number of the people that tasted the chocolate.

The most important information in this problem is that some of those people tasted both. It means that [tex]A \cap B = x[/tex], and x is the value we want to find.

The problem states that 97 people tasted the vanilla sample of frozen yogurt.  This includes the people that tasted both samples. It means that x people tasted the chocolate and vanilla twist and 97-x people tasted only the vanilla twist.

72 people tasted the chocolate, also including the people that tasted both samples. It means that x people that tasted the chocolate and vanilla twist and 72-x that tasted only the chocolate twist.

So, recapitulating, there are 120 people, and

97-x  tasted only the vanilla twist.

72 - x tasted only the chocolate twist

x people tasted both

So

97 - x + 72 - x + x = 120

-x = 120 - 72 - 97

-x = -49 *(-1)

x = 49

There were 49 people that chose chocolate and vanilla twist.

To find out how many people chose the chocolate and vanilla twist, we need to subtract the number of people who tasted only vanilla and only chocolate from the total number of people who tasted the frozen yogurt.

To find out how many people chose the chocolate and vanilla twist, we need to subtract the number of people who tasted only vanilla and only chocolate from the total number of people who tasted the frozen yogurt. We know that 97 people tasted vanilla and 72 people tasted chocolate. However, some people chose the chocolate and vanilla twist, so we need to subtract the overlapping cases.

To calculate the number of people who chose the chocolate and vanilla twist, we can use the principle of inclusion-exclusion. We add the number of people who tasted only vanilla and the number of people who tasted only chocolate, and then subtract the total number of people who tasted the frozen yogurt.

Using the formula:

(# of people who tasted vanilla) + (# of people who tasted chocolate) - (# of people who tasted both) = Total # of people who tasted the frozen yogurt

97 + 72 - X = 120

X = 97 + 72 - 120

X = 169 - 120

X = 49

Therefore, 49 people chose the chocolate and vanilla twist.

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Answer:

P= -0.05q+22

Step-by-step explanation:

To find the linear equation that relates price with quantity demanded, first we must find the slope. Because the independent variable is the quantity demanded and the dependent variable is the price, the slope represents how the price changes when there is an extra unit of quantity demanded. The problem gives this information: "for every additional card (extra unit) they need to drop the price by $0.05". The slope (m) in this case is negative because an extra unit, reduces the price: -0.05

The second step is to use this formula:

Y-y1= m*(X-x1)

y1 and x1 is a point of the demand curve, in this case it is y1= $7 and x1=300

Y-$7= -$0.05*(X-300)

Y-7=-0.05X+15

Y= -0.05X+15+7

Y= -0.05X-22

Price= -0.05 quantity demanded +22

Answer: she must walk for 72.8 s

Hi!

Lets say that with the cart she rides a time T1 (28 s) for a distance D1, then the average speed in the cart is V1 = D1 / T1 =  3.10 m/s. We can calculate D1 = (28 s )* (3.10 m/s) = 86.8 m

She then walks a time T2 for a distance D2, with average speed

V2 = D2 / T2 = 1.30 m/s

For the entire trip, we have average speed:

V3 = (D1 + D2) / (T1 + T2) = 1.80 m/s

We can solve for T2:

(1.8 m/s) *( 28s + T2) = 86.8 m  +  D2 = 86.8 m + (1.3 ms) * T2

Doing the algebra we get: T2 = 72,8 m/s

This question involves an application of the concept of average speed. Knowing that the average speed for the entire trip was 1.80 m/s, we first determined the distance covered while riding the golf cart. Using this, we set up an equation that allowed us to solve for the time spent walking to maintain the given average speed.

In order to solve this problem, we'll have to apply the formula for average speed, which is total distance covered (d) divided by the total time (t) taken.

Firstly, let's determine the distance covered while riding the golf cart. The golfer rides at an average speed of 3.10 m/s for 28.0 s. Therefore, she covers a distance of (average speed)x(time) = (3.1 m/s)(28.0 s) = 86.8 m.

Let's denote the time she walks as 't2'. The total time of the trip equals the sum of the time spent in the cart and the time spent walking: 28.0 s + t2.

Similarly, the total distance covered equals distance covered with the cart plus distance covered walking, which is 86.8 m + 1.30 m/s * t2.

Given the average speed for the entire trip is 1.80 m/s, we can write:

1.80 m/s = (total distance) / (total time)

1.80 m/s = (86.8 m + 1.30 m/s * t2) / (28.0 s + t2).

This equation could be solved for t2 to calculate how long the golfer needs to walk.

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Answer:

The level curves F(t,z) = C for any constant C in the real numbers

where

[tex]F(t,z)=z^3t^2+e^{tz}-4t+2z[/tex]

Step-by-step explanation:

Let's call

[tex]M(t,z)=2tz^3+ze^{tz}-4[/tex]

[tex]N(t,z)=3t^2z^2+te^{tz}+2[/tex]

Then our differential equation can be written in the form

1) M(t,z)dt+N(t,z)dz = 0

To see that is an exact differential equation, we have to show that

2) [tex]\frac{\partial M}{\partial z}=\frac{\partial N}{\partial t}[/tex]

But

[tex]\frac{\partial M}{\partial z}=\frac{\partial (2tz^3+ze^{tz}-4)}{\partial z}=6tz^2+e^{tz}+zte^{tz}[/tex]

In this case we are considering t as a constant.

Similarly, now considering z as a constant, we obtain

[tex]\frac{\partial N}{\partial t}=\frac{\partial (3t^2z^2+te^{tz}+2)}{\partial t}=6tz^2+e^{tz}+zte^{tz}[/tex]

So, equation 2) holds and then, the differential equation 1) is exact.

Now, we know that there exists a function F(t,z) such that

3) [tex]\frac{\partial F}{\partial t}=M(t,z)[/tex]  

AND

4) [tex]\frac{\partial F}{\partial z}=N(t,z)[/tex]

We have then,

[tex]\frac{\partial F}{\partial t}=2tz^3+ze^{tz}-4[/tex]

Integrating on both sides

[tex]F(t,z)=\int (2tz^3+ze^{tz}-4)dt=2z^3\int tdt+z\int e^{tz}dt-4\int dt+g(z)[/tex]

where g(z) is a function that does not depend on t

so,

[tex]F(t,z)=\frac{2z^3t^2}{2}+z\frac{e^{tz}}{z}-4t+g(z)=z^3t^2+e^{tz}-4t+g(z)[/tex]

Taking the derivative of F with respect to z, we get

[tex]\frac{\partial F}{\partial z}=3z^2t^2+te^{tz}+g'(z)[/tex]

Using equation 4)

[tex]3z^2t^2+te^{tz}+g'(z)=3z^2t^2+te^{tz}+2[/tex]

Hence

[tex]g'(z)=2\Rightarrow g(z)=2z[/tex]

The function F(t,z) we were looking for is then

[tex]F(t,z)=z^3t^2+e^{tz}-4t+2z[/tex]

The level curves of this function F and not the function F itself (which is a surface in the space) represent  the solutions of the equation 1) given in an implicit form.

That is to say,

The solutions of equation 1) are the curves F(t,z) = C for any constant C in the real numbers.

Attached, there are represented several solutions (for c = 1, 5 and 10)

The diameter of 4.3 cm equals 1.677 inches and 43 millimeters. This is calculated by using the conversion factors of 0.39 for inches and 10 for millimeters.

To convert diameter from centimeters to inches and millimeters, we use the conversion factors that 1 cm equals 0.39 inches and 1 cm equals 10 millimeters.

First, let's convert into inches. Multiply the given diameter (4.3 cm) by the conversion factor (0.39). 4.3 cm * 0.39 = 1.677 inches.

Next, let's convert into millimeters. Multiply the given diameter (4.3 cm) by the conversion factor (10) for millimeters. 4.3 cm * 10 = 43 millimeters.

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The sum E (4 + 6 + 8 + 10 + 106) is greater than the sum O (-3 + 5 + 7 + 9 + 105) by 11. E equals 134 and O equals 123.

To compare the sums O and E without computing each sum directly, let's analyze each expression:

For O: -3 + 5 + 7 + 9 + 105
For E: 4 + 6 + 8 + 10 + 106

Group the pairs of numbers for simplicity:

Comparing the two:

Therefore, the sum E is greater than the sum O by 11.

Answer:

1/6

Step-by-step explanation:

Slope is [tex]\frac{\text{rise}}{\text{run}}=\frac{1}{6}[/tex].

Convert 55% to a decimal by moving the decimal point two places to the left:

55% = 0.55

Now multiply:

3650.00 x 0.55 = 2,007.50

Answer:

The EMV of the project is $25,500.

Step-by-step explanation:

The EMV of the project is the Expected Money Value of the Project.

This value is given by the sum of each expected earning multiplied by each probability

So, in our problem

[tex]EMV = P_{1} + P_{2} + P_{3}[/tex]

The problem states that the project has a 60% of super success earning $50,000. So

[tex]P_{1} = 0.6*50,000 = 30,000[/tex]

The project has a 15% chance of mediocre success earning $20,000. So

[tex]P_{2} = 0.15 * 20,000 = 3,000[/tex]

The project has a 25% probability of failure losing $30,000. So

[tex]P_{3} = 0.25*(-30,000) = -7,500[/tex]

[tex]EMV = P_{1} + P_{2} + P_{3} = 30,000 + 3,000 - 7,500 = 25,500[/tex]

The EMV of the project is $40,500.

Answer:

For a monthly cost of at least $7 and at most $8, you can have between 100 and 110 calling minutes.

Step-by-step explanation:

The problem states that the monthly cost of a celular plan is modeled by the following function:

[tex]C(x) = 4 + 0.10(x-70)[/tex]

In which C(x) is the monthly cost and x is the number of calling minutes.

How many calling minutes are needed for a monthly cost of at least $7?

This can be solved by the following inequality:

[tex]C(x) \geq 7[/tex]

[tex]4 + 0.10(x - 70) \geq 7[/tex]

[tex]4 + 0.10x - 7 \geq 7[/tex]

[tex]0.10x \geq 10[/tex]

[tex]x \geq \frac{10}{0.1}[/tex]

[tex]x \geq 100[/tex]

For a monthly cost of at least $7, you need to have at least 100 calling minutes.

How many calling minutes are needed for a monthly cost of at most 8:

[tex]C(x) \leq 8[/tex]

[tex]4 + 0.10(x - 70) \leq 8[/tex]

[tex]4 + 0.10x - 7 \leq 8[/tex]

[tex]0.10x \leq 11[/tex]

[tex]x \leq \frac{11}{0.1}[/tex]

[tex]x \leq 110[/tex]

For a monthly cost of at most $8, you need to have at most 110 calling minutes.

For a monthly cost of at least $7 and at most $8, you can have between 100 and 110 calling minutes.

Answer:

840

Step-by-step explanation:

Total number of gifts (teddy bears)= 4

Total number of nieces = 7

We need to find the number of ways to give the 4 teddy bears to 4 of the 7 nieces, where no niece gets more than one teddy bear.

Number of possible ways to give first teddy = 7

It is given that no niece gets more than one teddy bear.

The remaining nieces are = 7 - 1 = 6

Number of possible ways to give second teddy = 6

Now, the remaining nieces are = 6 - 1 = 5

Similarly,

Number of possible ways to give third teddy = 5

Number of possible ways to give fourth teddy = 4

Total number of possible ways to distribute 4 teddy bears is

[tex]Total=7\times 6\times 5\times 4=840[/tex]

Therefore total possible ways to distribute 4 teddy bears are 840.

Final answer:

There are 35 different ways to give 4 identical teddy bears to 4 of the 7 nieces where no niece receives more than one teddy bear. The calculation is done using combinations formula C(7, 4).

Explanation:

To determine the number of different ways the 4 teddy bears can be given to 4 out of 7 nieces where each niece gets only one teddy bear, we use combinations. Combinations are a way of selecting items from a group, where the order does not matter. In mathematics, this is denoted as C(n, k), which represents the number of combinations of n items taken k at a time.

In this case, we want to find C(7, 4), because we have 7 nieces (n=7) and we are choosing 4 of them (k=4) to each receive one teddy bear. This is calculated by:

C(7, 4) = 7! / (4! * (7-4)!) => C(7, 4) = (7 * 6 * 5 * 4!) / (4! * 3!). Since 4! in the numerator and denominator cancel each other out, it simplifies to:

C(7, 4) = 7 * 6 * 5 / (3 * 2 * 1) = 35

Therefore, there are 35 different ways to give the 4 identical teddy bears to 4 of the 7 nieces when no niece gets more than one teddy bear.

Answer:

Area: 180 units2 (units 2 is because since the are no specific unit given but every area should have a unit  of measurement)

Step-by-step explanation:

The area enclosed by the graph of the function, the horizontal axis, and vertical lines is the integral of the function between thos two points (x=2 and x=4)

So , let's solve the integral of f(x)

Area =[tex]\int\limits^2_4 3{x}^3 \, dx = 3*x^4/4[/tex]+C

C=0

So if we evaluate this function in the given segment:

Area= 3* (4^4)/4-3*(2^4)/4= 3*(4^4-2^4)/4=180 units 2

Goos luck!

Answer: 38760

Step-by-step explanation:

Given : The number of employees in the company = 20

The number of employees will be selected by company owner to give bonus = 6

We know that the combination of n things taking r at a time is given by :-

[tex]^nC_r=\dfrac{n!}{r!(n-r)!}[/tex]

Then, the number of different sets of employees could receive bonuses is given by :-

[tex]^{20}C_6=\dfrac{20!}{6!(20-6)!}\\\\=\dfrac{20\times29\times18\times17\times16\times15\times14!}{(720)14!}=38760[/tex]

Hence, the number of different sets of employees could receive bonuses is  38760.

Answer:

The set of solutions is [tex]\{\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}12\\-7-r\\r\end{array}\right]: \text{r is a real number}  \}[/tex]

Step-by-step explanation:

The augmented matrix of the system is [tex]\left[\begin{array}{ccccc}3&6&6&-9\\-2&-3&-3&3\end{array}\right][/tex].

We will use rows operations for find the echelon form of the matrix.

Now we solve for the unknown variables:

The system has a free variable, the the system has infinite solutions and the set of solutions is [tex]\{\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{c}12\\-7-r\\r\end{array}\right]: \text{r is a real number}  \}[/tex]

Linear pair makes a straight line.

A. are vertical angles.

B. are vertical angles

C. make a right angle

D. makes a straight line of TR

The answer is D.

Answer:  a)   2.27 g     and     b)  15384

Step-by-step explanation:

Given : An effervescent tablet has the following formula:

acetaminophen 325 mg,

calcium carbonate 280 mg,

citric acid 900 mg,

potassium bicarbonate 300 mg, and

sodium bicarbonate 465 mg.

a) When we add all quantities together , we get

The total weight of the ingredients in each tablet = [tex]325 +280+900+300+465=2270[/tex]

Since, 1 gram = 1000 mg

Then, [tex]1\ mg=\dfrac{1}{1000}\ g[/tex]

Now, [tex]2270\ mg=\dfrac{2270}{1000}\ g=2.27\ g[/tex]

∴ The total weight of the ingredients in each tablet = 2.27 g

b. 1 kg = 1000g and 1 g = 1000 mg

Then, 1 kg = [tex]1000\times1000=1000,000\ mg[/tex]

⇒ 5 kg = 5000,000 mg

Now, The number of  tablets could be made with a supply of 5 kg of acetaminophen will be :

[tex]\dfrac{5000000}{325}=15384.6153846\approx15384[/tex]

Hence, the number of  tablets could be made with a supply of 5 kg of acetaminophen= 15384

The total weight of the ingredients in the effervescent tablet is 2.27 g. With a supply of 5 kg of acetaminophen, you could produce approximately 15,385 tablets.

To answer the student's questions, we start by calculating the total weight of the tablet:
acetaminophen: 325 mg, calcium carbonate: 280 mg, citric acid: 900 mg, potassium bicarbonate: 300 mg, and sodium bicarbonate: 465 mg.
Adding all these quantities together gives a total of 2270 mg or 2.27 g per tablet.

Now for the second question, to find out how many tablets you can make from 5 kg of acetaminophen, we need to determine how much acetaminophen is in a single tablet. We know that each tablet contains 325 mg of acetaminophen, so if we have 5 kg of it, we first convert the 5 kg into milligrams (since the amount in each tablet is given in milligrams).
There are 1,000,000 milligrams in a kilogram, so 5 kg = 5 x 1,000,000 = 5,000,000 mg.
We then divide this total quantity by the amount of acetaminophen in each tablet: 5,000,000 mg / 325 mg/tablet = approximately 15,385 tablets.

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Answer:

you need 100ml of 5X TAE and 400ml of water.

Step-by-step explanation:

You need to use a rule of three:

[tex]C_1V_1=C_2V_2[/tex]

where:

[tex]\left \{ {{C_1= 5X} \atop {C_2=1X}} \right.[/tex]

and

[tex]\left \{ {{V_1 = V_{TAE}} \atop {V_2=500ml}} \right.[/tex]

Therefore:

[tex]V_{TAE} = \frac{V_2*C_2}{C_1}[/tex]

[tex]V_{TAE} = 100ml[/tex]

Then just rest the TAE volume to the final Volume and you get the amount of water that you need to reduce the concentration.

Answer:

Step-by-step explanation:

It shall be 100xl times the number of 1x tae

Answer:

The resultant velocity of the airplane is 213.41 m/s.

Step-by-step explanation:

Given that,

Velocity of an airplane in east direction, [tex]v_1=210\ mph[/tex]

Velocity of wind from the north, [tex]v_2=38\ mph[/tex]

Let east lies in the direction of the positive x-axis and north in the direction of the positive y-axis.

We need to find the resultant velocity of the airplane. Let v is the resultant velocity. It can be calculated as :

[tex]v=\sqrt{v_1^2+v_2^2}[/tex]

[tex]v=\sqrt{(210)^2+(38)^2}[/tex]

v = 213.41 m/s

So, the resultant velocity of the airplane is 213.41 m/s. Hence, this is the required solution.

Final answer:

The resultant velocity of the airplane, combining its eastward direction and the northward wind, is approximately 213.4 miles per hour at an angle of 10.3 degrees north of east.

Explanation:

The student's question relates to the concept of resultant velocity, which is a fundamental topic in Physics. When two velocities are combined, such as an airplane's velocity and wind velocity, the outcome is a vector known as the resultant velocity. To calculate this, one must use vector addition.

The airplane has a velocity of 210 miles per hour due east, which can be represented as a vector pointing along the positive x-axis. The wind has a velocity of 38 miles per hour from the north, represented as a vector along the positive y-axis. To find the resultant velocity, these two vectors must be combined using vector addition.

Mathematically, the resultant vector [tex]\\(R)[/tex] can be found using the Pythagorean theorem if the vectors are perpendicular, as in this case:
[tex]\[ R = \sqrt{V_{plane}^2 + V_{wind}^2} \][/tex]

Where \\(V_{plane}\\) is the velocity of the airplane and [tex]\(V_{wind}\)[/tex] is the velocity of the wind.

The direction of the resultant vector can be determined by calculating the angle [tex]\(\theta\)[/tex] it makes with the positive x-axis using trigonometry, specifically the tangent function:
[tex]\[ \theta = \arctan\left(\frac{V_{wind}}{V_{plane}}\right) \][/tex]

By substituting the given values:

The resultant velocity (magnitude) is then calculated by:

[tex]\[ R = \sqrt{(210)^2 + (38)^2} = \sqrt{44100 + 1444} = \sqrt{45544} \][/tex]

This yields a resultant speed of approximately 213.4 miles per hour.

The direction \\(\theta\\) will be:

[tex]\[ \theta = \arctan\left(\frac{38}{210}\right) \][/tex]

Using a calculator, one finds that [tex]\(\theta\)[/tex] is approximately 10.3 degrees north of east.

Answer: The required value would be 0.000013.

Step-by-step explanation:

Since we have given that

0.001 % of [tex]\dfrac{4}{3}[/tex]

As we know that

To remove the % sign we should divide it by 100.

Mathematically, it would be expressed as

[tex]\dfrac{0.001}{100}\times \dfrac{4}{3}\\\\=\dfrac{0.004}{300}\\\\=0.000013[/tex]

Hence, the required value would be 0.000013.

Final answer:

To calculate the difference in earnings between the two investments, we can use the compound interest formula to find the future value of each investment. The first investment would earn $2,353,121.65 more than the second investment.

Explanation:

To calculate the difference in earnings between the two investments, we need to calculate the future value of each investment. For the first investment, we have $22,000 invested for 40 years at an annual interest rate of 14%. Using the compound interest formula:

FV = PV * (1 + r)^n

FV = $22,000 * (1 + 0.14)^40 = $2,889,032.39

For the second investment, we have $22,000 invested for 40 years at an annual interest rate of 7%. Using the compound interest formula:

FV = PV * (1 + r)^n

FV = $22,000 * (1 + 0.07)^40 = $535,910.74

The difference in earnings between the two investments is:

$2,889,032.39 - $535,910.74 = $2,353,121.65

Answer:

[tex]x = 0[/tex] or [tex]x = \pi[/tex].

Step-by-step explanation:

How are tangents and secants related to sines and cosines?

[tex]\displaystyle \tan{x} = \frac{\sin{x}}{\cos{x}}[/tex].

[tex]\displaystyle \sec{x} = \frac{1}{\cos{x}}[/tex].

Sticking to either cosine or sine might help simplify the calculation. By the Pythagorean Theorem, [tex]\sin^{2}{x} = 1 - \cos^{2}{x}[/tex]. Therefore, for the square of tangents,

[tex]\displaystyle \tan^{2}{x} = \frac{\sin^{2}{x}}{\cos^{2}{x}} = \frac{1 - \cos^{2}{x}}{\cos^{2}{x}}[/tex].

This equation will thus become:

[tex]\displaystyle \frac{1 - \cos^{2}{x}}{\cos^{2}{x}} \cdot \frac{1}{\cos^{2}{x}} + \frac{2}{\cos^{2}{x}} - \frac{1 - \cos^{2}{x}}{\cos^{2}{x}} = 2[/tex].

To simplify the calculations, replace all [tex]\cos^{2}{x}[/tex] with another variable. For example, let [tex]u = \cos^{2}{x}[/tex]. Keep in mind that [tex]0 \le \cos^{2}{x} \le 1 \implies 0 \le u \le 1[/tex].

[tex]\displaystyle \frac{1 - u}{u^{2}} + \frac{2}{u} - \frac{1 - u}{u} = 2[/tex].

[tex]\displaystyle \frac{(1 - u) + u - u \cdot (1- u)}{u^{2}} = 2[/tex].

Solve this equation for [tex]u[/tex]:

[tex]\displaystyle \frac{u^{2} + 1}{u^{2}} = 2[/tex].

[tex]u^{2} + 1 = 2 u^{2}[/tex].

[tex]u^{2} = 1[/tex].

Given that [tex]0 \le u \le 1[/tex], [tex]u = 1[/tex] is the only possible solution.

[tex]\cos^{2}{x} = 1[/tex],

[tex]x = k \pi[/tex], where [tex]k\in \mathbb{Z}[/tex] (i.e., [tex]k[/tex] is an integer.)

Given that [tex]0 \le x < 2\pi[/tex],

[tex]0 \le k <2[/tex].

[tex]k = 0[/tex] or [tex]k = 1[/tex]. Accordingly,

[tex]x = 0[/tex] or [tex]x = \pi[/tex].

Answer:

Step-by-step explanation:

Answer:

a) 0.2

b) 0.2

c) 0.5

Step-by-step explanation:

Let [tex]S[/tex] be the event "the car stops at the signal.

In the attached figure you can see a tree describing all possible scenarios.

For the first question the red path describes stopping at the first light but not stopping at the second. We can determine the probability of this path happening by multiplying the probabilities on the branches of the tree, thus

[tex]P(a)=0.4\times0.5=0.2[/tex]

For the second one the blue path describes the situation

[tex]P(b)=0.4\times 0.5=0.2[/tex]

For the las situation the sum of the two green path will give us the answer

[tex]P(c)=0.6\times 0.5 + 0.4\times 0.5=0.3+0.2=0.5[/tex]

Answer:

a) 0.3

b) 0.1

c) 0.3

Step-by-step explanation:

Lets call:

a = stop at first signal,  b = stop at second signal

The data we are given is P(a) = 0.4, and P(b)=0.5

Stoping at least at one is the event (a or b) = a ∪ b

P(a U b) = 0.6 is the other data we are given

a) Stoping at both signals is the event (a and b = a ∩ b)

The laws of probability say that:

P(a ∪ b)= P(a) + P(b) - P( a ∩ b) = 0.4 + 0.5  - P( a ∩ b) = 0.6

Then we get P( a ∩ b) = 0.3

b) The event is (a and not b) = a ∩(¬b).

P( a ∩(¬b) ) = P(a) - P( a ∩ b) = 0.1

c) The event is (a or b) without the cases in which (a and b)

P(a ∪ b) - P( a ∩ b) = 0.3

The Venn diagram can help you understand how the events are related to each other

Answer:

0.681

Step-by-step explanation:

Let's define the following events:

S: a Carter office worker is rated satisfactory

U : a Carter office worker is rated unsatisfactory

ND: a Carter office worker is placed by Nancy Dwyer

DN: a Carter office worker is placed by Darla Newberg

We have from the original text that

P(S | ND) = 0.8, this implies that P(U | ND) = 0.2.

P(S | DN) = 0.65, this implies that P(U | DN) = 0.35. Besides

P(DN) =  0.55 and P(ND) = 0.45, then we are looking for

P(DN | U), using the Bayes' formula we have

P(DN | U) = [tex]\frac{P(U | DN)P(DN)}{P(U | DN)P(DN) + P(U | ND)P(ND)}[/tex] = [tex]\frac{(0.35)(0.55)}{(0.35)(0.55)+(0.2)(0.45)}[/tex]=0.681

Final answer:

The probability that a Carter office worker rated unsatisfactory was placed by Darla is approximately 0.346.

Explanation:

To find the probability that a Carter office worker rated unsatisfactory was placed by Darla, we can use Bayes' theorem. Let's denote the event that the worker is placed by Darla as D and the event that the worker is rated unsatisfactory as U. We are given the following probabilities:

P(Darla places) = 55% = 0.55

P(Nancy places) = 45% = 0.45

P(Satisfactory | Nancy places) = 80% = 0.80

P(Satisfactory | Darla places) = 65% = 0.65

We want to find P(D | Unsatisfactory), which is the probability that the worker was placed by Darla given that they are rated unsatisfactory. Using Bayes' theorem, we have:

P(D | U) = (P(D) * P(U | D)) / (P(D) * P(U | D) + P(N) * P(U | N))

Substituting the given probabilities, we get:

P(D | U) = (0.55 * (1 - 0.65)) / (0.55 * (1 - 0.65) + 0.45 * (1 - 0.80))

P(D | U) ≈ 0.346

Therefore, the probability that a Carter office worker rated unsatisfactory was placed by Darla is approximately 0.346.

Answer:

PROPORTION.

Step-by-step explanation:

The relative frequency in a relative frequency histogram refers to                                  PROPORTION.

A relative frequency histogram uses the same information as a frequency histogram but compares each class interval with the number of items. The difference between frequency and relative frequency histogram is that the vertical axes uses the relative or proportional frequency rather than simple frequency

Answer:

See below.

Step-by-step explanation:

2−4÷2+23 =

= 2 - 2 + 23

= 0 + 23

= 23

This is the answer of the problem you posted, where 23 is the number twenty-three. 23 is not an answer choice, so perhaps 23 is not the number twenty-three, but rather 2 to the 3rd power, 2^3.

2−4÷2+2^3 =

= 2 - 2 + 8

= 0 + 8

= 8

8 is one of the choices.

Answer:

Step-by-step explanation:

Perfect number is the positive integer which is equal to sum of proper divisors of the number.

Aliquot part is also called as proper divisor which means any divisor of the number which isn't equal to number itself.

Number : 6

Perfect divisors / Aliquot part = 1, 2, 3

Sum of the divisors = 1 + 2 + 3 = 6

Thus, 6 is a perfect number.

Number : 28

Perfect divisors / Aliquot part = 1, 2, 4, 7, 14

Sum of the divisors = 1 + 2 + 4 + 7 + 14 = 28

Thus, 28 is a perfect number.

Answer:

Y=32.67°

Step-by-step explanation:

Supplement condition:

Y+(10x+4°)=180°     (1)

Complement condition:

Y+4x=90°      (2)

5*(2)-2*(1):

5Y +20x - 2Y -20x -8° =450°-360°

3Y=98°

Y=32.67°

Answer:

1052.944 g

Step-by-step explanation:

Given:

Initial temperature of water = 15° C

Final temperature of water = 1° C

Mass of ice = 185 grams

Now,

Heat of fusion of ice = 333.55 J/g

Thus,

The heat required to melt ice = Mass of ice × Heat of fusion

or

The heat required to melt ice = 185 × 333.55 = 61706.75 J

Now,

for water the specific heat capacity= 4.186 J/g.°C

Heat provided = mass × specific heat capacity × Change in temperature

or

61706.75 = mass × 4.186 × (15 - 1)

or

61706.75 = mass × 58.604

or

mass = 1052.944 g

Hence, the mass that can be heated 1052.944 g