A student uses the following steps to prove the sine sum identity. The proof is incorrect.

Answer:

Minus 9

Step-by-step explanation:

The theoretical probability of Nadir being chosen as part of the group of four students for the student council is 4/6, which simplifies to 2/3.

To calculate the theoretical probability, we need to determine the number of favorable outcomes (Nadir being chosen) and divide it by the total number of possible outcomes (all combinations of four students out of the six).

First, let's calculate the number of favorable outcomes. Since we want Nadir to be chosen, we can consider Nadir as one of the positions to be filled. This leaves us with three remaining positions to be filled by the remaining five students (Michelle, Olivia, Parvi, Quinn, and Richard). Therefore, the number of favorable outcomes is the number of ways to choose three students out of the five, which is given by the combination formula: 5 choose 3 = 5! / (3! * (5-3)!) = 10.

Next, let's determine the total number of possible outcomes, which is the number of ways to choose four students out of the six. Using the combination formula again, we have 6 choose 4 = 6! / (4! * (6-4)!) = 15.

Finally, we divide the number of favorable outcomes by the total number of possible outcomes: 10 / 15 = 2/3. Therefore, the theoretical probability of Nadir being chosen as part of the group is 2/3.

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The weighted average of a data set is 36.

Here,

Given data set;

20 has a weight of 3, 40 has a weight of 5, and 50 has a weight of 2.

We have to find the weighted average of a data set.

What is Weighted Average?

Weighted average is a calculation that takes into account the varying degrees of importance of the numbers in a data set.

Now,

Given data set;

20 has a weight of 3, 40 has a weight of 5, and 50 has a weight of 2.

The weighted average is given by the formula;

\(x = \frac{f_{1} x_{1} + f_{2} x_{2}+ f_{3} x_{3}+ ...... f_{n} x_{n}}{f_{1} +f_{2} + f_{2}}\)

Hence,

The weighted average of a data set;

x = 20 x 3 + 40 x 5 + 50 x 2 / 3+5+2

x = 360/10 = 36

Hence, The weighted average of a data set is 36.

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Based on mathematical operations, the number of the students surveyed who live less than 1 mile from school and use a bus to get to school is 21.

The basic mathematical operations are addition, subtraction, division, and multiplication.

Mathematical operations combine mathematical operands and the equation symbol (=) to solve mathematical problems.

                               Transportation to School

Distance from          Walk         Bus      Subway    Total

Home to school

Less than 1 mile         16            21               6           43

1 or more miles           3             13             21           37

Total                          19            34             27          80

Total = 27 (80 - 19 - 34)

Less than 1 mile = 6 (27 - 21)

Less than 1 mile = 21 (43 - 6 - 16)

1 or more miles = 13 (37 - 3 - 21)

                               Transportation to School

Distance from          Walk         Bus      Subway    Total

Home to school

Less than 1 mile         16                                            43

1 or more miles           3                              21           37

Total                          19              34                           80

Thus, 21 students surveyed by the teacher live less than 1 mile from school and use a bus to get to school.

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If x < 3, then the square root of  (x^2 - 6x + 9) can be simplified to (3-x).

First we factorise the quadratic expression:

x^2 - 6x + 9 = (x - 3)^2 ..(i)

(Since the expression is a perfect square trinomial, it can be factored as the square of a binomial.)

Then we will simplify the square root:

√(x^2 - 6x + 9) = √((x - 3)^2).

Now, since x - 3 is squared, taking the square root will eliminate the square, resulting in the absolute value of x - 3.

Final simplified form: √((x - 3)^2) = |x - 3|.

Therefore, the simplified square root expression is |x - 3| when x < 3 which equals to 3-x.

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The centre line of a control chart is calculated by computing the average (mean) of the data for every period.

In control chart analysis, the centre line represents the central tendency or average value of the process being monitored. It is typically obtained by calculating the mean of the data points collected over a specific period. The purpose of the centre line is to provide a reference point against which the process performance can be compared. Any data points falling within acceptable limits around the centre line indicate that the process is stable and under control.

The calculation of the centre line involves summing up the values of the data points and dividing it by the number of data points. This average is then plotted on the control chart as the centre line. By monitoring subsequent data points and their distance from the centre line, deviations and trends in the process can be identified. Deviations beyond the control limits may indicate special causes of variation that require investigation and corrective action. Therefore, the centre line is a critical element in control chart analysis for understanding the baseline performance of a process and detecting any shifts or changes over time.

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The rate at which the length of a side is decreasing when the area of the triangle is 300 cm² is approximately -0.083 cm/min, or -1/12 cm/min, accurate to 3 decimal places.

Let's use the formula for the area of an equilateral triangle to relate the rate of change of the area to the rate of change of the side length.

The area of an equilateral triangle with side length s is given by:

A = (√3/4) s²

Taking the derivative of both sides with respect to time t, we get:

dA/dt = (√3/2) s ds/dt

where ds/dt is the rate at which the side length is changing.

We know that dA/dt = -5 cm²/min (since the area is decreasing at a rate of 5 cm²/min), and we want to find ds/dt when A = 300 cm².

So we have:

-5 = (√3/2) s ds/dt

Solving for ds/dt, we get:

ds/dt = -10/(√3s)

When A = 300 cm², the side length can be found by rearranging the formula for the area:

s² = (4/√3) A

s² = (4/√3) (300)

s = 20√3 cm

Substituting this value into the expression for ds/dt, we get:

ds/dt = -10/(√3(20√3))

ds/dt = -1/12 cm/min

Therefore, the rate at which the length of a side is decreasing when the area of the triangle is 300 cm² is approximately -0.083 cm/min, or -1/12 cm/min, accurate to 3 decimal places.

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Answer:

then answer is never going to give you up never going to let you down

Answer:

Step-by-step explanation:

45342

Answer:

1.y=2x-1

2.y=-3/5x+4

3.y=-3x+2

4.y=-x+7

5.y=1/4x

6.y=-5/2x-3

Step-by-step explanation:

Answer:

33.75 :)

Step-by-step explanation:

Answer: 33.75

Step-by-step explanation:

45 x 0.75 = 33.75

A number is a restriction on a rational expression if the number makes the denominator of the rational expression equal to zero.

Hence, option (b) is correct choice.

Simply put, a rational expression is the product of two polynomials. Or to put it another way, it is a fraction using polynomials as the numerator and denominator.

A rational expression's denominator is undefined when it equals zero, hence it cannot be employed in the expression.

For that particular value of the number, the limitation renders the value of the rational statement undefinable.

To put it another way, the number is a limitation since it cannot be used as the input for the rational expression because the output would be an undefined value.

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The missing option may be:

(a) 1

(b) 0

(c) always positive

(d) Indefinite

The theoretical probability of choosing List A is less than the experimental probability of choosing List A.

The theoretical probability of choosing List B is greater than the experimental probability of choosing List A.

For a fair coin, the theoretical probability of obtaining a head or tail is 0.5 or 50% each.

Therefore, theoretically, the probability of choosing List A is 0.5, and the probability of choosing List B is also 0.5.

The experimental probability of choosing List A or List B is obtained using the formula below:

Experimental probability of List A = 37/70

Experimental probability of List A = 0.529

The experimental probability of choosing List B is:

Experimental probability of List B = 33/70

Experimental probability of List B = 0.471

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No.1

Set those two equations equal to each other.

-7/4 x - 4 = -1/4 x + 2

x on one side, constant on other side.

-6/4 x = 6

x = -4

now put x value into any of the two equations given

y = -7/4 * -4 - 4 = 7 - 4 = 3

(-4,3)

No.2

Set those two equations equal to each other.

x + 2 = -1/5 x - 4

x on one side, constant on other side.

6/5 x = -6

x = -5

now put x value into any of the two equations given

y = -5 + 2 = -3

(-5,-3)

The length of x in the triangle is 8 units.

A midsegment is the line segment connecting the midpoints of two sides of a triangle.

The midsegment theorem states that a line segment connecting the midpoints of any two sides of a triangle is parallel to the third side of a triangle and is half of it

Therefore, DE is the mid segment of the triangle ABC. The value of x can be found as follows:

Using the mid segment principle,

BE = EC

BD = DA

Therefore,

x = 8 units

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The solution is: 1/7 is the probability of picking an even number and then picking an even number.

Here, we have,

we know that,

Probability = required outcome /all possible outcome

given that,

You pick a card at random. Without putting the first card back, you pick a second card at random. 1 2 3 4 5 6 7.

number of  possible outcome = 7

Let:

A = 1st time pick an even number

B =2nd time pick an even number

so, we get,

There are 3 even numbers in total,

for the 2nd time it will be 2 even numbers in total,

we have,

P(A) = 3/7

and,

P(B) =2/6

we know that,

The intersection between both events is equal to the product of both probabilities since the events are independent.

so, we get,

P (A∩B) = P(A). P(B)

             = 3/7 * 2/6

             = 1/7

Hence, 1/7 is the probability of picking an even number and then picking an even number.

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Answer:

Tn=a+(n-1)d

Tn=29+(n-1)9

Tn=29+9n-9

Tn=20+9n

Tn=20+9 (21)

T21=580

Step-by-step explanation:

a=first term

d=common difference

Tn=a+(n-1)d .....general formula for an arithmetic number pattern

Answer:

Step-by-step explanation:

I assume the 8 is the question number and the sequence is

29,38,47...

This is an arithmetic sequence as any term minus the previous term is a constant, called the common difference, which in this case is 47-38=38-29=9. An arithmetic sequence has the form

an=a+d(n-1)

Where an=nth term, a=initial term, d=common difference, and n=term number.

In this case a=29 and d=9 so

an=29+9(n-1)

an=9n+20, then the 21st term is

a(21)=9(21)+20

a(21)=189+20

a(21)=209

So the 21st term is 209.

The answer is option B, the other two sides are 5.4 cm and 7.1 cm.


- We can use the Law of Sines to solve for the other two sides of the triangle.
- The Law of Sines states that for any triangle with sides a, b, c and angles A, B, C:
 a/sin(A) = b/sin(B) = c/sin(C)
- We are given angle A (106°) and side a (10 cm), so we can set up the following proportion:
 10/sin(106°) = b/sin(31°)
- Solving for b, we get:
 b = (10 sin(31°)) / sin(106°)
- Using a calculator, we get b ≈ 5.4 cm.
- To find the third side, we can use the Law of Sines again:
 10/sin(106°) = c/sin(C)
- We know that the sum of angles in a triangle is 180°, so we can find angle C by subtracting angles A and B from 180°:
 C = 180° - 106° - 31° = 43°
- Plugging in the values, we get:
 c = (10 sin(43°)) / sin(106°)
- Using a calculator, we get c ≈ 7.1 cm.
- Therefore, the other two sides are approximately 5.4 cm and 7.1 cm.
- The answer is option B.


To solve this problem, we can use the Law of Sines. The Law of Sines is a formula that relates the sides and angles of any triangle. It states that for any triangle with sides a, b, c and angles A, B, C:

a/sin(A) = b/sin(B) = c/sin(C)

In this problem, we are given angle A (106°) and side a (10 cm), and we need to find the other two sides. We can use the Law of Sines to set up a proportion and solve for one of the unknown sides.

First, we need to find the measure of angle B. We know that the sum of angles in a triangle is 180°, so we can find angle B by subtracting angles A and C from 180°:

B = 180° - A - C
B = 180° - 106° - C
B = 74° - C

Now we can set up the proportion:

a/sin(A) = b/sin(B)

Plugging in the values we know:

10/sin(106°) = b/sin(74° - C)

We can solve for b by cross-multiplying:

10 sin(74° - C) = b sin(106°)

Dividing both sides by sin(106°):

b = (10 sin(74° - C)) / sin(106°)

We can use a calculator to find that b ≈ 5.4 cm.

Now we need to find the third side, c. We can use the Law of Sines again, this time with angle C and side c:

a/sin(A) = c/sin(C)

Plugging in the values we know:

10/sin(106°) = c/sin(C)

We can solve for c by cross-multiplying:

c = (10 sin(C)) / sin(106°)

We already know the value of angle B (74° - C), so we can find the value of angle C:

B + C = 180°
74° - C + C = 180°
C = 106°

Plugging in the value of angle C, we get:

c = (10 sin(106°)) / sin(106°)

Using a calculator, we find that c ≈ 7.1 cm.

Therefore, the other two sides are approximately 5.4 cm and 7.1 cm, and the answer is option B.

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Answer:

-22

Step-by-step explanation:

Hope that helped!

Answer:

y = -1/2x + 5/2.

Step-by-step explanation:

If a line is perpendicular to another line, the slope of the line is the negative reciprocal of the other. In this case, the line is perpendicular to y = 2x + 3. That means the slope will be -1/2.

Now, we have y = -1/2x + c.

To solve for c, put in the points given.

2 = (-1/2) * 1 + c

c -1/2 = 2

c = 2 + 1/2

c = 4/2 + 1/2

c = 5/2

So, your final answer is that the equation of the line is y = -1/2x + 5/2.

Hope this helps!

Answer:

\(y = -\frac{1}{2} x+\frac{5}{2}\)

Step-by-step explanation:

Perpendicular => So, it would have a slope of negative reciprocal for this slope

Slope = m = -1/2

Now,

Point = (x,y) = (1,2)

So, x = 1, y = 2

Putting in the slope intercept equation to get c

=> \(y = mx+c\)

=> 2 = (-1/2)(1) + c

=> 2+1/2 = c

=> c = \(\frac{4+1}{2}\)

=> c = \(\frac{5}{2}\)

Now, Putting m and c in the slope-intercept equation:

=> \(y = mx+c\)

=> \(y = -\frac{1}{2} x+\frac{5}{2}\)

tte corresponding angle is angle 5

Given :-

To prove :-

Construction :-

Proof :-

In ∆ ABC , we have ,

As sum of two interior angles of a triangle is equal to the opposite exterior angle .

Also ,

( linear pair )

Hence from above two ,

Hence Proved !

Thus, the correct answer is: C. The Stafford loan will have a lower balance of $431.17 at the time of repayment.

In order to evaluate the Unsubsidized Stafford Loan against the PLUS Loan, a comparison of their balances upon repayment after 6 months of graduation is necessary.

Having taken out a loan of $9,529 with an annual interest rate of 5.95% compounded monthly, and after an interest accumulation period of 18 months, the Stafford Loan balance at this point in time stands roughly around $10,365.86.

On the other hand, the PLUS Loan's equilibrium at graduation is recorded at $10,172.23, but including a 6-month interest accumulation period along with an annual incrementation percentage of 6.55% that is compounded monthly equals to approximately $10,797.03 when due for repayment.

To summarize, based on the balance comparisons, it is concluded that the Stafford loan owes a lower amount at repayment which is different by $431.17 compared to the PLUS Loan.

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Answer:

its slope is 1

Step-by-step explanation:

For `C = 0, d = 1 therefore C + d = 0 +1 = 1

For C = 27, d =  0.2161 therefore C+d = 27 + 0.2161 = 27.2161

We can use the substitution method to find the values of x and y.

We can rearrange the first equation to solve for x in terms of y:

Cx + dy = 12

Cx = 12 - dy

\(x = \frac{ (12 - dy)}{C}\)

This expression for x can then be substituted into the second equation:

2x + 7y = 4

2(\(\frac{(12 - dy)}{C}\)) + 7y = 4

To eliminate the denominator, multiply both sides by C:

2(12 - dy) + 7Cy = 4C

Increasing the size of the brackets:

24 - 2dy + 7Cy = 4C

Rearranging and calculating y:

-2dy + 7Cy = 4C - 24

y(7C - 2d) = 4C - 24

y = \(\frac{(4C - 24)}{(7C - 2d)}\)

We can then plug this y expression back into the first equation to find x:

Cx + dy = 12

C(\(\frac{(4C - 24)}{(7C - 2d)}\)) + d(\(\frac{(4C - 24)}{(7C - 2d)}\)) = 12

Multiplying to eliminate the denominator, multiply both sides by (7C - 2d):

12(7C - 2d) = C(4C - 24) + d(4C - 24).

Increasing the size of the brackets:

84C - 24d = 4C2 - 24C + 4Cd - 24C

Simplifying:

\(4C^2 - 108C = 0\)

Taking 4C into account:

4C(C - 27) = 0

As a result, either C = 0 or C = 27.

If C is equal to zero, the first equation becomes:

dy = 12

The second equation is as follows:

2x + 7y = 4

Adding dy = 12 to the first equation:

d(12) = 12

d = 1

Adding d = 1 to the second equation:

2x + 7(12) = 4

2x = -80

x = -40

As a result, if C = 0, x = -40, and y = 1.

If C = 27, the first equation is as follows:

27x + dy = 12

The second equation is as follows:

2x + 7y = 4

Adding dy = 12 - 27x to the first equation:

27x + d(12 - 27x) = 12

-27dx + 27x = 12 - 27x

d = (12 - 27x)/-27x + 1

In the second equation, substitute d = (12 - 27x)/-27x + 1:

2x + 7((12 - 27x)/-27x + 1) = 4

To eliminate the denominator, multiply both sides by -27x:

-54x + 84 - 7x(-27x + 27x + 1) = -108x

Simplifying:

-54x + 84 + 7x = -108x

-47x = -84

x = 84/47

Adding x = 84/47 to the formula for d:

d = (12 - 27(84/47))/-27(84/47) +1

d = (12 - 1.7872)/ -27(1.7872) +1

d = 10.2128/-47.2544

d = 0.2161

For `C = 0, d = 1 therefore C + d = 0 +1 = 1

For C = 27, d =  0.2161 therefore C+d = 27 + 0.2161 = 27.2161

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I can help you with that problem i just did that on my own.

Answer:

Slope = -1

Step-by-step explanation:

m = second y - first y / second x - first x

second y = 1

first y = 9

second x = 9

first x = 1

1 - 9 = -8

9 - 8 = 8

-8/8 = -1

Find the slope of a line.

We mark the points: (x₁ = 1, y₁=9) and (x₂ = 8, y₂ = 1)

We have that the formula is:

          \(\boldsymbol{\sf{m=\dfrac{Y_{2}-Y_{1} }{X_{2}-X_{1}} }}\)

Now we apply the formula and solve:

                    \(\boldsymbol{\sf{m=\dfrac{1-9}{8-1} =\dfrac{-8}{7} }}\)

                   \(\boldsymbol{\sf{m=\dfrac{-8}{7} }}\)

The slope of the line through (1, 9) and (8, 1) is -8/7.