Suppose a curve is traced by the parametric equations x = 5 ( sin(t) + cos(t)) y = 47-15 cos2 ()-30 sin(t) as t runs from 0 to π. At what point (x,y) on this curve is the tangent line horizontal?

Answer:

Step-by-step explanation:

Pythagorean theorem: base ² +altitude² = hypotenuse²

1) hypotenuse² = base² + altitude²

          x² = 6² + 3²

               = 36 + 9

               = 45

         x = √45

         x = 6.7 cm

4) base ² +altitude² = hypotenuse²

x ² + 3.7 ² =  4.2 ²

x ² + 13.69 = 17.64

             x ² = 17.64 - 13.69 = 3.95

             x = √3.95

            x = 1.99 cm

10)c ² = a ² + b ²

           = 3 ² +4 ²

         = 9 + 16

  c ² = 25

c = √25 = 5

11) a ² + b ² = c ²

9 ² +b² = 13 ²

81 + b² = 169

     b² = 169 - 81 = 88

b = 9.38

12) a² + b² = c²

12² + b² = 22²

  144 + b² = 484

           b² = 484 - 144 = 340

           b =√340 = 18.44

A different term for a quadratic function's zeros is "roots." The zeros or roots of a quadratic function are the points on the x-axis where the function meets or crosses it.

They stand in for the x values at which the quadratic function equals zero. Depending on the discriminant of the quadratic equation, the zeros or roots of a quadratic function can either be real or complex integers. The quadratic has two unique real roots if the discriminant is positive. The quadratic has a single real root, also referred to as a double root or repeating root, if the discriminant is zero. The quadratic has two complex roots that can be represented as a + bi if the discriminant is negative.

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Answer:

45%

Step-by-step explanation:

$78 is 65% of $120, therefore, the price was decreased by 45%. To prove 78 is 65% of 120, multiply 120x0.65 and you'll get 78. The reason you use 0.65 is because 65% is 0.65 in it's decimal form.

Hope you could understand.

If you have any query, feel free to ask.

Expanding the given expression and substituting the given values of \(\dfrac{p}{q}\) with \(\dfrac{q}{r}\) proves that the given equation

Correct response:

\(The \ expression \ p^3 + q^3 + r^3 \ is \ equal \ to \ \left(\dfrac{1}{p^3} + \dfrac{1}{q^3} +\dfrac{1}{r^3} \right) \cdot p^2 \cdot q^2 \cdot r^2 \ by \ subtituting\)

\(\dfrac{p}{q} = \dfrac{q}{r}\)

The given relation is;

\(\dfrac{p}{q} = \mathbf{\dfrac{q}{r}}\)

The given equation is presented as follows;

\(p^3 + q^3 + r^3 = \mathbf{\left(\dfrac{1}{p^3} + \dfrac{1}{q^3} + \dfrac{1}{r^3} \right) \cdot p^2 \cdot q^2 \cdot r^2}\)

Expanding the right hand side gives;

\(\dfrac{p^2 \cdot q^2 \cdot r^2}{p^3} + \dfrac{p^2 \cdot q^2 \cdot r^2}{q^3} + \dfrac{p^2 \cdot q^2 \cdot r^2}{r^3} = \mathbf{ \dfrac{q^2 \cdot r^2}{p} + \dfrac{p^2 \cdot r^2}{q} + \dfrac{p^2 \cdot q^2 }{r}}\)

\(\dfrac{q^2 \cdot r^2}{p} + \dfrac{p^2 \cdot r^2}{q} + \dfrac{p^2 \cdot q^2 }{r} = \mathbf{ \dfrac{q}{p} \cdot q \cdot r^2 + \dfrac{p}{q} \cdot p \cdot r^2+\dfrac{q}{r} \cdot p^2 \cdot q }\)

\(\dfrac{q}{p} \cdot q \cdot r^2 + \dfrac{p}{q} \cdot p \cdot r^2+\dfrac{q}{r} \cdot p^2 \cdot q } = \dfrac{r}{q} \cdot q \cdot r^2 + \dfrac{q}{r} \cdot p \cdot r^2+\dfrac{p}{q} \cdot p^2 \cdot q } = \mathbf{ r^3 + q \cdot p \cdot r + p^3}\)

From the given relation, we have;

p·r = q²

Therefore;

q·p·r = q × q² = q³

Which gives;

r³ + q·p·r + p³ = r³ + q³ + p³

Which gives;

\(\left(\dfrac{1}{p^3} + \dfrac{1}{q^3} + \dfrac{1}{r^3} \right) \cdot p^2 \cdot q^2 \cdot r^2 = p^3 + q^3 + r^3\)

By symmetric property, therefore;

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The average speed on the winding road was 45 mph.

The bus traveled for 3 hours on the winding road, so the distance covered can be calculated using the formula: Distance = Speed × Time. Let's assume the average speed on the winding road as 'x' mph. Therefore, the distance covered on the winding road is 3x miles.

On the straight road, the bus traveled for 3 hours at an average speed that was 12 mph faster than its average speed on the winding road. So the average speed on the straight road can be expressed as 'x + 12' mph. The distance covered on the straight road can be calculated as 3(x + 12) miles.

The total distance covered in the entire trip is given as 210 miles. Therefore, we can write the equation:

3x + 3(x + 12) = 210

Simplifying the equation:

3x + 3x + 36 = 210

6x + 36 = 210

6x = 174

x = 29

So the average speed on the winding road was 29 mph.

The problem states that the bus traveled for 3 hours on both the winding road and the straight road. Let's assume the average speed on the winding road as 'x' mph. Since the bus traveled for 3 hours on the winding road, the distance covered can be calculated as 3x miles.

On the straight road, the average speed was 12 mph faster than on the winding road. Therefore, the average speed on the straight road can be expressed as 'x + 12' mph. The distance covered on the straight road can be calculated as 3(x + 12) miles.

The total distance covered in the entire trip is given as 210 miles. This allows us to set up the equation 3x + 3(x + 12) = 210 to solve for 'x'. Simplifying the equation leads to 6x + 36 = 210. Solving for 'x', we find that the average speed on the winding road was 29 mph.

In summary, the average speed on the winding road was 29 mph.

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Two additional polar representations of the point with coordinates (-4, 3π/2) within the interval [-2π, 2π] are (-4, 7π/2) and (4, 5π/2).

You find two additional polar representations of the point with polar coordinates (-4, 3π/2), keeping the angle in the interval [-2π, 2π].
First, let's understand that there can be multiple representations of a point in polar coordinates by adding or subtracting multiples of 2π to the angle while keeping the radius the same or by negating the radius and adding or subtracting odd multiples of π to the angle.
Representation 1:
Keep the radius the same and add 2π to the angle:
(-4, 3π/2 + 2π) = (-4, 3π/2 + 4π/2) = (-4, 7π/2)
Representation 2:
Negate the radius and add π to the angle:
(4, 3π/2 + π) = (4, 3π/2 + 2π/2) = (4, 5π/2)
So, two additional polar representations of the point with coordinates (-4, 3π/2) within the interval [-2π, 2π] are (-4, 7π/2) and (4, 5π/2).

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Answer: (x,y) = (5,2)

In other words, x = 5 and y = 2

====================================================

Explanation:

Subtract the equations straight down.

We end up with 6y = 12 which solves to y = 2 after dividing both sides by 6.

Now use this y value to find x.

If we used the first equation, then we have these steps

3x+4y = 23

3x+4(2) = 23

3x+8 = 23

3x = 23-8

3x = 15

x = 15/3

x = 5

Or we could use the second equation

3x-2y = 11

3x-2(2) = 11

3x-4 = 11

3x = 11+4

3x = 15

x = 15/3

x = 5

We get the same x value either way.

The solution is (x,y) = (5,2)

To check the answer, plug those coordinates into each equation. After simplifying, you should get the same number on both sides.

------------------

Check:

3x+4y = 23

3(5)+4(2) = 23

15+8 = 23

23 = 23 .. that works

and,

3x-2y = 11

3(5)-2(2) = 11

15 - 4 = 11

11 = 11 .. that works as well

Both original equations are true for those x and y values.

The solution is fully confirmed.

The values of the inverse functions are f-¹ (3) = -1 and f-¹ (8) = 7

From the question, we have the following parameters that can be used in our computation:

x   f(x)

-1   3

1     6

2    5

3    7

7   8

Using the above as a guide, we have the following:

f-¹ (x) means that we calculate the value of x from the table when y = x in the function notation

So, we have

f-¹ (3) = -1 and f-¹ (8) = 7

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(a) An example of a matrix A for which the number of solutions to Ax = b is 0 or 1, depending on b, is a 2x2 matrix with either a nonzero determinant or a zero determinant and a nonzero nullspace.

(b) An example of a matrix A for which the number of solutions to Ax = b is ∞, regardless of b, is a matrix with a zero determinant and a nonzero nullspace.

(c) An example of a matrix A for which the number of solutions to Ax = b is 0 or ∞, depending on b, is a 2x2 matrix with a zero determinant and a nonzero nullspace.

(d) An example of a matrix A for which the number of solutions to Ax = b is 1, regardless of b, is a matrix with a nonzero determinant and a nullspace of zero.

If the determinant of A is nonzero, then there is only one solution to Ax = b, regardless of b.

This type of matrix is known as a singular matrix. The nullspace of such a matrix is nonempty, and since any vector in the nullspace is a solution to Ax = b, there are infinitely many solutions.

If the right hand side of the equation b is orthogonal to the nullspace of A, then the equation Ax = b has no solutions. However, if b lies in the nullspace of A, then the equation Ax = b has infinitely many solutions.

A matrix A for which solutions to Ax = b is 1, regardless of b, is a matrix with a nonzero and a nullspace of zero. This type of matrix is known as a nonsingular matrix, and in this case there is only one solution to Ax = b, regardless of the value of b.

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Step-by-step explanation:

if x = -2

= 9x² - 4x - 3

= 9 ( -2)² - 4 ( -2 ) - 3 [ x = -2 ]

= 9 × 4 + 8 - 3

= 36 + 8 - 3

= 44 - 3

= 41

\(\huge{\boxed{{ x = -2}}}\)

\( 9x² - 4x - 3\)

\(↠ 9 (-2)² -4 (-2) - 3\)

\(↠ 9×4+8-3\)

\(↠ 36 + 8 -3\)

\(↠ 44-3\)

\(↠ 41\)

\(\large{{\bold{ \underline{ \underline{ \overline{ \overline{ \pink{9{x}^{2} - 4x – 3 = 41}}}}}}}}\)

Answer:

Thanks for the points! <3

CCCCCC"CCCCCCCCCCC"CCCCCCCCCCCCCCCCCCCCC

Answer:

Wheres the picture that gose with it ?

Step-by-step explanation:

Answer:

x= 17

Step-by-step explanation:

Go to

Answer:

\(9^3\)

Step-by-step explanation:

Rules Used

\(a ^x \cdot a^y = a^{x+y}\)

\(\frac{b^x}{b^y} = b^{x-y}\)

\(\frac{9^6 \cdot 9^0 \cdot 9^4}{9^7} = 9^{6 + 0 + 4 -7} = 9^{10-7} = 9^3\)

Answer:

9^3

Step-by-step explanation:

When we want to multiply exponents, we use the formula a^m x a^n= a^m+n.

In this case, we can see that 9 is our a, and their respective exponents are mn and n.

9^6 x 9^0 x 9^4 = 9^6+0+4= 9^10

When dividing exponents we subtract the exponents for each number from the top. For example, 9^10/9^7= 9^10-7= 9^3

If we want to simplify this even further, 9^3 = 729.

Hope this helps!

:)

Answer:

I also need help please help guys

To determine customer opinion of their musical variety, sony randomly selects 120 concerts during a certain week and surveys all concertgoers. The type of sampling is used is "Cluster."

A survey is a technique of gathering information from a sample of people by asking relevant questions with the goal of understanding populations as a whole.

Some key features regarding the survey are-

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Answer:

Holly is incorrect. The decimal part follows a pattern but does not actually repeat.

so D is the correct option

Step-by-step explanation:

The numbers follow a pattern but if you take a closer look you will see that the numbers do not repeat even though they still use the numbers 1, and 3 doesn't mean there a rational number.

Holly is incorrect. The decimal part follows a pattern but does not actually repeat.

The numbers follow a pattern but if you take a closer look you will see that the numbers do not repeat even though they still use the numbers 1, and 3 doesn't mean there is a rational number.

If the value of a numerical expression is terminating then they are the rational number then they are called the rational number and if the value of a numerical expression is non-terminating then they are called an irrational number.

Hence, the decimal part follows a pattern but does not actually repeat.

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Answer:

x ≈ 2.8

Step-by-step explanation:

Using the tangent ratio in the right triangle

tan26° = \(\frac{opposite}{adjacent}\) = \(\frac{CD}{DE}\) = \(\frac{x}{5.7}\) ( multiply both sides by 5.7 )

5.7 × tan26° = x , then

x ≈ 2.8 ( to the nearest tenth )

Answer:

2.8

Step-by-step explanation:

1. c = a · sin(C) / sin(A)  

This is the equation to find x. Let's see what we have.

c = x

a = 5.7

C = 26

A = ?

Let's find A. We know B is 90 and C is 26.  Triangle Sum Theorem time!

2. 180 = 90 + 26 + A

180 = 116 + A

64 = A

A = 64

We found A. Let's see what we have now!

c = x

a = 5.7

C = 26

A = 64

Let's solve the equation now.

3. x = 5.7 · sin26 / sin64

Let's use a calcalator to find the answer.

4. 2.78008 =  5.7 · sin26 / sin64

5.7 · sin26 / sin64 =  2.78008

We should round this to the nearest tenth. 7 is in the tenth place. Though 8 is behind 7. 8 is greater than 4 which means 7 must be rounded up.

5. 2.8 is the answer.

Let x be the total number of questions on the test; then,

Which is equivalent to

Solve for x as shown below

The answer is 20 questions in total on the test

By Quadratic Formula factors of \(a^2+4b^2\) are \((a+2bi),(a-2bi)\).

The quadratic formula in elementary algebra is a formula that gives the solution(s) to a quadratic equation. In addition to the quadratic formula, there are various methods for solving quadratic equations, including factoring (direct factoring, grouping, and the AC technique), completing the square, graphing, and others.

The following general quadratic equation is given:

\(ax^2+bx+c=0\)

The quadratic formula is:

\(x=\frac{-b+\sqrt{b^2-4ac} }{2a}, \frac{-b-\sqrt{b^2-4ac} }{2a}\);

where x stands for an unknown, a, b, and c are constants, and a≠0.

Each of these two answers is also referred to as a quadratic equation root (or zero). These roots, which are clearly stated as \(y = ax^2 + bx + c\), are the x-values at which any parabola crosses the x-axis geometrically.

Given;

\(y=a^2+4b^2;\\\)

By comparing this with general equation \(x=a,a=1,b=0,\)\(c=4b^2\)

By substituting in the formula;

\(x=\frac{-0+\sqrt{(0)^2-4(1)(4b^2)} }{2(1)} =2bi\\(or) x=\frac{-0-\sqrt{(0)^2-4(1)(4b^2)} }{2(1)} =-2bi\)

That gives us;

\(y=(a+2bi)(a-2bi)\)

By Quadratic Formula factors of \(a^2+4b^2\) are \((a+2bi),(a-2bi)\).

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The value of IQR is 90 when box plot data where Q₁ = 200, Q₂ = 250, and Q₃ = 290.

Given that,

For box plot data where Q₁ = 200, Q₂ = 250, and Q₃ = 290

We have to find the data of IQR.

We know that,

IQR is define as the variation in the distribution of your data's middle quartile is measured by the interquartile range (IQR). It is the range that corresponds to your sample's middle 50%. Measure the variability where the majority of your numbers are by using the IQR.

IQR Formula is Q₃ - Q₁

So,

Q₃ = 290

Q₁ = 200

Then ,

IQR = Q₃ - Q₁

IQR = 290 - 200

IQR = 90

Therefore, The value of IQR is 90.

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the answer is A ;)))))

Answer:

Travels 4.25 miles in thirty minutes

Step-by-step explanation:

In order to find the answer to this question you need to first find how many miles the boat travels in a hour and then divide it once more to find how many miles it traveled in half a hour.

\(17\div2\)

\(`7\div2=8.5\)

\(2=17\)

\(1=8.5\)

\(8.5\div2=4.25\)

\(=4.25\)

Hope this helps.

The table of probabilities for K given that N=2 is:\(P(K=0) = 0.25P(K=1) = 0.5P(K=2) = 0.25\)

In an experiment, a fair four-sided die (its faces are labeled 0, 1, 2, 3) is thrown once. The outcome of the throw determines how many times a fair coin is to be flipped: if N is the number that results from throwing the die, we flip the coin N times. Let K be the total number of heads resulting from the coin flips. Suppose the die roll results in a 2. What form does the conditional distribution of distribution along with its parameters calculated the probabilities in the table a) K have given that N-2?The conditional distribution of K given that N=2 is a binomial distribution with parameters n=2 and p=0.5.

This is because if N is the result of the die roll and N=2, we flip the coin twice. Each flip of the coin is a Bernoulli trial with a probability of success (heads) of 0.5. Since there are two independent flips, the total number of heads K follows a binomial distribution with parameters n=2 and p=0.5.

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Answer:

4. 12√3

5. -35√2

6. 5√2/2

Step-by-step explanation:

4. 3√12+2√27

= 3√(4×3) + 2√(9×3)

= 6√3+6√3

= 12√3

5. 5√8-9√50

= 5×2√2-9×5√2

= 10√2-45√2

= -35√2

6. 5/√2

= 5×√2/(√2×√2)

= 5√2/2

Answered by GAUTHMATH

The equation of a line passing through the point (5,-3) and parallel to 6x+3y=-12 is y =-2x+7.

Parallel lines are those that never intersect. As a result, two parallel lines must have the same slope but different intercepts (if they had the same intercepts, they would be identical lines).

The equation of the line is 6x+3y=-12.

6x+3y=-12​

3y =-12-6x

y = -2x-4    

The slope of this line is -2.

Because parallel lines have the same slope, the new line will also have a slope of -2.

You now have a point (5,-3) and a slope; thus, use the Point-Slope form to solve the equation of a line.

y-y₁= m(x-x₁)

y+3 = -2(x -5)

y+3 = -2x+10

y =-2x+7

The equation of a line passing through the point (5,-3) and parallel to 6x+3y=-12 is y =-2x+7.

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A bicycle repair shop that offers two service packages: a tune-up and a complete overhaul.

The shop operates for 60 hours per week and receives an average of 180 bicycles each week. To analyze the capacity and utilization of the process, we need to consider the time taken at each stage and the demand for each service option. We'll break down the problem into multiple parts and provide a detailed explanation using mathematical terms.

a) Creating the Demand Matrix:

To create a demand matrix, we need to determine the number of bicycles going through each stage of the process. Let's denote the demand for tune-up as T and the demand for additional services as A.

Given that the average number of bicycles received per week is 180 and 25% of customers opt for additional services, we can calculate the demands as follows:

Demand for tune-up (T) = Total demand - Demand for additional services

T = 180 - (0.25 * 180)

T = 180 - 45

T = 135

Demand for additional services (A) = 0.25 * Total demand

A = 0.25 * 180

A = 45

Now, we can create a demand matrix based on the demand for each service option:

Demand Matrix:

Tune-up Additional Services Wheel Balancing

Tune-up  [135  0  0]

Additional [0  45  0]

Services

Total [ 135  45  0 ]

The demand matrix shows the number of bicycles flowing through each stage of the process.

b) Daily Capacity at Each Stage:

To calculate the daily capacity at each stage, we need to consider the time taken for each service option. Given that the shop operates for 60 hours per week, we can calculate the daily capacity at each stage:

Tune-up technician time per bicycle (\(T_{tuneup}\)) = 75 minutes

Additional services technician time per bicycle (\(T_{additional}\)) = 72 minutes

Wheel balancing specialist time per bicycle (\(T_{balancing}\)) = 20 minutes

Daily Capacity (C) = (60 hours * 60 minutes) / (\(T_{tuneup}\) + \(T_{additional}\) + \(T_{balancing}\))

Substituting the given values:

C = (60 * 60) / (75 + 72 + 20)

C = 21600 / 167

C ≈ 129.34 bicycles per day

Therefore, the daily capacity at each stage of the process is as follows:

Tune-up: 129 bicycles per day

Additional Services: 129 bicycles per day

Wheel Balancing: 129 bicycles per day

c) Implied Utilizations:

To find the implied utilizations, we need to compare the demand and the capacity at each stage of the process. Utilization can be calculated as the demand divided by the capacity.

Implied Utilization (U) = Demand / Daily Capacity

For the Tune-up stage:

\(U_{tuneup}\) = 135 / 129 ≈ 1.05

For the Additional Services stage:

\(U_{additional}\) = 45 / 129 ≈ 0.35

For the Wheel Balancing stage:

\(U_{balancing}\) = 0 / 129 = 0

The implied utilizations show how efficiently each stage of the process is being utilized. Utilization values greater than 1 indicate that the stage is operating beyond its capacity.

d) Weekly Capacity of the Process:

To calculate the weekly capacity of the process, we multiply the daily capacity by the number of days the shop is open per week:

Weekly Capacity = Daily Capacity * Number of days shop is open per week

Given that the shop is open for 60 hours per week, the number of days the shop is open per week can be calculated as follows:

Number of days shop is open per week = 60 hours / 24 hours per day = 2.5 days

Therefore, the weekly capacity of the process is:

Weekly Capacity = Daily Capacity * Number of days shop is open per week

Weekly Capacity = 129 bicycles per day * 2.5 days

Weekly Capacity = 322.5 bicycles per week

e) Flow Rate and Constraint Analysis:

To determine if the flow rate of the process is capacity-constrained or demand-constrained, we compare the weekly capacity to the demand for each service option.

Demand for Tune-up (\(T_{demand}\)) = 135 bicycles per week

Demand for Additional Services (\(A_{demand}\)) = 45 bicycles per week

Comparing the demands with the weekly capacity:

\(T_{demand}\) < Weekly Capacity (135 < 322.5)

\(A_{demand}\) < Weekly Capacity (45 < 322.5)

Since both the demands for tune-up and additional services are less than the weekly capacity, the flow rate of the process is demand-constrained. This means the shop has the capacity to handle the current demand without operating beyond its limits.

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Answer:

1. False

2. True

3. True

Step-by-step explanation:

The value of the expression tan(-5π/6) can be determined using trigonometric properties and  identities.

First, let's consider the unit circle. The angle -5π/6 is measured in the clockwise direction from the positive x-axis. In this position, the reference angle is π/6, which is the positive angle formed with the x-axis.

The tangent function (tan) is defined as the ratio of the sine (sin) of an angle to the cosine (cos) of the angle. Using the reference angle π/6, we can determine the value of tan(π/6) as the ratio of sin(π/6) to cos(π/6). sin(π/6) equals 1/2, and cos(π/6) equals √3/2. Therefore, tan(π/6) is (1/2) divided by (√3/2), which simplifies to 1/√3 or √3/3. Since tan is an odd function, tan(-5π/6) is equal to the negative of tan(5π/6). Therefore, tan(-5π/6) has the same value as -√3/3.

In conclusion, the value of tan(-5π/6) is -√3/3, which means that the tangent of the angle -5π/6 is negative and equal to the ratio of -√3 to 3.

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Answer:

A line graph, because it would show the association or relationship between two variables

Step-by-step explanation: