find an equation of the tangent plane to the given surface at the specified point. z = y cos(x − y), (7, 7, 7)

The formula for permutations with repetition is a useful tool for counting the number of distinct sequences we can make when we have objects that are repeated.

It allows us to account for the fact that these objects can be arranged in different orders. The formula for permutations with repetition is:

n! / (n_1! * n_2! * ... * n_k!)

where n is the total number of objects in the sequence, n_1 is the number of objects of type 1, n_2 is the number of objects of type 2, and so on until n_k is the number of objects of type k.

This formula works because when we have objects that are repeated, we need to account for the fact that the same objects can be arranged in different orders. The denominator of the formula takes care of this by dividing out the number of permutations for each type of object.

To understand this better, let's consider an example. Suppose we have a sequence of 6 objects, consisting of 2 As, 2 Bs, and 2 Cs. We want to find the number of distinct sequences we can make.

Using the formula, we have:

6! / (2! * 2! * 2!) = 720 / 8 = 90

This means that there are 90 distinct sequences we can make using these 6 objects. To see why this works, let's consider one possible sequence:

A B C A B C

We can rearrange this sequence in different ways by swapping any of the As with each other, any of the Bs with each other, or any of the Cs with each other. Each of these rearrangements will give us a different sequence, but they will all have the same objects in them. The denominator of the formula takes care of this by dividing out the number of ways we can rearrange the As, the Bs, and the Cs separately.

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Answer:

Step-by-step explanation:

To solve the expression (7.2×10^(-1))×(4×10^(-7)), you can multiply the numerical values and combine the exponents.

First, multiply the numerical values:

7.2 × 4 = 28.8

Next, add the exponents of 10:

10^(-1) × 10^(-7) = 10^(-1-7) = 10^(-8)

Putting it all together, the expression simplifies to:

28.8 × 10^(-8)

This is the final answer in scientific notation.

Answer:

No

Step-by-step explanation:

"Looking" at our triangles

If Angle D is congruent to Angle G, and angle E is congruent to Angle H, lets call those the first and second vertices of each triangle.  If we're trying to prove the triangles congruent, we're trying to prove Triangle DEF is congruent to Triangle GHJ (in that order, since we're already given that the first and second angles correspond between triangles).

Could these triangle be congruent?  Yes!  As a quick example, imagine two Equilateral triangles with side lengths 5.  They are congruent, and they do have the first angle and second angle corresponding pairs congruent, and one of those sides from triangle 1 does match the length of one of the sides in triangle 2.  ...and they are congruent triangles.

The problem with the given scenario is that the side they give for triangle 1 (between vertex 1 and 2) is not the corresponding side of triangle 2 (between vertex 2 and 3).  This makes it so that we cannot guarantee that the triangles are congruent.

As can be seen in the attached image, I've made Side DE length 2, and put it on a coordinate plane so it's easy to see.  Side HJ is also definitely length 2.  Clearly, those triangles are not congruent.

When would triangles be congruent?

Going back to the definition of congruent triangles, three corresponding angles and three corresponding sides must be congruent.  That's 6 pairs of parts needed to be proven congruent to prove triangles congruent!

Fortunately, there are a number of theorems that can be proven to lessen the amount of pairs of parts needed to be proven congruent and still guarantee that the resulting triangles must be congruent.

There are exactly 5 cases:

It looks like a bunch of alphabet soup, but notice that all of them require 3 pieces of information (even HL, which is Hypotenuse-Leg... since there's a Hypotenuse, it must be a right triangle, so there are a pair of corresponding congruent right angles that are congruent that they don't talk about)

In order to remember which items work, remember that you MUST know 3 parts, and think about all of the way you could know 3 parts:

Scenario 1: Have all 3 side pairs (SSS)

There is only one way that all three sides can be equal, and yes, that is one of the triangle congruence shortcuts.

Scenario 2: Have 2 side pairs & 1 angle pair

There are two option here.  

Two sides with the angle between them is SAS (note the angle is between them).

Two sides with the angle not between them is ... (well, that can get you in trouble).  Conveniently, that is NOT one of the triangle congruence theorems.  So, stay out of trouble, and don't use this for triangle congruence.

Interestingly, HL, is a special case of this "trouble case".  Note that if you have the Hypotenuse and a leg, that's two Sides.  The right Angle is NOT the angle between those two sides (because the hypotenuse is always across from the right angle).  When that angle is 90 degrees, then the set of two sides and the angle not between them DOES work as sufficient to prove triangle congruence, and it's the only time that it's sufficient.

Scenario 3: Have 1 Side and 2 Angles

Again, there are two option here:

In either scenario, this is sufficient to prove the triangles are congruent.

Scenario 4: Have all 3 angles (AAA)

This definitely won't be enough to prove congruence.  Zoom in.  It's got the same angles, but not the same lengths.

So, only 5 short ways to prove triangles are congruent, and all of them require 3 parts of one triangle corresponding to 3 parts of another, each of the 3 pairs must be congruent.

Answer:

10:23 am

Step-by-step explanation:

We Know

Ms. Brown left at 9:00 am

Her plane landed 1 hour and 23 minutes later.

What time does her plane land?

We Take

9:00 + 1:23 = 10:23 am

So, her plane landed at 10:23 am

The arc length of the curve  x = a(θ − sin θ), y = a(1 − cos θ) from 0 to 2π is 8a.

To find the arc length of the given curve, we need to use the formula:

L = ∫ [a to b] √[1 + (dy/dx)²] dx

Here, a = 0 and b = 2π, and the given parametric equations are:

x = a(θ − sin θ), y = a(1 − cos θ)

So, we need to find dx/dθ and dy/dθ and then substitute these in the above formula. On solving and simplifying, we get L = 8a as the final answer.

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1+b²+b⁴ ​can be resolved into factors (1+b²+b),(1+b²- b).

We will evaluate  1+b²+b⁴​ with the help of algebraic identities,

We can write 1+b²+b⁴ as,

1+b²+b⁴ = 1+(b)²+(b²)²

             = 1+(b)²+(b)²+(b²)²-b² (Adding and subtracting b²)

             = 1+2(b²)+(b²)²-(b)²

             = (1)²+2(b²)(1)+(b²)²- (b)²

Now using the identity (a + b)²= a²+b²+2ab, we have,

1+b²+b⁴ = (1+b²)²- (b)²

             = (1+b²+b)(1+b²- b)  [By using identity a²-b²= (a+b)(a-b)]

Therefore, the correct answer is  1+b²+b⁴​= (1+b²+b)(1+b²- b).

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Answer:

D = 5

Step-by-step explanation:

The line from center of the circle perpendicularly biscets chord. Hence, you can form a right-angled triangle with the radius as the hypotenuse (See picture)

So now you use Pythagoras theorem to solve for d:
a^2 +b^2 =c^2

12^2 +b^2 = 13^2

b^2 = 13^2 - 12^2

b = [tex]\sqrt{13^{2}-12^{2} }[/tex]

b= 5

Hence, d = 5

The measure of the angle S is 60 degrees

To determine the angle, we need to know the six different trigonometric identities in mathematics;

These trigonometric identities are;

We also have that these identities have their ratios, we have that;

sin θ = opposite/hypotenuse

cos θ = adjacent/hypotenuse

tan θ = opposite/adjacent

From the information given, we have that;

Adjacent = 2√3

Hypotenuse= 4√3

Then,

cos S = 2√3/4√3

Divide the values

cos S = 0. 5

find the inverse

S = 60 degrees

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The correct one is Eden, the exponents should be added.

The mistakes are that Erin multiplies the exponents and Emma multiplies the bases.

Remember that when we have the product of two powers with the same base, we just need to add the exponents, so:

[tex]x^n*x^m = x^{n + m}[/tex]

With that in mind, we can see that the one who did the operation correctly is Eden, because:

[tex]6^2*6^5 = 6^{2 + 5} = 6^7[/tex]

The mistake for Erin is that she multiplied the exponents, while the problem for Emma is that she also multiplied the bases.

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The height of the pyramid from the base and the volume is h = 3V/B

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

Volume and base area are known

This means that we make the height the subject of the formula

The volume of a pyramid is calculated as

V = 1/3Bh

Where

B  Base area

h - height

So, we have

h = 3V/B

By making the height the subject of the formula

Hence, the height of the pyramid from the base and the volume is h = 3V/B

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Answer:

D

Step-by-step explanation:

Usr the equation they gave you and plug in the number given for x and y.

y=2x+4

4=2(0)+4. 4=0+4. 4=4

8=2(2)+4. 8=4+4. 8=8

12=2(4)+4. 12=8+4. 12=12

16=2(6)+4. 16=12+4. 16=16

If you did that method with the other options they don't equal each other such as option B for example

0=2(0)+4. 0=0+4. 0=4

4=2(2)+4. 4=4+4. 4=8

These don't equal each other .

If a level of significance is used in a two-tail hypothesis test, and zstat= 1.58, we will fail to reject the null hypothesis.

In a two-tail hypothesis test, we test for the possibility of the sample mean being significantly different from the population mean in both directions, i.e., in both tails of the normal distribution. The level of significance is the probability of rejecting the null hypothesis when it is true. If the calculated test statistic (in this case, zstat= 1.58) falls outside the critical region, i.e., the region of rejection, then we fail to reject the null hypothesis. In other words, we cannot conclude that the sample mean is significantly different from the population mean. Therefore, the correct decision in this case is to fail to reject the null hypothesis.

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Based on the given assumptions, we can assume that the model predicts the number of disasters at a given temperature rise linearly. However, it is important to note that this model is based on certain assumptions that may or may not be accurate.

These assumptions include the mean value being linear in temperature rise and the actual number of disasters being normally distributed with constant variance.

Given the true value of 16 that was left out, we can use the standard error of prediction and the prediction from the regression line to calculate a t statistic. The null hypothesis is that the point belongs to this line under these assumptions.

To calculate the t statistic, we can use the formula:

t = (observed value - predicted value) / standard error of prediction

Using the given information, we can calculate the t statistic as:

t = (16 - predicted value) / standard error of prediction

Once we have the t statistic, we can calculate the p-value using a t-distribution table or a statistical software. The p-value represents the probability of getting a t statistic as extreme or more extreme than the one we calculated under the null hypothesis.

Based on the p-value, we can determine if we reject or fail to reject the null hypothesis. If the p-value is less than the significance level (usually 0.05), we reject the null hypothesis and conclude that the point does not belong to the line. If the p-value is greater than the significance level, we fail to reject the null hypothesis and conclude that the point belongs to the line.

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Therefore, the equation of the set of all points equidistant from A and B is 5x - 3y - 3z = 0. This represents a plane perpendicular to the line segment AB.

The set of all points equidistant from points A(-1, 4, 2) and B(4, 1, -1) forms a plane perpendicular to AB.

To find the equation of this plane, we can use the midpoint formula to find the coordinates of the midpoint M between A and B, and then use the vector AB as the normal vector for the plane.

Midpoint M:

M = ((-1 + 4) / 2, (4 + 1) / 2, (2 - 1) / 2) = (1.5, 2.5, 0.5)

Vector AB:

AB = B - A = (4 - (-1), 1 - 4, -1 - 2) = (5, -3, -3)

Now, we can write the equation of the plane in point-normal form:

(x - 1.5, y - 2.5, z - 0.5) · (5, -3, -3) = 0

Expanding the dot product, we get:

5(x - 1.5) - 3(y - 2.5) - 3(z - 0.5) = 0

Simplifying:

5x - 7.5 - 3y + 7.5 - 3z + 1.5 = 0

5x - 3y - 3z = 0

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The amount of the substance that will be left after 5 half life is 5.94 grams

Radioactive decay is the process by which an unstable atomic nucleus loses energy by radiation.

The time required for half of the original population of radioactive atoms to decay is called the half-life.

Half life = 1/2

5 half life = 1/2 × 1/2 × 1/2 × 1/2 × 1/2

= 1/32

This means 1/32 of tht original mass will be left after 5 half life.

Therefore;

The mass of the substance left = 190 × 1/32

= 5.94 grams

therefore 5.94g of the mass of the substance will be left.

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However, if the decisions drawn from both methods differ (e.g., one rejects the null hypothesis while the other does not), then the conclusions are not the same.

In order to compare the conclusions obtained using the p-value method with your initial conclusion, it is essential to first understand the p-value's role in hypothesis testing. The p-value is a probability that helps in determining the statistical significance of your test results. A smaller p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, leading to its rejection in favour of the alternative hypothesis. Conversely, a larger p-value (typically > 0.05) suggests weak evidence against the null hypothesis, and therefore, it cannot be rejected.
When comparing the conclusions, assess if both methods lead to the same decision regarding the null hypothesis. If your initial conclusion rejects the null hypothesis and the p-value method yields a p-value ≤ 0.05, then the conclusions are the same. Alternatively, if your initial conclusion does not reject the null hypothesis and the p-value method results in a p-value > 0.05, the conclusions also align.
However, if the decisions drawn from both methods differ (e.g., one rejects the null hypothesis while the other does not), then the conclusions are not the same. In such cases, it is crucial to revisit the methodology and ensure the correctness of the approach taken, as it is vital to have consistent and accurate results in any statistical analysis.

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The percentage of people that spend between 41 and 56 minutes watching TV shows on this streaming service is: 13.5%

The formula for the z-score here is:

z = (x' - μ)/σ

where:

x' is sample mean

μ is population mean

σ is standard deviation

Thus, for:

σ = 15

μ  = 71

x' = 41, we have:

z = (41 - 71)/15

z = -30/5

z = -6

for:

σ = 15

μ  = 71

x' = 56, we have:

z = (56 - 71)/15

z = -15/5

z = -3

From online p-value from 2 z-scores calculator, we have the p-value as:

P(-6 < z < -3) = 0.135

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Answer:

Angles 4 and 8 do not equal 180 degrees (supplementary), therefore letter A is the answer.

Step-by-step explanation:

Looking at the other angles they all equal 180 degrees except for A which would appear to equal more than 180.

The relation ∼ on R × R, given by (x1,y1) ∼ (x2,y2) ⇔ (x1)²(y1)² = (x2)²(y2)², is an equivalence relation.

To show that the given relation is an equivalence relation, we need to prove that it satisfies the three conditions of reflexivity, symmetry, and transitivity.

Reflexivity: For any (x,y) in R × R, we have (x)²(y)² = (x)²(y)², which implies that (x,y) ∼ (x,y).

Symmetry: If (x1,y1) ∼ (x2,y2), then (x1)²(y1)² = (x2)²(y2)². This implies that (x2)²(y2)² = (x1)²(y1)², and hence (x2,y2) ∼ (x1,y1).

Transitivity: If (x1,y1) ∼ (x2,y2) and (x2,y2) ∼ (x3,y3), then (x1)²(y1)² = (x2)²(y2)² and (x2)²(y2)² = (x3)²(y3)². Multiplying these equations, we get (x1)²(y1)² = (x3)²(y3)², which implies that (x1,y1) ∼ (x3,y3).

Since the given relation satisfies all three conditions, it is an equivalence relation.

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If the actual temperature of steam is 99 °C and is assumed to be 100 °C in calculations, the error introduced can be significant depending on the specific calculations being performed.

Similarly, if the actual temperature of steam is 101 °C and is assumed to be 100 °C, the error introduced can also be significant.

The amount of error introduced when assuming the temperature of steam to be 100 °C instead of the actual temperature of 99 °C or 101 °C depends on the specific calculations being performed.

Similarly, in industrial processes, assuming an incorrect temperature of steam could result in inefficiencies or even safety hazards. Therefore, the difference between the actual temperature of steam and the assumed temperature of 100 °C should not be considered negligible in many cases.

In conclusion, the error introduced by assuming a temperature of 100 °C instead of the actual temperature of steam depends on the specific calculations being performed and can be significant in some cases. It is always best to use accurate and precise measurements in scientific and engineering calculations to minimize the potential for error.

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Complete Question:

1. If you use 100 °C for the temperature of steam in your calculations, how much error do you introduce if it is actually 99 °C or 101 °C? Is this difference negligible?

Answer: $11

Step-by-step explanation:

88/8 = 11

Answer:

m^2-2 and -6x^3+7x^2-x+1

Answer

D. 126 inches squared

Step-by-step explanation:

8 x 17 = 136

2 x 5 = 10

136-10= 126

Answer:

1. D.

2. b

Step-by-step explanation:

1.Change to 2.375 to a fraction

2.735=19/8

2. 6^2=36

    8^2=64

    36+64=10^2

36+64=100

100=100

Question 2:

Choose D.

[tex]\frac{19}{8} = 2\frac{3}{8} =2.375[/tex]

Look at the fraction 19/8.

The numerator (19) is greater than the denominator (8). That means we have a number > 1.

So how many 8's are in 19?

Well, 8 x 2 = 16. So that would be 16/8 = 2.

Then we'd have 3 leftover because 19-16 = 3.

That's how we know that 19/8 = 2 3/8.

Are you allowed to use a calculator? Because if so, 3/8 = 0.375. That's the fastest way to figure it out!

If not, we can do it the old fashioned way (ha). 3/8 is halfway between 2/8 and 4/8.

2/8 = 1/4 = 0.25 (1/4 = one quarter, just like a quarter worth 0.25)

4/8 = 1/2 = 0.50 (1/2 = a half, a half dollar, 0.50)

0.25 + 0.50 = 0.75 >>> Now divide that by 2 to find the halfway between 0.25 and 0.50 = 0.75/2 = 0.375.

^^^^ That's just a workaround way of figuring out 3/8 as a decimal if you can't use a calculator.

Question 3: Use the pythagorean theorem. a^2 + b^2 = c^2.

C is the hypotenuse, aka the LONGEST length of the right triangle.

If the numbers work in that formula, it's a right triangle. If not, it's NOT a right triangle.

Marissa: 5, 12, 13

Does 5^2 + 12^2 = 13^2?

25 + 144 = 169

13^2 = 169

So YES, Marissa can make a right triangle.

Berta: 6, 8, 10

6^2 + 8^2 = 100 = 10^2

So YES, Berta can make a right triangle.

Corrin: 12, 16, 20

12^2 + 16^2 = 400 = 20^2

So YES, Corrin can make a right triangle.

So the answer is D, all 3 can make right triangles.

Answer:

Step-by-step explanation:If triangle BCD is dilated by a scale factor of 3/4 to form triangle B’C’D’, and side CD measures 10, then the measure of side C’D’ would be 7.5.

The probability of randomly selecting a marble that is NOT red is C. P(not red) = 11/20.

There are 9 red marbles out of a total of 20 marbles. Thus, the probability of selecting a red marble is 9/20.

To find the probability of selecting a marble that is NOT red, we can subtract the probability of selecting a red marble from 1 (since the probability of selecting any marble must be 1).

P(not red) = 1 - P(red) = 1 - 9/20 = 11/20

Therefore, the probability of randomly selecting a marble that is NOT red is C. P(not red) = 11/20.

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The second term of the sequence is given as follows:

1.5.

We use a recursive rule for the sequence, as we have the rule to obtain the (n + 1)th term considering the nth term.

The (n+1)th term rule of a sequence is add 7 then multiply by 2, hence the recursive formula is given as follows:

[tex]a_{n+1} = 2(a_n + 7)[/tex]

The third term for this problem is given as follows:

[tex]a_3 = 17[/tex]

Hence the second term is obtained as follows:

[tex]a_{3} = 2(a_2 + 7)[/tex]

[tex]2a_2 + 14 = 17[/tex]

[tex]2a_2 = 3[/tex]

[tex]a_2 = 1.5[/tex]

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One macroeconomic problem that could cause the Federal Reserve to become aggressive in raising interest rates is inflation.

Inflation is the increase in the prices of goods and services over time, which erodes the purchasing power of money. When inflation is high, people tend to spend less and save more, which slows down economic growth.

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the degree 5 Taylor polynomial of f(x) = cos(x) at a = 0 is T₅(x) = 1 - (1/2!)x² + (1/4!)x⁴.

What is Taylor series?

The Taylor series is a representation of a function as an infinite sum of terms involving the function's derivatives at a particular point. It provides a way to approximate a function with a polynomial expression.

To find the degree 5 Taylor polynomial, denoted as T₅(x), of the function f(x) = cos(x) at a = 0, we need to compute the polynomial using the derivatives of f evaluated at x = 0.

The Taylor polynomial of degree n for a function f(x) centered at a is given by:

Tₙ(x) = f(a) + f'(a)(x - a) + (f''(a)/2!)(x - a)² + ... + (fⁿ(a)/n!)(x - a)ⁿ

In this case, a = 0, so the Taylor polynomial simplifies to:

T₅(x) = f(0) + f'(0)x + (f''(0)/2!)x² + (f'''(0)/3!)x³ + (f⁴(0)/4!)x⁴ + (f⁵(0)/5!)x⁵

Now, let's calculate the derivatives of f(x) = cos(x) up to the 5th order:

f'(x) = -sin(x)

f''(x) = -cos(x)

f'''(x) = sin(x)

f⁴(x) = cos(x)

f⁵(x) = -sin(x)

Evaluating these derivatives at x = 0:

f(0) = cos(0) = 1

f'(0) = -sin(0) = 0

f''(0) = -cos(0) = -1

f'''(0) = sin(0) = 0

f⁴(0) = cos(0) = 1

f⁵(0) = -sin(0) = 0

Substituting these values into the Taylor polynomial formula:

T₅(x) = 1 + 0x + (-1/2!)x² + (0/3!)x³ + (1/4!)x⁴ + (0/5!)x⁵

Simplifying the expression:

T₅(x) = 1 - (1/2!)x² + (1/4!)x⁴

Therefore, the degree 5 Taylor polynomial of f(x) = cos(x) at a = 0 is T₅(x) = 1 - (1/2!)x² + (1/4!)x⁴.

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The coordinates for E from the parallelogram CDEH are (5, 1).

We know that diagonals of parallelograms bisect each other. Therefore coordinates of the midpoint of CE and DH will be the same.

Here, (x, y) = [(2+8)/2, (5+1)/2]

= (5, 3)

Let the coordinates of E be (a, b)

Now, (5, 3) = [(5+a)/2, (5+b)/2]

(5+a)/2 =5 and (5+b)/2 = 3

5+a=10  and  5+b=6

a=10-5           b=6-5

a=5                b=1

Therefore, the coordinates for E from the parallelogram CDEH are (5, 1).

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