For the following exercises, use substitution to solve the system of equations. Show work and give answer as and order pair.

. 5x + 6y = 14
4x + 8y = 8

A
(-4,1)

B
(4,-1)

C
(-4,-1)

D
(-4,1)

E
None of the above

The trapezoid has only one line of reflectional symmetry.

When we divide the image, the mirror image of one side of the image to the other is known as reflectional symmetry. We can say that one half of the image is the reflection of the other half. Reflection symmetry is also known as mirror symmetry.

In an isosceles trapezoid, the length of the sides is the same which means that the left and the right sides are equal. But, the bases of an isosceles trapezoid are not the same. When a vertical line is drawn in the middle of the isosceles trapezoid, the left side of the image becomes the reflection of the right side. So there is only one line of reflection symmetry.

Therefore, there is only one line of reflectional symmetry in this figure of an isosceles trapezoid.

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The complete question is "The figure is an isosceles trapezoid. How many lines of reflectional symmetry does the trapezoid have? The image is given below"

Answer:

9(6+11)=(9/6)*(9/11)

Step-by-step explanation:

The left side of the equation can be simplified as follows:

9(6+11) = 9(17) = 153

On the right side, we use the fact that the product of two fractions is the product of their numerators over the product of their denominators. So:

(9/6)[6/(9/11)] = (9/6) * (611/9) = 11

Therefore, the equation becomes:

153 = 11

which is not true for any value of the missing numbers in the equation. So there is no solution for the missing numbers.

you have a good day too and you're welcome!

The ONB for the given plane in R3 is ū = [-5/√(26), 1/√(26), 0] and ī = [25/(√(26/13)), -5/(√(26/13)), 0].

To find an orthonormal basis for the plane x + 5y + 4z = 0, we first solve the system and get the parametric solution

x = -5t - 4s

y = t

z = s

Assigning parameters s and t to the free variables and writing the solution in vector form as su + tv, we get

[-5t - 4s, t, s] = t[-5, 1, 0] + s[-4, 0, 1]

Taking u = [-5, 1, 0] and v = [-4, 0, 1], we normalize u to have norm 1 by dividing it by its length

||u|| = √(26)

ū = [-5/√(26), 1/√(26), 0]

To find the component of v orthogonal to u, we take the dot product of v and u, and divide it by the dot product of u and u, and then multiply u by this scalar

v - ((v · u) / (u · u))u

v · u = -5

u · u = 26

v - (-5/26)[-5, 1, 0]

v - [25/26, -5/26, 0]

Finally, we normalize this vector to have norm 1

||v - proj_u v|| = √(26/13)

ī = [25/(2√(26/13)), -5/(2√(26/13)), 0]

Therefore, the orthonormal basis for the plane x + 5y + 4z = 0 is ū = [-5/√(26), 1/√(26), 0] and ī = [25/(√(26/13)), -5/(√(26/13)), 0].

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We find 6 possible 2-digit numbers "ab" that satisfy the conditions: 14, 19, 26, 37, 58, and 69.

To find the number of 2-digit numbers "ab" such that "ab" "ba" is a perfect square and a < b, follow these steps:

1. Iterate through all possible 2-digit numbers "ab" with a < b (e.g., a = 1, b = 2, a = 1, b = 3, etc.).
2. For each "ab", form the 4-digit number "ab" "ba".
3. Check if the 4-digit number is a perfect square (i.e., its square root is an integer).
4. Count the number of "ab" that satisfy the condition.

After performing these steps, we find 6 possible 2-digit numbers "ab" that satisfy the conditions: 14, 19, 26, 37, 58, and 69.

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It is proved that if g is not cyclic, all elements of g have order 1, 2, or 3, and there must be an element of order 3.

To prove that if g is not cyclic, all elements of g have order 1, 2, or 3, and show that there must be an element of order 3, follow these steps,

1. Assume that g is a finite group and is not cyclic.
2. Recall that the order of an element a in group g is the smallest positive integer n such that a^n = e, where e is the identity element in g.
3. If g were cyclic, it would have an element a with order equal to the order of the group itself (|g|). However, we are given that g is not cyclic, so the order of any element in g must be less than |g|.
4. We now consider the possibilities for the order of elements in g. If all elements of g have order 1, then g is the trivial group, which is cyclic, contradicting our assumption.
5. If there is an element of order 2, there must be an element of order 3 as well. This is because, according to Cauchy's theorem, if a prime number p divides the order of a finite group g, then g has an element of order p. Since we have assumed that g is not cyclic, |g| must be divisible by at least two prime numbers. The smallest possible case is when |g| is divisible by the primes 2 and 3.
6. By Cauchy's theorem, since 2 and 3 both divide |g|, there must be elements in g of order 2 and order 3.
7. Therefore, if g is not cyclic, all elements of g have order 1, 2, or 3, and there must be an element of order 3.

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The surface area of the given cylinder is 200.96 square centimeters.

Given that the radius of the cylinder is 4 cm and the height of the cylinder is also 4 cm,

The surface area of the cylinder can be found using the formula:

SA = 2πr² + 2πrh, where r is the radius of the circular base and h is the height of the cylinder.

Substitute given values into the formula to get:

SA = 2π(4)² + 2π(4)(4)

= 2π(16) + 2π(16)

= 32π + 32π

= 64π

= 64(3.14)

= 200.96

Therefore, the surface area of the cylinder is 200.96 square centimeters.

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The complete question is as follows

What is the surface area of the cylinder?

Here, the radius of the cylinder is 4 cm and the height of the cylinder is 4 cm

Apply the formula SA = 2πr² + 2πrh.

The statements that is correct is: C. Both functions are decreasing and have different rates of change.

A function is said to be decreasing if the value of y decreases for every value of x that increases.

In the first function given, as x values increased from 3 to 4, the y value decreases from 3 to 0. So it is a decreasing function.

Rate of change = 3 - 0 / 3 - 4

= 3/-1

= -3.

In the second function, as x values increases from -4 to 0, the y value decreases from 0 to -1. It is also a decreasing function.

Rate of change = change in y / change in x = 0 - (-1) / -4 - 0

= 1/-4

= -1/4.

Therefore, they both have the same rate of change.

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The graph is  both graph A and graph B, because both curves are above the horizontal axis, and their areas are positive

Density curves are visuals that demonstrate the probability distribution of a data set.

It is a liquid, uninterrupted line that illustrates the variability of a constant haphazard element - with the entire locality below the curve accounting for 1.

In other words, the region underneath the curve denotes the likelihood of registering a precise or scope of values inside the depth of the grouping.

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The correct relationship is n² = 2n.

To finish the combinatorial argument by completing Method 2, we should consider the following:

Method 2:
1. As you mentioned, there are n ways for Arshpreet and Meixuan to choose the same flavor.
2. If they pick different flavors, there are a total of n*(n-1)/2 unique combinations, since this accounts for all the possible flavor pairings without double-counting.

Now, let's combine both parts of Method 2:
Total ways = Ways of choosing the same flavor + Ways of choosing different flavors
Total ways = n + n*(n-1)/2

Since both methods should result in the same number of total ways to order ice cream, we set Method 1 equal to Method 2:

n² = n + n*(n-1)/2

By solving this equation, we can verify if the given partial combinatorial argument holds true:

n² = n + n*(n-1)/2
2n² = 2n + n*(n-1)
2n² = 2n + n² - n
n² = 2n

This result shows that the given partial combinatorial argument (n² = n + 2 - 1 (i – 1)) is incorrect, as the correct relationship is n² = 2n.

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The probability of a standard normal variable being less than -2.65 is 0.0040. Therefore, P(x < 979) = 0.0040.

To solve this problem, we use the central limit theorem since we have a large enough sample size.

a) Probability that a single randomly selected value is less than 979 dollars

To find the probability that a single randomly selected value is less than 979 dollars, we standardize the value and use the standard normal distribution:

z = (979 - 1046) / 206 = -0.3233

Using a standard normal distribution table or calculator, we find that the probability of a standard normal variable being less than -0.3233 is 0.3736. Therefore, P(X < 979) = 0.3736.

b) Probability that a sample of size n = 66 is randomly selected with a mean less than 979 dollars

To find the probability that a sample of size n = 66 is randomly selected with a mean less than 979 dollars, we use the central limit theorem.

The mean of the sampling distribution of the sample means is the same as the population mean, which is 1046 dollars. The standard deviation of the sampling distribution of the sample means is the standard error, which is:

SE = σ / sqrt(n) = 206 / sqrt(66) = 25.23

To standardize the sample mean, we use the formula:

z = (x - μ) / SE = (979 - 1046) / 25.23 = -2.65

Using a standard normal distribution table or calculator, we find that the probability of a standard normal variable being less than -2.65 is 0.0040. Therefore, P(x < 979) = 0.0040.

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We used the chain rule and derivative of an inverse tangent function to find [tex]\frac{dy}{dx} &= \frac{d}{dx} \left[\tan^{-1}\left(\frac{2x}{1+x^2}\right)\right][/tex] and got [tex]$\frac{dy}{dx} = 2\left(\frac{1}{1+4x^2}\right)\left(\frac{1}{1+x^2}\right)$[/tex].

To find [tex]\frac{dy}{dx}[/tex] for [tex]$y = \tan^{-1}\left(\frac{2x}{1+x^2}\right)$[/tex], we can use the chain rule and the derivative of the inverse tangent function:

[tex]$\frac{dy}{dx} = \frac{d}{dx}\left[\tan^{-1}\left(\frac{2x}{1+x^2}\right)\right]$[/tex]

[tex]$= \frac{1}{\left(\frac{2x}{1+x^2}\right)^2+1} \cdot \frac{d}{dx}\left[\frac{2x}{1+x^2}\right] \quad $[/tex] [chain rule]

[tex]$= \frac{1}{\left(\frac{2x}{1+x^2}\right)^2+1} \cdot \frac{(1+x^2)\cdot2 - 2x\cdot2x}{(1+x^2)^2} \quad[/tex] [quotient rule]

[tex]$= \frac{2}{1+4x^2} \cdot \frac{1}{1+x^2}$[/tex]

Therefore, [tex]\frac{dy}{dx} = \frac{2}{1+4x^2} \cdot \frac{1}{1+x^2}$[/tex].

(a) To find [tex]\frac{dy}{dx} = (e^x\sqrt{2x})^4$[/tex], we can use the chain rule and the power rule:

[tex]\frac{dy}{dx} = \frac{d}{dx}\left[(e^x\sqrt{2x})^4\right][/tex]

[tex]= 4(e^x\sqrt{2x})^3 \frac{d}{dx}[e^x\sqrt{2x}][/tex]

[tex]= 4(e^x\sqrt{2x})^3 \left(e^x\frac{1}{2}(2x)^{-1/2} + e^x\sqrt{2x}\frac{1}{2}(2x)^{-3/2}\right) \quad[/tex] [chain rule]

[tex]= 2(e^x\sqrt{2x})^2\frac{1+\sqrt{2x}}{x}[/tex]

Therefore, dy/dx = [tex]= 2(e^x\sqrt{2x})^2\frac{1+\sqrt{2x}}{x}[/tex]

(b) To find [tex]\frac{dy}{dx} \quad[/tex] for [tex]\quad e^{xy} + \ln(xy) = 0[/tex] we can use implicit differentiation:

[tex]\quad e^{xy} + \ln(xy) = 0[/tex]

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

[tex](e^{xy})(y + x\frac{dy}{dx}) + \frac{1}{xy}(xy' + y) = 0[/tex]

Simplifying and solving for [tex]\frac{dy}{dx}[/tex], we get:

[tex]\frac{dy}{dx} = \frac{-e^{xy} - \frac{y}{x^2y+1}}{xe^{xy}}[/tex]

Therefore, [tex]\frac{dy}{dx} = \frac{-e^{xy} - \frac{y}{x^2y+1}}{xe^{xy}}[/tex].

To find the indefinite integral of [tex]e^x sin2x[/tex], we can use integration by parts:

Let u = sin2x and [tex]\frac{dv}{dx} = e^x[/tex]. Then du/dx = 2cos2x and v = e^x.

Using the formula for integration by parts, we get:

[tex]\int e^x \sin 2x \ dx = e^x \sin 2x - \int 2e^x \cos 2x \ dx[/tex]

We can now integrate by parts again, letting u = cos2x and dv/dx = e^x. Then du/dx = -2sin2x and v = e^x.

Using the formula again, we get:

[tex]\int e^x \sin 2x \ dx = e^x \sin 2x - 2e^x \cos 2x + 4\int e^x \sin 2x \ dx[/tex]

Rearranging terms and dividing by 5, we get:

[tex]\int e^x \sin 2x \ dx = \frac{e^x}{5} (\sin 2x - 2\cos 2x) + C[/tex]

Therefore, the indefinite integral of e^x sin2x is [tex]\frac{e^x}{5} (\sin 2x - 2\cos 2x) + C[/tex]

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Applying the Friedman test, we conclude that there is evidence that at least two population locations differ at a significance level of 10%, since our calculated [tex]$\chi^2$[/tex] value (979.5) is greater than the critical value (7.81).

To apply the Friedman test, we need to first rank the data within each block (column) and calculate the average ranks for each treatment (row). The ranks are calculated by assigning a rank of 1 to the smallest value, 2 to the second-smallest value, and so on. Ties are given the average rank of the tied values.

Treatment Block 1 Block 2 Block 3 Block 4 Ranks

1 2 1.5 3 3.5 10

2 1 3 5 5 14

3 3 2.5 4 2 11.5

4 2 4 2 1 9

5 1 4.5 1 4.5 11

The Friedman test statistic is calculated as:

[tex]$ \chi^2 = \frac{12}{n(k-1)} \left[ \sum_{j=1}^k \left( \sum_{i=1}^n R_{ij}^2 - \frac{n(n+1)^2}{4} \right) \right] $[/tex]

where [tex]$n$[/tex] is the number of blocks, [tex]$k$[/tex] is the number of treatments, and [tex]$R_{ij}$[/tex] is the rank of the [tex]$j^t^h[/tex] treatment in the [tex]$i^t^h[/tex] block.

In this case, [tex]$n=4$[/tex] and [tex]$k=5$[/tex], so:

[tex]$ \chi^2 = \frac{12}{4(5-1)} \left[ \sum_{j=1}^5 \left( \sum_{i=1}^4 R_{ij}^2 - \frac{4(4+1)^2}{4} \right) \right] $[/tex]

[tex]$ \chi^2 = \frac{3}{2} \left[ (10^2 + 14^2 + 11.5^2 + 9^2 + 11^2) - \frac{4(5^2)}{4} \right] $[/tex]

[tex]$ \chi^2 = \frac{3}{2} \left[ 727 - 50 \right] = 979.5 $[/tex]

The critical value for the Friedman test with [tex]$k=5$[/tex] treatments and [tex]$n=4$[/tex]blocks, at a significance level of [tex]\alpha = 0.1$,[/tex] is:

[tex]$ \chi_{0.1}^2 = 7.81 $[/tex]

Since our calculated [tex]$\chi^2$[/tex] value (979.5) is greater than the critical value (7.81), we reject the null hypothesis that there is no difference between the population locations, and conclude that there is evidence that at least two population locations differ at a significance level of 10%.

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Answer:

Step-by-step explanation:

Solve for x.

4x - 9 = 2x + 5

4x - 2x = 5 + 9

2x = 14

x = 14 : 2

x = 7

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

check   (replace "x" with "7")

4 * 7 - 9 = 2 * 7 + 5                  (remember PEMDAS)

28 - 9 = 14 + 5

19 = 19

the answer is good

Answer:

hence the required value of x is 7.

The statements that are true about the triangle are

a) m∠5 + m∠6 = 180°

b) ∠ 2+ ∠ 3 = ∠ 6

c) m∠2 + m∠3 + m∠5 = 180°

Given data ,

Let the triangle be represented as ΔABC

Now , An exterior angle of a triangle is equal to the sum of the opposite interior angles.

For Exterior ∠ 1 we have

∠ 1 = ∠ 5 + ∠ 3 ( Exterior angle Property of Triangle )

Similarly,

For Exterior ∠ 4 we have

∠ 4 = ∠ 5 + ∠ 2 ( Exterior angle Property of Triangle )

Similarly,

For Exterior ∠ 6 we have

∠ 6 = ∠ 2 + ∠ 3 ( Exterior angle Property of Triangle )

From the triangle sum property , we get

Ina triangle sum of the measures of angles is equal to 180°

m∠2 + m∠3 + m∠5 = 180°

Hence , the triangle is solved

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The complete question is attached below :

A triangle is shown with its exterior angles. The interior angles of the triangle are angles 2, 3, 5. The exterior angle at angle 2 is angle 1. The exterior angle at angle 3 is angle 4. The exterior angle at angle 5 is angle 6. Which statements are always true regarding the diagram? Select three options.
m∠5 + m∠3 = m∠4
m∠3 + m∠4 + m∠5 = 180°
m∠5 + m∠6 =180°
m∠2 + m∠3 = m∠6
m∠2 + m∠3 + m∠5 = 180°

The x-coordinate of the point that the line passes through is x = 0.

We can use the point-slope form of a linear equation to solve this problem.

The slope of the line can be found using the two given points:

slope  (change in y) / (change in x)

slope = (-3 - (-9)) / (12 - x)

slope = 6 / (x - 12)

Now we can use the point-slope form of the linear equation, with the y-intercept of 6:

y - 6 = slope * (x - 0)

Substituting the slope we just found:

y - 6 = (6 / (x - 12)) * x

Simplifying:

y - 6 = 6x / (x - 12)

Multiplying both sides by (x - 12):

y(x - 12) - 6(x - 12) = 6x

Distributing:

xy - 12y - 6x + 72 = 6x

Moving the x terms to one side:

xy - 12y - 12x + 72 = 0

Now we can substitute the y-coordinate of the other given point, (-9), and solve for x:

x(-9) - 12(6) - 12x + 72 = 0

Simplifying:

-9x - 72 - 12x + 72 = 0

-21x = 0

x = 0

Therefore, the x-coordinate of the point that the line passes through is x = 0.

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If the French club is sponsoring a bake sale to raise at least $395. The number of  pastries they must they sell at $2.35 each in order to reach their goal  is: D. at least 168.

Set up an equation:

Total amount raised =Number of pastries x Price per pastry

Let x represent the number of pastries:

x × $2.35 = $395

To solve for x we need to isolate it on one side of the equation

x = $395 / $2.35

x = 168

Based on the above calculation the French club must sell at least 168 pastries to raise at least $395.

Therefore the correct option is D.

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The 2.65 × 10²⁵ atoms of Fluorine into moles of F atoms after conversions are 4.406 moles of F atoms.

To convert 2.65 × 10²⁵ atoms of Fluorine into moles of F atoms, we can use Avogadro's number, which is the number of particles (atoms or molecules) in one mole of a substance. Avogadro's number is approximately 6.022 × 10²³ particles per mole.

The conversion factor between atoms and moles is,

1 mole = 6.022 × 10²³ atoms

In order to convert Fluorine atoms to moles, we may do the following steps: multiply the number of atoms by Avogadro's number to obtain the molecular weight,

2.65 × 10²⁵ atoms / 6.022 × 10²³ atoms/mole

= 4.406 moles

Round the answer to an appropriate number of significant figures, if necessary. Therefore, 2.65 × 10²⁵ atoms of Fluorine is equal to 4.406 moles of F atoms.

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Complete question - Convert 2.65 × 10²⁵ atoms of Fluorine into moles of F atoms.

The perimeter of the square based on the dimensions of the diagonal is 145.27 millimeters.

We will begin with calculating the side of square from the diagonal of square. It will form right angled triangle and hence the formula will be represented as -

diagonal² = 2× side²

Keep the value of diagonal

(37✓2)² = 2× side²

Side² = 2638/2

Side² = 1319

Side = ✓1319

Side = 36.32 millimetres

Perimeter of the square = 4 × side

Perimeter = 145.27 millimeters

Thus, the perimeter of the square is 145.27 millimeters.

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Answer:

x+15 dollars

Step-by-step explanation:

x could be any number, but if you add 15 to x, it would be x+15. Since you don't know what x is, you can't do anything else.

1/5

you take 1/4÷ 1 1/4 and you get 1/5

a. Rounded to the nearest mile, the distance from New York City to Los Angeles is 2556 miles.
b. Rounded to one decimal place, we can expect to use 82.4 gallons of gas driving from New York to Los Angeles.

a. To convert the distance from kilometers to miles, we can use the given unit fraction: 1 mile ≈ 1.6 kilometers. First, set up the conversion using the given distance:

4090 kilometers × (1 mile / 1.6 kilometers)

The kilometers units will cancel out, leaving the result in miles:

4090 / 1.6 ≈ 2556.25 miles

Rounded to the nearest mile, the distance is approximately 2556 miles.

b. To calculate the number of gallons of gas needed, we can use the car's average of 31 miles per gallon. Set up the conversion using the distance in miles:

2556 miles × (1 gallon / 31 miles)

The miles units will cancel out, leaving the result in gallons:

2556 / 31 ≈ 82.5 gallons

Rounded to one decimal place, you can expect to use approximately 82.5 gallons of gas driving from New York to Los Angeles.

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The area of the pizza box measuring 2/3 feet wide by 4/5 feet long is 8/15 square feet.

The shape of the pizza box is a rectangle. The rectangle is a quadrilateral with opposite sides parallel and equal with an equal angle and of 90°.

The area of a rectangle is considered as:

A = L * B

where L is the length

B is the breadth

Given in the question,

L = 4/5 feet

B = 2/3 feet

The area is calculated by multiplying the fractions. For the multiplication of fractions, we multiply the numerators and denominators separately. And final answer is calculated by simplifying the resulting fraction.

A = 4/5 * 2/3

= 4*2 / 5*3

= 8/15 square feet

Thus, the pizza box has an area of 8/15 square feet

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The value of x in the parallelogram is 1/3.

PQRT is a parallelogram

PT = 8x, TR = 6y, QT = 2x + 2, TS = 2y

We have to find the value of x

In a parallelogram the opposite sides are equal

8x=2x+2

Subtract 2x from both sides

6x=2

Divide both sides by 6

x=2/6

x=1/3

Hence, the value of x in the parallelogram is 1/3.

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The value of j is equal to -9.

In Mathematics and Geometry, the slope of any straight line can be determined by using the following mathematical equation;

Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Slope (m) = rise/run

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

By substituting the given data points into the formula for the slope of a line, we have the following;

10 = (-5 - 5)/(-10 - j)

10(-10 - j) = -10

(-10 - j) = -1

j = -10 + 1

j = -9

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Complete Question:

A line with a slope of 10 passes through the points (j, 5) and (-10,-5). What is the value of j?

The equation for the temperature, d, in terms of t is:
d(t) = 9 * cos[(π/12) * (t - 17)] + 80

To create an equation for the temperature, d, in terms of t, we will use the information given: the high temperature of 89 degrees at 5 pm and the average temperature of 80 degrees.

Determine the amplitude (A) of the sinusoidal function.
Amplitude = (High Temperature - Average Temperature)
A = (89 - 80)
A = 9

Determine the period (P) of the sinusoidal function.
Since the temperature pattern repeats every 24 hours, the period is 24 hours.

Determine the horizontal shift (HS) of the sinusoidal function.
Since the high temperature occurs at 5 pm (17 hours since midnight), the horizontal shift is 17 hours.

Determine the vertical shift (VS) of the sinusoidal function.
The vertical shift is the average temperature, which is 80 degrees.

Write the sinusoidal equation for the temperature, d, in terms of t.
Since the temperature reaches its peak (high temperature) at 5 pm, we will use the cosine function, as it starts at its peak value. The general form of the cosine function is:

d(t) = A * cos[(2π/P) * (t - HS)] + VS

Now, plug in the values found in steps 1-4:

d(t) = 9 * cos[(2π/24) * (t - 17)] + 80

So, the equation for the temperature, d, in terms of t is:

d(t) = 9 * cos[(π/12) * (t - 17)] + 80

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1) The statement "When dummy coding qualitative variables, the level with the lowest mean value should always be the base level" is FALSE.

2) we will not divide the p-value by 2.

1. Dummy coding qualitative variables: The statement "When dummy coding qualitative variables, the level with the lowest mean value should always be the base level" is FALSE. The choice of the base level in dummy coding is arbitrary, and it does not have to be the level with the lowest mean value.

2. Testing a quadratic model: Since you are unsure whether the curvature would be negative or positive, the appropriate alternative hypothesis is that the beta for the quadratic term is not equal to zero (i.e., it has an effect). In this case, you will perform a two-tailed test, which means you will not divide the p-value by 2.

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The angles 75° and ∠2 are alternate interior angles.

Option D is the correct answer.

We have,

From the figure,

55°, ∠3, and 75° forms a straight angle.

Alternate angles are pairs of angles formed when a transversal line intersects two parallel lines.

Alternate angles are equal in measure, which means they have the same angle degree value.

So,

75° and ∠2 are alternate angles.

Thus,

75° and ∠2 are alternate angles.

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The distance from the ceiling to the reflection of the light bulb in the mirror is 6ft.

The tangent of the angle of incidence is equal to the tangent of the angle of reflection.

We have,

To find the distance from the ceiling to the reflection of the light bulb in the mirror we can use similar triangles.

Let's call the distance we're trying to find x.

First, we can find the length of the hypotenuse of the triangle formed by the light bulb, the mirror, and the ceiling.

This is equal to the distance from the light bulb to the mirror plus the distance from the mirror to the ceiling, which is:

= 4ft + 8ft

= 12ft.

Next, we can set up the following proportion:

2ft / x = 4ft / 12ft

Cross-multiplying, we get:

4ft x (x) = 2ft x 12ft

Simplifying, we get:

x = 6ft

So,

The distance from the ceiling to the reflection of the light bulb in the mirror is 6ft.

Now,

To prove that the tangent of the angle of incidence is equal to the tangent of the angle of reflection, we can use the law of reflection, which states that the angle of incidence is equal to the angle of reflection. Let's call these angles θ.

We can draw a diagram to represent the situation, with a ray of light hitting a mirror at an angle of incidence:

         |\

         | \

  d      |  \

--------->|   \

         |θ_i \

         |____\

Here, d is the distance from the light source to the mirror, and θi is the angle of incidence.

Using trigonometry, we can express the tangent of θi as:

tan(θi) = opposite / adjacent

In this case, the opposite side is the vertical distance from the light source to the mirror, which is "h" in the diagram.

The adjacent side is the distance from the mirror to the point where the ray of light reflects off the mirror, which is also d.

tan(θ_i) = h / d

After the light reflects off the mirror, it travels at the same angle as the angle of reflection, θr:

         |\

         | \

  d      |  \

--------->|   \

         |θ_i \

         |____\

           \ θ_r

            \

             \

Using the law of reflection, we know that θi = θr.

So we can write:

tan(θr) = opposite / adjacent

The opposite side is now the vertical distance from the mirror to the point where the ray of light reflects off the mirror, which is also h.

The adjacent side is still d since the distance from the mirror to the point where the ray of light reflects off the mirror is the same as the distance from the light source to the mirror.

So, we have:

tan(θr) = h / d

Since θi = θr, we can substitute tan(θi) for tan(θr) in the equation above:

tan(θi) = h / d = tan(θr)

This means,

The tangent of the angle of incidence is equal to the tangent of the angle of reflection.

Thus,

The distance from the ceiling to the reflection of the light bulb in the mirror is 6ft.

The tangent of the angle of incidence is equal to the tangent of the angle of reflection.

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Answer:

104-18=86 and also  18-104=-86

Step-by-step explanation: