Use a unit circle and a 30°-60°-90° triangle to find the degree measures of the angles.

angles whose cotangent is -√3

The area of the square is changing at 12√2 + 2√2t square inches/minute when the circle is increasing at 1 in/min.

Let us find the relationship between r and s using the Pythagorean theorem. Since the square is inscribed in the circle, the diameter of the circle is equal to the diagonal of the square.

2r = s√2
Squaring both sides, we get:
4r² = 2s²or s² = 2r²
Dividing by 2 on both sides, we get:
s²/2 = r²
Differentiating both sides with respect to t, we get:
ds²/dt = 2r (dr/dt)
Dividing both sides by 2s, we get:
ds/dt = r (dr/dt) / s

Substituting r² = s²/2,

\(ds/dt = r (dr/dt) / √2s2s ds/dt = r (dr/dt)s ds/dt = (r/2) (dr/dt)2s ds/dt = r (dr/dt) or dA/dt = 2s ds/dt = 2r (dr/dt)\)

Now, substituting r² = s²/2,

dA/dt = 2s ds/dt = 2(√2 s) (dr/dt) = 2(√2) r (dr/dt)

Now, substituting dr/dt = 1 in/min and r = 6 in (since the circle is increasing at 1 in/min, the radius after t minutes is 6 + t),

dA/dt = 2(√2) (6 + t) (1) = 12√2 + 2√2t square inches/minute

Therefore, the area of the square is changing at a rate of 12√2 + 2√2t square inches/minute.

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a. The teacher's age in seconds can be written in scientific notation as 1.5 × \(10^{9}\) seconds.

b. A more reasonable unit of measurement for this situation could be years, as it is a common unit used to express human age.

c. To convert the teacher's age from seconds to years, we can divide the number of seconds by the number of seconds in a year. There are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, and approximately 365 days in a year. So,

1.5 × \(10^{9}\) seconds ÷ (60 seconds/minute × 60 minutes/hour × 24 hours/day × 365 days/year) = approximately 47.5 years

Therefore, the teacher is approximately 47.5 years old when using the more reasonable unit of measurement.

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Answer:

1.) y = 2x -2

2.) y = 1/3x + 1

3.) y = 1x - 1

4.) y = -3x - 4

B. Yes, multiplying A by the vector writes u as a linear combination of the columns of A.

To check if u is in the subset of R3 spanned by the columns of A, we need to see if there exist scalars x, y, and z such that:
u = x * (1, 13, 7) + y * (2, 3, 2) + z * (1, 2, 1)
This can be written as a matrix equation:
A * x = u
where A is the matrix with columns (1, 2, 1), (13, 3, 2), and (7, 2, 1). Solving for x gives:
x = (1/2) * (u + 2z - 13y, -z + 3y, -u + y + z)
So, if we can find values of y and z such that x has all entries equal to (1/2), then u is in the subset of R3 spanned by the columns of A.
Alternatively, we can check if the augmented matrix [A | u] has a unique solution. We can row reduce this matrix to get:
1 0 1 | 0
0 1 -2 | 0
0 0 0 | 1
Since the last row corresponds to the equation 0 = 1, the system is inconsistent and u is not in the subset of R3 spanned by the columns of A.

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The magnitude of the charge on each sphere is approximately 4.47 x 10^-7 C. Each sphere has an excess of approximately 2.79 x 10^9 electrons.

To determine the magnitude of the charge on each sphere, we can use Coulomb's law. Coulomb's law states that the force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

First, let's convert the mass of each sphere to kilograms. Each sphere has a mass of 5.80 g, which is equal to 0.0058 kg. We know that the acceleration of the spheres is 190.0 m/s^2.

Using Newton's second law, F = ma, where F is the force between the spheres, m is the mass of each sphere, and a is the acceleration, we can calculate the force between the spheres. The force between the spheres is given by F = ma = (0.0058 kg) * (190.0 m/s^2) = 1.102 N.

Now, let's apply Coulomb's law. The force between the spheres is equal to the electrostatic force, given by F = k * (q1 * q2) / r^2, where k is the electrostatic constant, q1 and q2 are the charges on the spheres, and r is the distance between the spheres.

The electrostatic constant, k, is approximately equal to 8.99 x 10^9 N m^2/C^2. The distance between the spheres, r, is given as 5.5 cm, which is equal to 0.055 m.

Rearranging the equation, we can solve for the charges on the spheres, q1 and q2. Plugging in the known values, we have 1.102 N = (8.99 x 10^9 N m^2/C^2) * (q1^2) / (0.055 m)^2.

Solving for q1, we find q1 = sqrt((1.102 N * (0.055 m)^2) / (8.99 x 10^9 N m^2/C^2)). Evaluating this expression, we get q1 ≈ 4.47 x 10^-7 C.

To determine the number of excess electrons, we can use the elementary charge, which is approximately 1.602 x 10^-19 C. The number of excess electrons can be calculated by dividing the charge on each sphere by the elementary charge.

Dividing 4.47 x 10^-7 C by 1.602 x 10^-19 C, we find that each sphere has approximately 2.79 x 10^9 excess electrons.

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we know that 76% of the specialty soaps are made with fresh herbs, and we also know that there are a total of 350 specialty soap bars, so how many are made with fresh herbs?  well, just 76% of those 350

\(\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{76\% of 350}}{\left( \cfrac{76}{100} \right)350}\implies 266\)

Answer:

$69.40

Step-by-step explanation:

$9.50 × 6 = $57

$57 + $12.40 = $69.40

Answer:

The answer is quad 1 and quad 2 are congruent because quad 2 can be created rotating and then translating quad 1

Step-by-step explanation:

Answer:

B. 3/8

Step-by-step explanation:

Answer:

$25.20

Step-by-step explanation:

This is the answer because:

1) 10 + 10 + 5 = 25

2) Next, dimes are equal to 0.10

25 + 0.10 + 0.10 = 25.20

Hope this helps! :D

Answer:

→ First value is 1

→ Second value is 2

→ Third value is 5

Step-by-step explanation:

• let numbers be x, y and z

\({ \tt{z = 8y - 9 - - - (eqn \: 1)}} \\ \\ { \tt{10x = 8y - 7 - - - (eqn \: 2)}} \\ \\ { \tt{x + y + z = 8 - - - (eqn \: 3)}}\)

• from eqn 2, make x the subject:

\( { \tt{x = \frac{8y - 7}{10} }} \\ \)

• substitute all variables in eqn 3:

\({ \tt{ \frac{8y - 7}{10} + y + 8y - 9 = 8}} \\ \\ { \tt{8y - 7 + 10y + 80y - 90 = 80}} \\ \\ { \tt{98y = 177}} \\ \\ { \boxed{ \tt{ \: y = 1.8}}}\)

• find z

\({ \tt{z = 8y - 9}} \\ \\ { \tt{z = 8(1.8) - 9}} \\ \\ { \tt{z = 14.4 - 9}} \\ \\ { \boxed{ \tt{ \: z = 5.4 \: }}}\)

• find x:

\({ \tt{x = \frac{8(1.8) - 7}{10} }} \\ \\ { \tt{x = \frac{7.4}{10} }} \\ \\ { \boxed{ \tt{ \: x = 0.74 \: }}}\)

Rounding to nearest value:

\({ \boxed{ \rm{x = 1}}} \\ { \boxed{ \rm{y =2 }}} \\ { \boxed{ \rm{z = 5}}}\)

Answer:

Step-by-step explanation:

The sum of the angles in a triangle is always 180 degrees. Therefore,

2x + 3x + 22 = 180

Simplifying the equation,

5x = 158

Dividing by 5 on both sides,

x = 31.6

To find the actual measures of 2x and 3x, we substitute the value of x:

2x = 2(31.6) = 63.2 degrees

3x = 3(31.6) = 94.8 degrees

Therefore, the actual measures of 2x and 3x are 63.2 degrees and 94.8 degrees, respectively.

Performing a Chi-Square test is a statistical tool used to determine if there is a significant difference between observed and expected data. The test helps to analyze categorical data by comparing observed frequencies to the expected frequencies. The p-value in a Chi-Square test refers to the probability of obtaining the observed results by chance alone.

If a p-value lower than 0.01 is obtained in a Chi-Square test, it means that the results are statistically significant. In other words, there is strong evidence to reject the null hypothesis, which states that there is no significant difference between the observed and expected data. This means that the observed data is not due to chance alone, but rather to some other factor or factors.

The mean, or average, is not directly related to the Chi-Square test or the p-value. The Chi-Square test is specifically used to determine the significance of the observed data. However, the mean can be used as a measure of central tendency for continuous data, but it is not applicable to categorical data.

In conclusion, obtaining a p-value lower than 0.01 in a Chi-Square test means that there is strong evidence to reject the null hypothesis, and that the observed data is statistically significant.

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The y-coordinate of the robot's position is equal to 18.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):

y - y₁ = m(x - x₁)

Where:

First of all, we would determine the slope of the linear path (line 1);

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (6 - 5)/(6 - 2)

Slope (m) = 1/4

Since the robot turns 90° Counterclockwise and continues to move linearly, we know a perpendicular would be formed;

m₁ × m₂ = -1

1/4 × m₂ = -1

m₂ = -4

Slope, m₂ = -4

At data point (6, 6) and a slope of -4, the y-coordinate of the robot's position can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 6 = -4(x - 6)

y = -4x + 24 + 6

y = -4x + 30

y = -4(3) + 30

y = 18

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Answer:

523.33 ft^3

Step-by-step explanation:

d = 10 => r = 10/2 = 5

The formula for the volume of a sphere is V = 4/3 π r^3

V = 4/3 x 3.14 x 5^3

= 4/3 x 3.14 x 125 = 523.33

Answer:

If you pacifically asking for the fraction 115/12

Step-by-step explanation:

I also looked on mathw it helps

Answer:

0.35, 0.359, 1

Step-by-step explanation:

0.35 would be 0.350 meaning its less than 0.359

The length of RP is given as follows:

RP = 91.3 ft.

The three trigonometric ratios are the sine, the cosine and the tangent of an angle, and they are obtained according to the formulas presented as follows:

For the angle of 18º, we have that:

Hence the length RP is given as follows:

cos(18º) = RP/96

RP = 96 x cosine of 18 degrees

RP = 91.3 ft.

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The dual model for the given primal model is, Maximize W = 55y₁ + 80y₂ + 57y₃ and Subject to 11y₁ + 21y₂ + 31y₃ ≥ 10 and 12y₁ + 22y₂ + 33y₃ ≥ 9, with the nature restrictions: y₁ ≥ 0, y₂ unrestricted, y₃ unrestricted.

To obtain the corresponding dual model for the given primal model

The primal model is:

Minimize Z = 10x₁ + 9x₂ + 8x₃

Subject to:

11x₁ + 12x₂ = 55

21x₁ + 22x₂ + 23x₃ ≤ 80

31x₁ + 33x₃ ≤ 57

x₁ ≤ 0, x₂ ≥ 0, x₃ is unrestricted

a) The dual objective function is obtained by taking the coefficients of the primal model's constraints as the variables in the dual model's objective function. Thus, the dual objective function is:

Maximize W = 55y₁ + 80y₂ + 57y₃

where y₁, y₂, and y₃ are the dual variables corresponding to the primal constraints.

b) The dual constraints are obtained by taking the coefficients of the primal model's variables as the variables in the dual model's constraints. The primal inequalities become the dual equalities and vice versa. Additionally, the sign restrictions of the primal variables become sign restrictions for the dual variables. Thus, the dual constraints are:

11y₁ + 21y₂ + 31y₃ ≥ 10

12y₁ + 22y₂ + 33y₃ ≥ 9

where y₁, y₂, and y₃ are the dual variables corresponding to the primal variables.

c) Nature restrictions of the dual variables:

Since x₁ ≤ 0 in the primal model, the corresponding dual variable y₁ has the restriction y₁ ≥ 0 (non-negativity).

Since x₂ ≥ 0 in the primal model, the corresponding dual variable y₂ has no specific restriction.

Since x₃ is unrestricted in the primal model, the corresponding dual variable y₃ is also unrestricted.

Therefore, the complete dual model is:

Maximize W = 55y₁ + 80y₂ + 57y₃

Subject to:

11y₁ + 21y₂ + 31y₃ ≥ 10

12y₁ + 22y₂ + 33y₃ ≥ 9

y₁ ≥ 0, y₂ is unrestricted, y₃ is unrestricted

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Answer:

10^12

Step-by-step explanation:

The data of brand C suggests that brand C has the fastest clean time because it has the least average time

To do this, we simply calculate the average time of each brand.

From the complete question, we have:

Brand A: 20, 40, 60

Brand B: 30, 60, 90

Brand C: 15, 30, 45

The average is calculated using:

Average = Sum/Count

So, we have:

Brand A = (20 + 40 + 60)/3 = 40

Brand B = (30 + 60 + 90)/3 = 60

Brand C = (15 +30+ 45)/3 = 30

The least clean time is 30 (brand C).

Hence, the data of brand C suggests that brand C has the fastest clean time

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The operation that results in the simplified expression x⁵ + x⁴ - 5x³ - 3x² + x - 8 is; P + Q

We are given the Polynomials as;

P(x) = x⁴ + 3x³ + 2x² - x + 2

Q(x) = (x³ + 2x² + 3)(x² - 2)

We want to find the combination of P and Q that would yield;

x⁵ + x⁴ - 5x³ - 3x² + x - 8

Let us expand Q(x) to get;

Q(x) = x⁵ + 2x⁴ + 3x² - 2x³ - 4x² - 6

Q(x) = x⁵ + 2x⁴ - 2x³ - x² - 6

Now, the combined polynomial shows us that coefficient of x⁴ is 1 and coefficient of x³ is - 5.

By inspection, we can say that the combination that would produce the required result is;

Q(x) - P(x) = x⁵ + 2x⁴ - 2x³ - x² - 6 -  x⁴ - 3x³ - 2x² + x - 2

Q(x) - P(x) = x⁵ + x⁴ - 5x³ - 3x² + x - 8

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The standard form of the equation of the hyperbola is: (x^2 / 36) - (y^2 / (25/144)) = 1

The center of the hyperbola is the midpoint between the vertices, which is (0,0).

Since the transverse axis is horizontal, the standard form is:

(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1

where (h,k) is the center of the hyperbola, a is the distance from the center to a vertex, and b is the distance from the center to a point on one of the asymptotes.

From the given information, we know that a = 12.

To find b, we use the fact that the slope of the asymptotes is equal to b/a, so:

b/a = 5/12

b = (5/12)a

Substituting in the values of a and b, we get:

(x - 0)^2 / 12^2 - (y - 0)^2 / (5/12)^2 = 1

Simplifying, we get:

x^2 / 144 - y^2 / (25/144) = 1

Multiplying both sides by 25/144, we get:

x^2 / 144 * 25/144 - y^2 = 25/144

Simplifying, we get:

x^2 / 36 - y^2 / (25/144) = 1

Therefore, the standard form of the equation of the hyperbola is:

(x^2 / 36) - (y^2 / (25/144)) = 1

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Answer:

720 Should be the answer I think.

Answer:

base: 12

area: 84

Step-by-step explanation:

Please let me know if you want me to add an explanation as to why this is the answer. I can definitely do that, I just wouldn’t want to write it if you don’t want me to :)

If we want to increase  the width of a confidence interval , we need to increase the confidence interval which will increase our z value

What is confidence interval?

In statistics, the probability that a population parameter will fall between a set of values for a predetermined percentage of the time is referred to as the confidence interval. Analysts frequently employ confidence ranges that include 95% or 99% of anticipated observations.

E=zs√n. The margin of error is represented by the quantity E. The margin of error is equal to the confidence interval's width divided by two. Occasionally, confidence intervals are expressed as x±E, for instance, 9.95±0.43.

Thus if we want to increase  the width of a confidence interval , we need to increase the confidence interval which will increase our z value

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The computation given that the weight of steel that was used is 32835 pounds.

From the information given, it was stated that the total weight of the redesigned car was 42,695 pounds and the amount of aluminum used was 9,860 pounds and the remainder was steel.

Therefore, the amount used for steel will be:

= 42695 - 9860

= 32835 pounds.

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