Many trig equations have an infinite number of solutions (ie an infinite number of angles θ that satisfy the equation). An example to consider is the equation sin θ= 1/2. Choose an equation that corresponds to f(θ)=cosθ that also has an infinite number of solutions.
a) What are the solutions?
b) How can you locate all of them?
c) Make connections to angles in standard position, as well as to graphical representations of trig functions.

9514 1404 393

Answer:

  (2, 3)

Step-by-step explanation:

The coefficients of x are opposites, so the x-terms can be eliminated by adding the two equations together:

  (2x+3y) +(-2x+5y) = (13) +(11)

  8y = 24 . . . simpify

  y = 3 . . . . . divide by 8

Only one answer choice has y = 3: (x, y) = (2, 3).

__

If you like, you can finish the solution by substituting into one of the equations.

  2x +3(3) = 13

  2x = 4 . . . . . . . subtract 9

  x = 2 . . . . . . . . divide by 2

The solution is (x, y) = (2, 3).

_____

A graph confirms this solution.

Answer:

\(m=\frac{-5}{4}\)

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Algebra I

Step-by-step explanation:

Step 1: Define

Find points from graph.

Point (0, 2)

Point (-4, 7)

Step 2: Find slope m

Answer: x = -2

Step-by-step explanation: vertical lines are the x axis

Answer:

Step-by-step explanation:

A vertical line is of the form x=k where k is a constant of the value of x.

In this case we have the point (-2,-9) so

x=-2

To solve the congruence 4x ≡ 5 (mod 9), we need to find the inverse of 4 modulo 9, which we found in part (a) of exercise 5 to be 7.

Multiplying both sides of the congruence by the inverse of 4, we get:

4x * 7 ≡ 5 * 7 (mod 9)

28x ≡ 35 (mod 9)

Since 28 ≡ 1 (mod 9), we can simplify the left side of the congruence:

x ≡ 35 (mod 9)

Now we need to find the smallest non-negative integer solution for x. We can do this by repeatedly subtracting 9 from 35 until we get a number less than 9:

35 - 9 = 26
26 - 9 = 17
17 - 9 = 8

So x ≡ 8 (mod 9) is the smallest non-negative integer solution to the congruence 4x ≡ 5 (mod 9) using the inverse of 4 modulo 9 found in part (a) of exercise 5.

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The equivalent expression for the new price of the item is 0.57b. With b be the original price of the item.

The original price of the item is represented by "b". Paul received a coupon for "43 percent" off one item at a clothing store. This means that the new price of the item will be the "original price" minus the discount. The discount is calculated by multiplying the original price by the percentage discount.
So, the discount is:
43 percent of b = 0.43 * b

The new price of the item is the original price minus the discount:
b - 0.43 * b = b - 0.43b

This expression can be simplified by combining the like terms. The like terms in this expression are b and -0.43b.
To combine like terms, we add the coefficients of the like terms:
1b + (-0.43)b = 0.57b

So, the equivalent expression for the new price of the item is:
0.57b

This means that the new price of the item is 57% of the original price. Therefore, the equivalent expression for the new price of the item is 0.57b.

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Answer:

8x+30

Step-by-step explanation:

The required probabilities are as follows:

(a) P(A') = 0.6

(b) P(A∪B) = 0.5

(c) P(A'∩B) = 0.1

(d) P(A∩B') = 0.3

(e) P[(A∪B)'] = 0.5

(f) P(A'∪B) = 0.7

Let's determine the probabilities using the provided information:

(a) P(A'): This represents the probability of the complement of event A, which is everything that is not in A.

P(A') = 1 - P(A) = 1 - 0.4 = 0.6

(b) P(A∪B): This represents the probability of either event A or event B occurring.

P(A∪B) = P(A) + P(B) - P(A∩B) = 0.4 + 0.2 - 0.1 = 0.5

(c) P(A'∩B): This represents the probability of the intersection of the complement of event A and event B.

P(A'∩B) = P(B) - P(A∩B) = 0.2 - 0.1 = 0.1

(d) P(A∩B'): This represents the probability of the intersection of event A and the complement of event B.

P(A∩B') = P(A) - P(A∩B) = 0.4 - 0.1 = 0.3

(e) P[(A∪B)']: This represents the probability of the complement of the union of event A and event B.

P[(A∪B)'] = 1 - P(A∪B) = 1 - 0.5 = 0.5

(f) P(A'∪B): This represents the probability of the union of the complement of event A and event B.

P(A'∪B) = P(A') + P(B) - P(A'∩B)

          = 0.6 + 0.2 - P(A'∩B)  (Note: P(A'∩B) is obtained in part (c))

Substituting the value of P(A'∩B) from part (c):

P(A'∪B) = 0.6 + 0.2 - 0.1 = 0.7

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Answer:

5/9

Step-by-step explanation:

we add all of the values and divide by the total number in this case 9, all of them add to make 5 so we do 5÷9 to get 5/9

Step 1. You need x isolated.
\(x < 13+7\)

When you move a number from the left to right (and its addition or subtraction, the sign stays (< ≤ > and ≥ is signs).
x<20 or in words, x is less than 20, and cannot be 20.

Answer:

x < 20

Any number less than 20 is a solution

Step-by-step explanation:

x-7 < 13

The first step to solve this inequality is to add 7 to each side

x-7+7 < 13+7

x < 20

Any number less than 20 is a solution

False. The expected cell frequency in statistical analysis, specifically in the context of contingency tables and chi-square tests, is not based on the researcher's opinion. Instead, it is determined through mathematical calculations and statistical assumptions.

In contingency tables, the expected cell frequency refers to the expected number of observations that would fall into a particular cell if the null hypothesis of independence is true (i.e., if there is no relationship between the variables being studied). The expected cell frequency is calculated based on the marginal totals (row totals and column totals) and the overall sample size.

The expected cell frequency is computed using statistical formulas and is not influenced by the researcher's opinion or subjective judgment. It is a crucial component in determining whether the observed frequencies in the cells significantly deviate from what would be expected under the null hypothesis.

By comparing the observed cell frequencies with the expected cell frequencies, statistical tests like the chi-square test can assess the association or independence between categorical variables in a data set.

Thus, the statement "the expected cell frequency is based on the researcher's opinion" is false. The expected cell frequency is derived through statistical calculations and is not subject to the researcher's subjective input.

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Answer:

Step-by-step explanation:

It drops 235 feet per move.

235 * 4 = 1140 feet below sea level.

-1140 is also possible.

Answer: C

Step-by-step explanation:

You would need to divide the 5z and 3 by 4. Putting 5z and 3 as the numerator in a fraction with 4 as the denominator is the same because you’re dividing both values by 4. I think I may have explained it badly, so if someone has a better explanation, can you add that in the comments or in an answer?

Answer:

Two packs of buns and three hotdog packages

Step-by-step explanation:

The total area under a probability density function is always equal to 1.

This means that the probability of any possible outcome occurring in the given probability distribution is 1, or 100%. The area under the probability density function represents the likelihood of a random variable falling within a particular range of values, and the total area represents the probability of all possible outcomes.

The probability density function can be used to calculate the probability of a random variable taking on a specific value, as well as the probability of a random variable falling within a certain range of values. The area under the probability density function can be calculated using integration techniques.

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The range of k for real roots is k ≥ 0.

For the characteristic equation s^2 + 4s + k = 0, the range of k should be greater than or equal to zero to ensure all the roots are real.

The characteristic equation of a control system is given as s^2 + 4s + k = 0, where s represents the complex variable and k is a constant term. To have real roots, the discriminant of the equation (b^2 - 4ac) must be greater than or equal to zero. In this case, the discriminant is 4^2 - 4(1)(k) = 16 - 4k. For real roots, this should be greater than or equal to zero. Solving the inequality 16 - 4k ≥ 0, we find k ≤ 4. Therefore, the range of k for real roots is k ≥ 0.

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The values of the given expression having exponent with negative fractional base i.e. \(-2^{(3/2)^2}\\\) is evaluated out to be 16/9.

First, we need to simplify the expression inside the parentheses using the rule that says "exponents with negative fractional base can be rewritten as a fraction with positive numerator."

\(-2^{(3/2)^2}\) = (-2)² × (2/3)²

Now, we can simplify the expression  further by solving the exponent of (-2)² and (2/3)²:

(-2)² × (2/3)² = 4 × 4/9 = 16/9

Therefore, the value of the given expression \(-2^{(3/2)^2}\\\)  is 16/9.

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The question is :

What is the value of the expression \(-2^{(3/2)^2}\) ?

Answer:

He has a lot of money

Step-by-step explanation:

a)  The direction in which you should walk to descend fastest is: (-12, -16)

b) The slope of your path is: -88

c) The direction of the greatest increase in temperature at the point (−2, 2) is: (-4, 4)

The maximum rate of change is: 4√2

d) The direction of the greatest decrease is: (4, -4).

The most negative rate of change is: 4√2

(a) The equation on the surface is:

z = 15 - (3x² + 2y²)

The gradient of this surface will be the partial derivatives of the equation. Thus:

Gradient of the surface z:

∇z = (-6x, -4y)

Since you are standing above the point (2,4), then the direction to descend fastest is:

∇z(2,4) = (-6(2), -4(4))

∇z(2,4) = (-12, -16)

That gives us the direction to descend fastest is in the direction.

(b) If you start to move in the direction (-12, -16) above, then slope of your path (rate of descent) is given by the dot product expressed as:

Slope = ∇z(2,4) · (-12, -16)

= (2)(-12) + (4)(-16)

= -24 - 64

= -88

(c) We want to find the direction of the greatest increase in temperature at the point (−2,2).

Thus, the gradient of T(x,y) is given by:

∇T = (2x, 2y).

The direction is:

∇T(-2, 2) = (2(-2), 2(2))

∇T(-2,2) = (-4, 4)

The maximum rate of change is:

∇T(-2,2) = √((-4)² + 4²)

= √(16 + 16)

= √(32)

= 4√2

(d) The direction of the greatest decrease is:

(-∇T(-2, 2)) = (-(-4), -4)

= (4, -4).

The most negative rate of change is:

∇T(-2, 2) = √(4² + (-4)²)

= √(16 + 16)

= √(32)

= 4√2

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The dimensions that will produce the maximum enclosed area for two adjacent rectangular safe play areas of equal size with 200 feet of fencing is 100 feet by 100 feet.

To solve this problem, use the area of a rectangle formula A = lw, where l is the length and w is the width.

Since the two play areas must be of equal size, let's call the length and width l and w, respectively.

We know that the total length of fencing is 200 feet, so l + w = 200 feet.

We can then solve for l by rearranging the equation as l = 200 - w.

We then substitute this into the area equation to get A = (200 - w)w.

To maximize the area, differentiate this equation to get A' = w - 200.

Set this to 0 and solve for w to get w = 100. Since l + w = 200, l = 100 as well.

Therefore, the maximum enclosed area is produced by dimensions of l = 100 feet and w = 100 feet.

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Answer:

Dimensions: 11 meters by 27 meters

2. \(76=2w+2(2w+5)\)

Step-by-step explanation:

Finding Dimensions:

The perimeter of a rectangle can be written as \(2l+2w\) where \(l\) is the length and \(w\) is the width. It is given that the length is 5 more than twice the length of width and the perimeter of the rectangle is 76 meters.

With this information we can write:

Length = \(2w+5\)

\(2(2w+5)+2w = 76\) (perimeter of the rectangle)

\(4w+10+2w=76\)

\(6w=66\)

\(w=11\)

Length = \(2(11)+5\)

Length = \(27\)

∴ The dimensions of the rectangle is 11 meters by 27 meters.

Perimeter

Since the length can be written as \(2w+5\), the equation that represents the perimeter of the rectangle is \(76=2w+2(2w+5)\)

The distance between the two points (5,6) and (10,12) is \(\sqrt{61}\)

What is the distance between two points formula?

The length of the line segment bridging two points on a plane is known as the distance between the points. d=((x2 - x1)2 + (y2 - y1)2) is a standard formula to calculate the distance between two points. This equation can calculate the separation between any two locations on an x-y plane.

Solution explained:

First, Write the distance formula:

\(\sqrt{(x2 - x1)^{2} + (y2 - y1)^{2}}\)

Now,

Substitute (5,6) and (10,12) into \(\sqrt{(x2 - x1)^{2} + (y2 - y1)^{2}}\) :

\(\sqrt{(10 - 5)^{2} + (12 - 6)^{2}}\)      Calculate

\(\sqrt{(5)^{2} + (6)^{2}}\)                      Calculate

\(\sqrt{25 + 36}\)                            Calculate

Hence. the distance between points (5,6) and (10,12) is \(\sqrt{61}\).

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The pharmaceutical company uses the quantitative variable to conducts an experiment.

Variables:

A variable is a characteristic of an object. Their values may occur more than once for a set of data. We consider just two main types of variables in this course.

Quantitative Variables - Variables whose values result from counting or measuring something.

Qualitative Variables - Variables that are not measurement variables. Their values do not result from measuring or counting.

Given,

A pharmaceutical company conducts an experiment in which a subject takes 75 mg of a substance orally.

And  the researchers measure how many seconds it takes for one quarter of the substance to exit the bloodstream.

Here we need to find the kind of variable is the company​ studying.

Here the company uses the Qualitative Variable because here they measure the effects of the medicine not the measurement.

So, the experiment uses the Qualitative Variable.

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The frequency of A allele after a single generation of mutation can be found as follows: Frequency of A allele after a single generationp(A) = p(A) x (1 - m) + q(a) x m

where,
m = mutation rate = 10^-6p(A) = frequency of A allele in initial generation = 0.75q(a) = frequency of a allele in initial generation = 0.25Thus,p(A) = 0.75 x (1 - 10^-6) + 0.25 x 10^-6 = 0.74999925

And the frequency of a allele will beq(a) = 1 - p(A) = 1 - 0.74999925 = 0.25000075

Therefore, the frequency of the A and a alleles in the next generation will beA: 0.74999925 and a: 0.25000075.

This is option D.

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ANSWER

EXPLANATION

We want to finish the synthetic division given:

-2| 2 2 -2 4

Step 1:

Drop down 2:

-2| 2 2 -2 4

|

2

Step 2: Mutliply -2 by the 2 that was dropped down:

-2| 2 2 -2 4

|+ -4

2

Step 3: Add 2 and -4:

-2| 2 2 -2 4

|+ -4

2 -2

Step 4: Repeat (2) and (3) with -2 and -2

-2| 2 2 -2 4

|+ -4 4

2 -2 2

Step 5: Repeat (2) and (3) with -2 and 2:

-2| 2 2 -2 4

|+ -4 4 -4

2 -2 2 0

It is important to note that since we started with 4 coefficients (2, 2, -2 and 4), the largest power of the expression we divided was 3 i.e.:

This means that the largest power of our quotient is 2.

So, the answer is:

Option D

The intervals are nested, each subsequent interval is contained within the previous one. Mathematically, this means I₁ ⊇ I₂ ⊇ ... In ⊇ ... . Therefore, we have:

1. I₁ ⊇ I₂ implies [a₁; b₁] ⊇ [a₂; b₂], which means a₁ ≤ a₂ and b₁ ≥ b₂.
2. I₂ ⊇ I₃ implies [a₂; b₂] ⊇ [a₃; b₃], which means a₂ ≤ a₃ and b₂ ≥ b₃.

To show that a1 ≤ a2 ≤ ... ≤ an ≤ ..., we need to use the fact that the sequence of intervals is nested, meaning that each interval is contained within the next one.

First, we know that I1 contains I2, so every point in I2 is also in I1. That means that a1 ≤ a2 and b1 ≥ b2.

Now consider I2 and I3. Again, every point in I3 is also in I2, so a2 ≤ a3 and b2 ≥ b3.

We can continue this process for all the intervals in the sequence, until we reach In. So we have:

a1 ≤ a2 ≤ ... ≤ an-1 ≤ an

and

b1 ≥ b2 ≥ ... ≥ bn-1 ≥ bn

This shows that the endpoints of the intervals are ordered in the same way.

Given that I₁ ⊇ I₂ ⊇ ... In ⊇ ... is a nested sequence of intervals and In = [an; bn], we can show that a₁ ≤ a₂ ≤ ... ≤ an ≤ ... and b₁ ≥ b₂ ≥ ... ≥ bn ≥ ... as follows:

Since the intervals are nested, each subsequent interval is contained within the previous one. Mathematically, this means I₁ ⊇ I₂ ⊇ ... In ⊇ ... . Therefore, we have:

1. I₁ ⊇ I₂ implies [a₁; b₁] ⊇ [a₂; b₂], which means a₁ ≤ a₂ and b₁ ≥ b₂.
2. I₂ ⊇ I₃ implies [a₂; b₂] ⊇ [a₃; b₃], which means a₂ ≤ a₃ and b₂ ≥ b₃.

Continuing this pattern for all intervals in the sequence, we can conclude that a₁ ≤ a₂ ≤ ... ≤ an ≤ ... and b₁ ≥ b₂ ≥ ... ≥ bn ≥ ... .

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The answer to the given question is the angular acceleration of the wheel is 0.7742 rad/s².

We can use the formula for rotational dynamics:

τ = Iα

where τ is the torque applied, I is the moment of inertia, and α is the angular acceleration.

Rearranging this equation, we get:

α = τ / I

Plugging in the given values, we have:

α = 48 Nm / 62 kg⋅m²

Simplifying, we get:

α = 0.7742 rad/s²

Rotational dynamics deals with the motion of objects that are rotating around an axis. The key concept in rotational dynamics is torque, which is a measure of how much a force can cause an object to rotate. The moment of inertia is another important quantity in rotational dynamics, and it describes an object's resistance to rotational motion. The moment of inertia depends on the distribution of mass around the axis of rotation. The angular acceleration of a rotating object is directly proportional to the torque applied and inversely proportional to the moment of inertia. Therefore, a greater torque or a smaller moment of inertia will result in a greater angular acceleration. These concepts are used in a variety of applications, from the motion of planets to the operation of engines and turbines.

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Given:

The triangles DEF is similar to GHF.

The objective is to find a similar ratio of DF/DE.

Explanation:

Using the basic proportionality theorem, for the similar triangles DEF and GHF,

Considering the first two ratios of equation (1),

On interchanging the segments further,

Hence, the required segment in the blanks is GF/GH.

Answer:

Annie will have $87.75 more than Ellie

Step-by-step explanation:

Annie :

2500(.024)7

Multiply all 3 together to get the answer

2500 x .024 x 7=  420

Ellie :

2500(.019)7

Multiply all three together to get the answer

2500 x .019 x 7 = 332.50

Answer:

332.50

Step-by-step explanation: