Solve the equation 120 + 1814
x = 180 − 134
x for x.

Solution:

Notice that for each function, they are simply in the form [tex]f(k(x))[/tex], where [tex]k(x)[/tex] is some function of [tex]x.[/tex] More specifically,[tex]g(x)=f(x-2), h(x)=f(2x), \text{ and } j(x) = f(x^2).[/tex]

Thus, because the Interval of Convergence of [tex]f(x)[/tex] is [tex]-3 < x < 3,[/tex] the Interval of Convergence of [tex]g(x)[/tex] should be [tex]-3 < x-2 < 3 \implies -1 < x < 5.[/tex]

Similarly, the Interval of Convergence of [tex]h(x)[/tex] is [tex]-3 < 2x < 3 \implies -\frac32 < x < \frac32.[/tex]

Finally, the Interval of Convergence of [tex]j(x)[/tex] should be [tex]-3 < x^2 < 3 \implies 0 < x < \sqrt{3}.[/tex]

The expression (cscx + cotx)² is the same as csc²x + 2(cscx)(cotx) + cot²x. (option a).

The given expression is (cscx + cotx)². To simplify this expression, we can use the formula for squaring a binomial, which is (a + b)² = a² + 2ab + b². In this case, a = cscx and b = cotx. Therefore, we can substitute these values into the formula to get:

(cscx + cotx)² = csc²x + 2(cscx)(cotx) + cot²x

So the expression (cscx + cotx)² is the same as csc²x + 2(cscx)(cotx) + cot²x.

Hence the correct option is (a).

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The direction and speed the plane traveling is at About 84.3° west of north at approximately 502.5 mph. Option C

We need to first  find the distance between points A and C using the distance formula; Distance AC = √((x2 - x1)² + (y2 - y1)²)

If we input the figures as seen in the diagram, it becomes

Distance AC = √((-30 - 20)² + (520 - 20)²)

which is 502.49. if we round it off, it becomes 502.5

We have to find find the angle θ that the plane is traveling using the law of cosines

cos(θ) = (AB² + BC² - AC²) / (2 x AB x BC)

cos(θ) = (500² + 50² - 502.5²) / (2 x 500 x 50)

which is -0.000125

θ = arccos( -0.000125)

θ = 90.0071621563 (in degrees)

Give than the wind is blowing west, the angle should be measured west of north.

180° - 90.01° = 90°

It only mean that the plane is travelling at approximately 84.3° west of north  

The answer is based on the question below;

A plane is set to fly due north, but it is pushes off course by crosswind blowing west. At 1 pm, the plane is located at point A and at 2pm, the plane is located at point C, as shown in the diagram. In what direction and at what speed is the plane traveling?

A. About 5.7° west of north at approximately 500.1 mph.

B. About 5.7° west of north at approximately 502.5 mph

C. About 84.3° west of north at approximately 500.1 mph.

D. About 84.3° west of north at approximately 502.5 mph

Point C coordinates (-30, 520)

Point A (20, 20)

Distance from A to B on a straight course is 500

B to C is 50

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The entries in both columns when matched are

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

The graph

When the line of best fit is drawn, we have

(8, 0) and (0, 8)

The equation is calculated as

y = mx + c

So, we have

8 = 0 * m + c

0 = 8 * m + c

This gives

c = 8

0 = 8 * m + 8

So, we have

m = -1

This means that the equation is

y = -x + 8

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

Equation of line of best fit: y = -x + 8

Slope = -1

y-intercept = 8

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Answer:

Amount of change: 4,000 - 3,600 = 400

Percent of change: 400/4,000 = 1/10 = 10%

There was a 10% decrease in the number of visitors:

Answer:

68.9 degrees

Step-by-step explanation:

To find this we can use the rule of sines.

It states [tex]\frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c}[/tex]

We will use 56 degrees and its complementary measurement, which is discovered by observing the opposite side from the angle, which is 4. Then, we will find the side that compliments x, which is 4.5. Then we can plug those values into the rule of sines.

[tex]\frac{sin56}{4}=\frac{Sinx}{4.5}[/tex]

Then, we want to get Sin x by itself.

[tex]\frac{sin56}{4}*4.5=sinx[/tex]

Then, we can solve for sin x.

[tex]0.932667269124=sinx[/tex]

finally, we need to take the inverse of sin to find our solution.

[tex]sin^{-1} (0.932667269124)=sin^-^1(sinx)\\x=68.85\\[/tex]

Which can be rounded to 68.9.

The rating of chips whose cost is $1.10 is obtained replacing the value of x on whichever's equation is correct by 1.10.

To calculate the numeric value of a function or of an expression, we substitute each instance of any variable or unknown on the function by the value at which we want to find the numeric value of the function or of the expression presented in the context of a problem.

The problem is incomplete, hence the procedure to estimate the rating of chips whose cost is $1.10 is presented.

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A dilation can be defined as a transformation that multiplies the distance between every point in an object and a fixed point, called the center of dilation, by a constant factor called the scale factor.

The lengths of the corresponding segments are given as follows:

Hence the scale factor is given as follows:

k = 4/2

k = 2.

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Okay, here are the steps to solve this problem:

* There are 10 sections on the spinner, 3 of which are green and 7 of which are yellow.

* So there is a 3/10 = 0.3 probability that the spinner will land on green.

* A fair coin has a 1/2 probability of landing on heads.

* For the spinner and coin toss to both have the desired outcome (green and heads), we multiply their individual probabilities:

* 0.3 * 1/2 = 0.15

Therefore, the probability that the spinner lands on green and the coin toss is heads is 0.15.

The space sample would include the lotion, spray and gel, with the SPF and there are 15 possible outcomes.

The enumeration of sample space and all potential results can be achieved by duly considering various combinations of SPF and sunscreen variety. It is achievable to list the entire gamut of possibilities when each type of sunscreen is matched with every value of SPF:

The sample space would look like this:

This shows that there are 15 possible outcomes in the sample space.

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Answer:

  (-4, 2)∪(8, ∞)

Step-by-step explanation:

Given a particle's position is described by x(t) = t³ -6t² -96t, you want the intervals where speed is increasing.

The speed of the particle is the magnitude of its rate of change of position.

The rate of change of position is ...

  x'(t) = 3t² -12t -96 = 3(t² -4t) -96

  x'(t) = 3(t -2)² -108

This describes a parabola that opens upward, with a vertex at (2, -108). It has zeros at x = 2 ± 6 = {-4, 8}.

The magnitude of the speed is shown by the blue curve in the attachment. Between t=-4 and t=8, it is the opposite of the parabola described by the above equation.

The rate of change of speed is the derivative of speed with respect to time. The green curve in the attachment shows the particle's rate of change of speed. Speed is increasing when the green curve is above the x-axis.

Between the point when speed is 0, at t=-4, and when it reaches a local maximum, at t=2, it is increasing. Speed is increasing again after it becomes 0 at t=8.

The intervals of increasing speed are (-4, 2) ∪ (8, ∞).

__

Additional comment

We have made the distinction between speed and velocity. Velocity is the signed rate of change of position. If position is plotted on a number line increasing to the right, then velocity is positive anytime the particle is moving to the right. Velocity is increasing if acceleration is to the right (positive).

Velocity of this particle is increasing on the interval (2, ∞).

The solutions to the equations are marked and matched

Given data ,

Let the equation be represented as A

Now , the value of A is

a)

4 + x = -8x - 5

Adding 8x on both sides , we get

9x + 4 = -5

Subtracting 4 on both sides , we get

9x = -9

Divide by 9 on both sides , we get

x = -1

b)

2 ( 5x + 2 ) = 10x - 4

On simplifying the equation , we get

10x + 4 = 10x - 4

Subtracting 10x on both sides , we get

4 ≠ 4

So , it has no solution

c)

7 + 2x = 2x - 7

Subtracting 2x on both sides , we get

7 ≠ -7

So , it has no solution

d)

-3x + 3 = 3 ( 1 - x )

-3x + 3 = 3 - 3x

x = all real numbers

Hence , the equations are solved

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The dimensions of cone B is radius, 6.8 units and height 7 units.

The possible dimensions of cone B is calculated as follows;

Volume of cone = ¹/₃πr²h

Volume of cone A = ¹/₃π(6²)(4) = 150.8 units³

Volume of cone B = 487 units³ - 150.8 units³ = 336.2 units³

The dimensions of cone B is calculated as;

¹/₃πr²h = 336.2 units³

r²h = 321

Let the height of cone B = 7, then the radius of the cone is calculated as;

7r² = 321

r² = 321/7

r² = 45.86

r = √45.86

r = 6.8 units

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Answer:

The resulting linear equation is 3d = 12, so d = 4 and e = 0.

Answer:

1 Mexican Peso = 0.0552 US Dollar Hope this helped! Have a great day! :)

Step-by-step explanation:

Answer:

1 United States Dollar is equivalent or equal to 18.24 (Mexican peso)

The equation of the tangent line to the graph of y = g(x) at x = 2 is y = 6x - 17.

The point-slope form of the equation of a line can be used to find the equation of the tangent line to the graph of y = g(x) at x = 2.

y - y1 = m(x - x1)

where m is the tangent line's slope and (x1, y1) is the point on the g(x) graph at x = 2.

Given that g(2) = -5, the point on the g(x) graph at x = 2 is (2, -5).

We are told that g'(2) = 6, which is the slope of the tangent line at x = 2.

As a result, the tangent line equation is: y - (-5) = 6(x - 2)

We get the following after simplifying and rearranging:

y + 5 = 6x - 12 y = 6x - 17

Hence, the equation of the tangent line to the graph of y = g(x) at x = 2 is y = 6x - 17.

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For statement 5/3 = slope, the reason is Definition of slope and for statement 5/3 = 15/9 , the reason is Triangle A is similar to triangle B.

If slope of triangles are equal than Triangle A is similar to triangle Therefore, in statement 5/3 = 15/9 when we simplify

we get 5/3 = 5/3.

Then the reason for this statement is "If slope of triangles are equal" and the definition of slope in general is m = y/x.

Therefore, for the statement 5/3 = slope, the reason is Definition of slope.

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Answer:

To convert 130 kph to mph, we can use the conversion factor 1 kph = 0.62 mph:

130 kph × 0.62 mph/kph ≈ 80.6 mph

So the maximum whole-number speed that Hannah can drive without exceeding the speed limit is 80 mph (option D).

Step-by-step explanation:

The intervals in which the function is decreasing. [tex][-\pi , -\frac{\pi }{3} ], [\frac{\pi }{3}, \pi ][/tex]. Option 3

We're given a function f(x) = 2 sin x - x, which describes a curve on a graph. We want to find the intervals where this curve is decreasing (going down) within the range of -π to π.

To find when the function is decreasing, we look at its slope. The slope tells us if the curve is going up or down. We find the slope by taking the first derivative of the function: f'(x) = 2 cos x - 1.

We now have an equation for the slope, f'(x) = 2 cos x - 1. A negative slope means the function is decreasing. So, we want to find where f'(x) is less than 0 (negative).

We set up the inequality: 2 cos x - 1 < 0. We solve it to find the x-values where the slope is negative. The solution is cos x < 1/2.

From the inequality cos x < 1/2, we find the intervals within the range of -π to π where the function is decreasing. These intervals are [-π, -π/3] and [π/3, π].

The above answer is in response to the question below as seen in the picture.

Determine the interval(s) in [tex][-\pi, \pi ][/tex] on

which f(x) =  2 sin x - x

is decreasing.

1. [tex][-\frac{\pi }{3}, \frac{\pi }{3} ][/tex]

2. [tex][-\frac{\pi }{6}, \frac{\pi }{6} ][/tex]

3. [tex][-\pi , -\frac{\pi }{3} ], [\frac{\pi }{3}, \pi ][/tex]

4. [tex][-\pi , -\frac{-2\pi }{3} ], [\frac{2\pi }{3}, \pi ][/tex]

5. [tex][-\pi , - \frac{5x}{6} ], [\frac{\pi }{6}, \pi ][/tex]

6. [tex][-\frac{\pi }{6}, \frac{5\pi }{6} ][/tex]

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The distance that the ball has traveled when the height of the ball is 80 feet is either about 9.86 feet or about 3.64 feet.

We are given that;

Velocity= 140feet

Inclination= 45degrees

Function  ​h(x)=(-32x^2/140^2)+x

Now,

To find the distance that the ball has traveled when the height of the ball is 80 feet, we need to solve the equation:

h(x) = 80

Substituting h(x) with the given function, we get:

(-32x2/1402) + x = 80

Multiplying both sides by 140^2 and simplifying, we get:

-32x^2 + 140x - 11200 = 0

Dividing both sides by -32 and simplifying, we get:

x^2 - (35/4)x + 350 = 0

Using the quadratic formula, we get:

x = [ (35/4) ± √( (35/4)^2 - 4(350) ) ] / 2 x ≈ 9.86 or x ≈ 3.64

Using a graphing utility, we can confirm that these are the approximate x-intercepts of the function h(x) - 80.

Therefore, by graphing the answer will be about 9.86 feet or about 3.64 feet.

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Thus, the height of the oak trees of found to be 98 feet.

In maths, addition is the process of adding two or more numbers together. The numbers that added are known as addends, while the outcome of the addition process, or the final response, is known as the sum.

Given data:

Height of the bird from the base of the oak tree: 33 feet.

Height flew by the bird to reach at the top of the tress: 65 feet.

So,

Height of the oak tree = 33 + 65

Height of the oak tree = 98 feet

Thus, the height of the oak trees of found to be 98 feet.

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Answer:

0.3788

Step-by-step explanation:

12 people want to teach math at a college.

6 people have a PhD.

5 people have a master's degree.

boss wants to interview 5 people for the job.

chance that all 5 of the people interviewed have a PhD.

total number of ways to select 5 applicants out of 12 is given by the combination:

combinations formula :

nCr = n! / r! * (n – r)!

C(12, 5) = 12! / (5! * 7!) = 792

number of ways to select 3 applicants with a PhD out of the 6 available is:

C(6, 3) = 6! / (3! * 3!) = 20

remaining 2 applicants can be selected from the remaining 6 applicants with a master's degree:

C(6, 2) = 6! / (2! * 4!) = 15

total number of ways to select 5 applicants with 3 having a PhD is:

20 * 15 = 300

P(3 PhDs) = 300 / 792

P(3 PhDs) = 0.3788 (rounded to four decimal places)

So, the probability of selecting five applicants for an interview where all three applicants have a PhD is approximately 0.3788

ChatGPT

The square root can be simplified as 7√2

Irrational numbers are numbers in the form of square roots, which when evaluated cannot be expressed in the form a/b e.g. √7, π, 2/√5, 3√11, etc. They are also called surd.

The opposite of an irrational number is called rational number. A rational number is a number that can be written in the form of a/b, where a and b are integers, and b is not equal to 0. e.g. 1/3, 2/3, √4 = 4/1, etc.

The square root can be simplified as follow:

√98 = √(49 * 2)

        = √49 * √2

        = 7 * √2

        = 7√2

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Answer:

-tan(x)/sin^2(x).

Step-by-step explanation:

To simplify the expression:

sec(x)sin(-x) + tan(-x)

1 + sec(-x)

We can use the fact that sin(-x) = -sin(x), cos(-x) = cos(x), and tan(-x) = -tan(x) to rewrite the expression as:

-sec(x)sin(x) - tan(x)

1 + sec(x)

Next, we can multiply both the numerator and denominator by the conjugate of the denominator, 1 - sec(x), to eliminate the square root in the denominator. This gives:

(-sec(x)sin(x) - tan(x))(1 - sec(x))

(1 + sec(x))(1 - sec(x))

Simplifying the numerator by distributing and combining like terms, we get:

-sec(x)sin(x) + sec(x)tan(x) + tan(x) - sec(x)tan(x)

1 - sec^2(x)

Simplifying further by canceling out the sec(x)tan(x) terms in the numerator, we get:

-tan(x)

1 - sec^2(x)

Finally, we can use the identity 1 - sec^2(x) = sin^2(x) to rewrite the denominator and simplify further:

-tan(x)

sin^2(x)

Therefore, the simplified expression is -tan(x)/sin^2(x).