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masha68 [24]
2 years ago
6

What are rational numbers? Why is it important to study rational numbers?

Mathematics
1 answer:
xxTIMURxx [149]2 years ago
3 0

Answer:

Any number that can be written as a fraction with integers is called a rational number . For example, 17 and −34 are rational numbers. (Note that there is more than one way to write the same rational number as a ratio of integers. For example, 17 and 214 represent the same rational number.)

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the function intersects its midline at (-pi,-8) and has a maximum point at (pi/4,-1.5) write an equation
Tcecarenko [31]

The equation that represents the <em>sinusoidal</em> function is x(t) = -8 + 6.5 \cdot \sin \left[\left(\frac{2}{3} \pm \frac{4\cdot i}{3}\right)\cdot t + \left(\frac{2\pi}{3} \pm \frac{7\pi \cdot i}{3}  \right)\right], i\in \mathbb{Z}.

<h3>Procedure - Determination of an appropriate function based on given information</h3>

In this question we must find an appropriate model for a <em>periodic</em> function based on the information from statement. <em>Sinusoidal</em> functions are the most typical functions which intersects a midline (x_{mid}) and has both a maximum (x_{max}) and a minimum (x_{min}).

Sinusoidal functions have in most cases the following form:

x(t) = x_{mid} + \left(\frac{x_{max}-x_{min}}{2} \right)\cdot \sin (\omega \cdot t + \phi) (1)

Where:

  • \omega - Angular frequency
  • \phi - Angular phase, in radians.

If we know that x_{min} = -14.5, x_{mid} = -8, x_{max} = -1.5, (t, x) = (-\pi, -8) and (t, x) = \left(\frac{\pi}{4}, -1.5 \right), then the sinusoidal function is:

-8 +6.5\cdot \sin (-\pi\cdot \omega + \phi) = -8 (2)

-8+6.5\cdot \sin\left(\frac{\pi}{4}\cdot \omega + \phi \right) = -1.5 (3)

The resulting system is:

\sin (-\pi\cdot \omega + \phi) = 0 (2b)

\sin \left(\frac{\pi}{4}\cdot \omega + \phi \right) = 1 (3b)

By applying <em>inverse trigonometric </em>functions we have that:

-\pi\cdot \omega + \phi = 0 \pm \pi\cdot i, i \in \mathbb{Z} (2c)

\frac{\pi}{4}\cdot \omega + \phi = \frac{\pi}{2} + 2\pi\cdot i, i \in \mathbb{Z} (3c)

And we proceed to solve this system:

\pm \pi\cdot i + \pi\cdot \omega = \frac{\pi}{2} \pm 2\pi\cdot i -\frac{\pi}{4}\cdot \omega

\frac{3\pi}{4}\cdot \omega = \frac{\pi}{2}\pm \pi\cdot i

\omega = \frac{2}{3} \pm \frac{4\cdot i}{3}, i\in \mathbb{Z} \blacksquare

By (2c):

-\pi\cdot \left(\frac{2}{3} \pm \frac{4\cdot i}{3}\right) + \phi =\pm \pi\cdot i

-\frac{2\pi}{3} \mp \frac{4\pi\cdot i}{3} + \phi = \pm \pi\cdot i

\phi = \frac{2\pi}{3} \pm \frac{7\pi\cdot i}{3}, i\in \mathbb{Z} \blacksquare

The equation that represents the <em>sinusoidal</em> function is x(t) = -8 + 6.5 \cdot \sin \left[\left(\frac{2}{3} \pm \frac{4\cdot i}{3}\right)\cdot t + \left(\frac{2\pi}{3} \pm \frac{7\pi \cdot i}{3}  \right)\right], i\in \mathbb{Z}. \blacksquare

To learn more on functions, we kindly invite to check this verified question: brainly.com/question/5245372

5 0
2 years ago
Enter numbers into the table so that the paired values are in a proportional relationship.
Alisiya [41]

Answer:

x- 5

y- 6

Step-by-step explanation:

A proportional relationship is a relationship for which a proportional constant y/x exists and for which it passes through (0,0) or the origin.

The proportional constant is 6.

18/3=6

42/7=7

54/9=6

Thus the filled in blanks should be x 1, 3, 5, 7, 9 and y 6, 18, 30, 42, 54.


8 0
3 years ago
Read 2 more answers
roberto made a 74 on last weeks math quiz. this week he made an 85 on the quiz. which best describes the percent of increase in
laila [671]

The percentage of increase in Roberto score is 14.87%, if Roberto made a 74 on last weeks math quiz. this week he made an 85 on the quiz.

Step-by-step explanation:

The given is,

               Roberto score on math quiz at last week is 74

               This week he made an 85 on the quiz

Step:1

            Formula to calculate the percentage of increase the score,

                  Percentage of increase = \frac{Final score-Starting score}{Starting score} × 100...........(1)

            From the given,

                  Final score = 85

                  Starting score = 74

           Equation (1) becomes,

                  Percentage of increase =( \frac{85-74}{74}) ×  100

                                                       = (\frac{11}{74}) ×  100

                                                       = 0.1486486  ×  100

                                                       = 14.86486 %

                                                      ≅ 14.87 %

              Percentage of increase = 14.87 %

Result:

        The percentage of increase in Roberto score is 14.87%, if Roberto made a 74 on last weeks math quiz. this week he made an 85 on the quiz.

                                                                   

         

7 0
3 years ago
What is slope and y-intercept of the equation on the graph?
san4es73 [151]
The y intercept is the point where the line crosses the y (vertical) axis. So the y intercept is the point (0,3). The slope is 3/2
5 0
3 years ago
Simplify the expression.
BigorU [14]
D) 1/m^18

Explanation:
(m^-2/m-1m5)^3
m^-6/m^-3m^15
1/m^6 / m^15/m^3
1/m^6 / m^12
1/m^6 (1/m^12)
1/m^18

let me know if you want clarification
4 0
2 years ago
Read 2 more answers
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