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2 years ago

Find the angle [OA] makes with the positive x-axis if the x-coordinate of the point A on the unit circle is 0.222

1 answer:

4 0

By using the concepts of <em>unit</em> circle and <em>trigonometric</em> functions, we find that the angle OA, whose x-coordinate is 0.222, has a measure of approximately 77.173°.

<h3>How to find an angle in an unit circle</h3>

<em>Unit</em> circles are circles with radius of 1 and centered at the origin of a Cartesian plane, which are used to determine angles and <em>trigonometric</em> functions related to them. If we use <em>rectangular</em> coordinate system and the definition of the <em>tangent</em> function, we find that the angle OA is equal to:

$\tan \theta = \frac{\sqrt{1 - x^{2}}}{x}$

$\tan \theta = \frac{\sqrt{1-0.222^{2}}}{0.222}$

tan θ ≈ 77.173°

By using the concepts of <em>unit</em> circle and <em>trigonometric</em> functions, we find that the angle OA, whose x-coordinate is 0.222, has a measure of approximately 77.173°.

To learn more on unit circles: brainly.com/question/12100731

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The exponential function f(x) = 3(5)x grows by a factor of 25 between x = 1 and x = 3. What factor does it grow by between x = 5

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<h2> Question # 1</h2>

Answer:

25 is the factor which grows by between x = 5 and x = 7.

Step-by-step explanation:

Considering the exponential function

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The growth factor between $x=1$ and $x=3$ is given by:

$\left[3\left(5\right)^3\right]\div \left[3\left(5\right)^1\right]$

$\mathrm{Calculate\:within\:parentheses}\:\left[3\left(5\right)^3\right]\::\quad 375$

$\mathrm{Calculate\:within\:parentheses}\:\left[3\left(5\right)^1\right]\::\quad 15$

So,

= $375\div \:15$

= $25$

Similarly, growth factor between $x=5$ and $x=7$ is given by:

$\left[3\left(5\right)^7\right]\div \left[3\left(5\right)^5\right]$

= $\frac{3\cdot \:5^7}{3\cdot \:5^5}$

$\mathrm{Divide\:the\:numbers:}\:\frac{3}{3}=1$

= $\frac{5^7}{5^5}$

$\mathrm{Apply\:exponent\:rule}:\quad \frac{x^a}{x^b}=x^{a-b}$

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= $5^{7-5}$

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Therefore, 25 is the factor which grows by between x = 5 and x = 7.

<h2> Question # 2</h2>

Answer:

The population be in 24 years will be 27000.

Also, the population growth modeled by an exponential function as $y=A\cdot \left(b\right)^t$ is an exponential function.

The graph for $y=A\cdot \left(b\right)^t$ is also shown in attached figure.

Step-by-step explanation:

If a city that currently has a population of 1000 triples in size every 8 years.
what will the population be in 24 years?
Is the population growth modeled by a linear function or an exponential function?

As the city that currently has a population of 1000 triples in size every 8 years.

So, for this case

$y=A\cdot \left(b\right)^t$

where

$A$ = Initial population amount

$b$ = growth rate

$t$ = time

Substituting the values in the function

$y=A\cdot \left(b\right)^t$

$y=1000\cdot \:\:3^{\frac{1}{8}t}$

So, the population be in 24 years

$y=1000\cdot \:\:3^{\frac{1}{8}24}$

$3^{\frac{1}{8}\cdot \:24}=3^3$

$\:y=3^3\cdot 1000$

$y=1000\cdot \:\:27$

$y=27000$

Therefore, the population be in 24 years will be 27000.

Also, the population growth modeled by an exponential function as $y=A\cdot \left(b\right)^t$ is an exponential function.

<h2 /><h2> Question # 3</h2>

Answer:

the graph of the function will translate horizontally 3/5 units right.

Step-by-step explanation:

We have to find the effect on the graph of the function $f(x)=2x$ when it is replaced by f(x- 3/5).

We already have an idea that rule for horizontal translation:

Given a function $f(x)$ , and a constant c > 0, the function $g(x) = f(x - a)$ represents a horizontal shift c units to the right from f(x). The function $h(x) = f(x + a)$ represents a horizontal shift c units to the left.

As 3/5 > 0, so the graph of the function will translate horizontally 3/5 units right.

Therefore, the graph of the function will translate horizontally 3/5 units right.

Keywords: exponential function, translation function, growth factor

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Step-by-step explanation:

We need to solve the equation f(x)=g(x)

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Solving to find f(x)=g(x)

Putting values of f(x) and g(x)

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So, value of x is: $x=\frac{-31}{12}$

The graph will be a straight line x= -2.58 or x = -31/12

Keywords: Composition of Functions

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Step-by-step explanation:

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