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suter [353]
3 years ago
14

What is the value of x

Mathematics
1 answer:
Anika [276]3 years ago
6 0

We know that the right triangle has one 90 degree angle and two acute (< 90 degree) angles. Since the sum of the angles of a triangle is always 180 degrees.

<u>To find x:</u>

(10x - 20) + (6x + 8) =  180

16x - 12 = 180

      +12     +12

16x = 192

-----    -----

16        16

x = 12

<u>Check:</u>

(10(12) - 20) + (6(12) + 8) = 180

(120 -  20) + (72 + 8) = 180

100 + 80 = 180

180 = 180

<u>Angle (10x - 20):</u>

(10(12) - 20)

(120 - 20)

100

<u>Angle (6x + 8):</u>

(6(12) + 8)

(72 + 8)

80

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The solution to the problem is as follows:


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3 years ago
Solve for yyy. a(n+y) = 10y+32a(n+y)=10y+32a
Afina-wow [57]
To solve for y, you need to separate it in one side and the other terms in the other side..... as follows:

<span>a(n+y)=10y+32a
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Read 2 more answers
Find the value of x in the triangle shown below.
Fittoniya [83]

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3 years ago
Evaluate the following integral (Calculus 2) Please show step by step explanation!
barxatty [35]

Answer:

\dfrac{1}{2} \left(25 \arcsin \left(\dfrac{x}{5}\right) -x\sqrt{25-x^2}\right) + \text{C}

Step-by-step explanation:

<u>Fundamental Theorem of Calculus</u>

\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

Given indefinite integral:

\displaystyle \int \dfrac{x^2}{\sqrt{25-x^2}}\:\:\text{d}x

Rewrite 25 as 5²:

\implies \displaystyle \int \dfrac{x^2}{\sqrt{5^2-x^2}}\:\:\text{d}x

<u>Integration by substitution</u>

<u />

\boxed{\textsf{For }\sqrt{a^2-x^2} \textsf{ use the substitution }x=a \sin \theta}

\textsf{Let }x=5 \sin \theta

\begin{aligned}\implies \sqrt{5^2-x^2} & =\sqrt{5^2-(5 \sin \theta)^2}\\ & = \sqrt{25-25 \sin^2 \theta}\\ & = \sqrt{25(1-\sin^2 \theta)}\\ & = \sqrt{25 \cos^2 \theta}\\ & = 5 \cos \theta\end{aligned}

Find the derivative of x and rewrite it so that dx is on its own:

\implies \dfrac{\text{d}x}{\text{d}\theta}=5 \cos \theta

\implies \text{d}x=5 \cos \theta\:\:\text{d}\theta

<u>Substitute</u> everything into the original integral:

\begin{aligned}\displaystyle \int \dfrac{x^2}{\sqrt{5^2-x^2}}\:\:\text{d}x & = \int \dfrac{25 \sin^2 \theta}{5 \cos \theta}\:\:5 \cos \theta\:\:\text{d}\theta \\\\ & = \int 25 \sin^2 \theta\end{aligned}

Take out the constant:

\implies \displaystyle 25 \int \sin^2 \theta\:\:\text{d}\theta

\textsf{Use the trigonometric identity}: \quad \cos (2 \theta)=1 - 2 \sin^2 \theta

\implies \displaystyle 25 \int \dfrac{1}{2}(1-\cos 2 \theta)\:\:\text{d}\theta

\implies \displaystyle \dfrac{25}{2} \int (1-\cos 2 \theta)\:\:\text{d}\theta

\boxed{\begin{minipage}{5 cm}\underline{Integrating $\cos kx$}\\\\$\displaystyle \int \cos kx\:\text{d}x=\dfrac{1}{k} \sin kx\:\:(+\text{C})$\end{minipage}}

\begin{aligned} \implies \displaystyle \dfrac{25}{2} \int (1-\cos 2 \theta)\:\:\text{d}\theta & =\dfrac{25}{2}\left[\theta-\dfrac{1}{2} \sin 2\theta \right]\:+\text{C}\\\\ & = \dfrac{25}{2} \theta-\dfrac{25}{4}\sin 2\theta + \text{C}\end{aligned}

\textsf{Use the trigonometric identity}: \quad \sin (2 \theta)= 2 \sin \theta \cos \theta

\implies \dfrac{25}{2} \theta-\dfrac{25}{4}(2 \sin \theta \cos \theta) + \text{C}

\implies \dfrac{25}{2} \theta-\dfrac{25}{2}\sin \theta \cos \theta + \text{C}

\implies \dfrac{25}{2} \theta-\dfrac{5}{2}\sin \theta \cdot 5 \cos \theta + \text{C}

\textsf{Substitute back in } \sin \theta=\dfrac{x}{5} \textsf{ and }5 \cos \theta = \sqrt{25-x^2}:

\implies \dfrac{25}{2} \theta-\dfrac{5}{2}\cdot \dfrac{x}{5} \cdot \sqrt{25-x^2} + \text{C}

\implies \dfrac{25}{2} \theta-\dfrac{1}{2}x\sqrt{25-x^2} + \text{C}

\textsf{Substitute back in } \theta=\arcsin \left(\dfrac{x}{5}\right) :

\implies \dfrac{25}{2} \arcsin \left(\dfrac{x}{5}\right) -\dfrac{1}{2}x\sqrt{25-x^2} + \text{C}

Take out the common factor 1/2:

\implies \dfrac{1}{2} \left(25 \arcsin \left(\dfrac{x}{5}\right) -x\sqrt{25-x^2}\right) + \text{C}

Learn more about integration by trigonometric substitution here:

brainly.com/question/28157322

6 0
2 years ago
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