Mathematics: there were 62 people in the movie theater for the matinee. ticket sales for that showing were $364. if adult tickets cost $7 and children tickets cost $3.50, how many of each were at that show. please solve with systems of equations.

Hello Carousel!Solve the integral.First, let s take I as ⇨ .Let, x + 1 = m² = dx = 2mdm.Now, divide the numerator & denominator by m²....we ll get it as...Now, let m - 1/m be t = (1 + 1/m²) dm = dtWe know, ...therefore...Now, substitute the value of t in eq. 1..we ll get..Now, substitute the value of m in eq. 2...we ll get...The correct answer is Option B.__________________The question is really long & tricky but once you get the hang of it you ll be good. Good luck!__________________Hope...

there were 62 people in the movie theater for the

Ответ:

ysabel0420

12.02.2021

Mathematics

Step-by-step explanation:

a+c=62, c=62-a

7a+3.5c=364, using c found above gives you

7a+3.5(62-a)=364 expand left side

7a+217-3.5a=364 group like terms on left side

3.5a+217=364 subtract 217 from both sides

3.5a=147 divide both sides by 3.5

a=42, since c=62-a

c=62-42=20

So 20 children and 42 adult tickets were sold.

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Ответ:

SynysterWxlf661

08.04.2022

Mathematics

Hello Carousel!

$\huge \boxed{\mathbb{QUESTION} \downarrow}$

Solve the integral.

$\large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}$

$\tt \:\int \: \frac{ x + 2 }{( {x}^{2} + 3x + 3) \sqrt{x - 1}} \: dx \\$

First, let's take I as ⇨ $\tt \:\int \: \frac{ x + 2 }{( {x}^{2} + 3x + 3) \sqrt{x - 1}} \: dx \\$ .

$\tt \:I = \int \: \frac{ x + 2 }{( {x}^{2} + 3x + 3) \sqrt{x - 1}} \: dx \\\tt \: I =\int \: \frac{ x + 2 }{( {x}^{2} + 2x + 1 + 3) \sqrt{x - 1}} \: dx \\ \tt \:I =\int \: \frac{ x + 2 }{( ({x + 1}^{2}) + x + 1 + 3) \sqrt{x - 1}}$

Let, x + 1 = m² => dx = 2mdm.

$\tt \:I =\int \: \frac{ {m}^{2} + 1}{ {m}^{4} + {m}^{2} + 1 \cdot \: m} 2mdm \\ \tt \:I =\int \: \frac{ {m}^{2} + 1}{ {m}^{4} + {m}^{2} + 1 \cdot \: \bcancel{ m}} 2 \bcancel{m}dm \\ \tt \: I = \: \int \: \frac{ {m}^{2} + 1}{ {m}^{4} + {m}^{2} + 1 } 2dm$

Now, divide the numerator & denominator by m²....we'll get it as...

$\tt \:I =2\int \: \frac{ 1 + \frac{1}{ {m}^{2} } }{ {m}^{2} + 1 + \frac{1}{ {m}^{2} } } \: dm \\\tt \: I =2\int \: \frac{ 1 + \frac{1}{ {m}^{2} } }{( {m}^{2} + \frac{1}{ {m}^{2} } - 2) + 3} \: dm \\\tt \: I =2\int \: \frac{ (1 + \frac{1}{ {m}^{2} }) \: dm }{ ({m} - \frac{1}{ m } ) ^{2} + 3}$

Now, let m - 1/m be t => (1 + 1/m²) dm = dt

$\tt \:I =2\int \: \frac{ dt}{ {t}^{2} + 3 } \\ \tt \:I = 2\int \: \frac{ dt}{ {t}^{2} + ( \sqrt{3}) ^{2} }$

We know, $\tt \:\int \: \frac{dx}{ {x}^{2} + a ^{2} } = \frac{1}{a} tan ^{ - 1} (\frac{x}{a} ) + c$ ...therefore...

$\tt \:I = \frac{2}{ \sqrt{3} } {tan}^{ - 1} ( \frac{t}{ \sqrt{3} } ) + c \: \rightarrow \boxed{ \tt \: eq. \: 1}$

Now, substitute the value of 't' in eq. 1..we'll get..

$\tt \:I = \frac{2}{ \sqrt{3} } {tan}^{ - 1} ( \frac{m - \frac{1}{m} }{ \sqrt{3} } ) + c \: \rightarrow \boxed{ \tt \: eq. \: 2}$

Now, substitute the value of 'm' in eq. 2...we'll get...

$\tt \: I = \frac{2}{ \sqrt{3} } {tan}^{ - 1} ( \frac{ \sqrt{x + 1} - \frac{1}{ \sqrt{x - 1} } }{ \sqrt{3} } ) + c \\ \tt \:I = \frac{2}{ \sqrt{3} } {tan}^{ - 1} ( \frac{x + 1 - 1}{ \sqrt{3} \sqrt{x - 1} } ) + c \\ \boxed{\boxed{ \bf \: I = \frac{2}{ \sqrt{3} } {tan}^{ - 1} ( \frac{x }{ \sqrt{3 (x - 1} )} ) + c }}$

The correct answer is Option B.

__________________

The question is really long & tricky but once you get the hang of it you'll be good. Good luck!

__________________

Hope it'll help you!

ℓu¢αzz ッ

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there were 62 people in the movie theater for the matinee. ticket sales for that showing were $364. if adult tickets cost $7 and children tickets cost $3.50, how many of each were at that show. please solve with systems of equations.

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