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Leni [432]
3 years ago
14

A movie theater has a seating capacity of 235. The theater charges $5 for children, $7 for students, and $12 for adults. There a

re half as many adults as there are children. If the total ticket sales was $1706, How many children, students, and adults attended?
Mathematics
1 answer:
Annette [7]3 years ago
7 0

Answer:

61 adults, 122 children, and 52 students

Step-by-step explanation:

Let a = amount of adults, c = amount of children, and s = amount of students.

Let's go over what we know:

total amount: 235

twice as many children as adults

children: $5

student: $7

adult: $12

Using this information, we can determine the following:

a = 1/2c

235 = c + s + a

1706 = 5c + 7s + 12a

Since we know a = 1/2c, plug 1/2c in for a. The equations become:

235 = c + s + 1/2c

1706 = 5c + 7s + 6c

Now, solve the inequalities by substitution.

235 = c + s + 1/2c

235 = 1 1/2c + s

-s + 235 = 1 1/2c

-s = -235 + 1 1/2c

s = 235 - 1 1/2c

Now, plug in the value for s into the other equation.

1706 = 5c + 7(235 - 1 1/2c) + 6c

-5c + 1706 =  7(235 - 1 1/2c) + 6c

-5c = -1706 + 7(235 - 1 1/2c) + 6c

-5c = -1706 + 1645 - 21c/2 + 6c

-5c = -61 - 21c/2 + 6c

-5c = -61 - 9c/2

c = 122

Now, plug in the value for c into the value of s.

s = 235 - 1 1/2(122)

s = 235 - 183

s = 52

Finally, plug in the value for c into the value of a.

a = 1/2(122)

a = 61

To check, add up all values to make sure they equal 235.

61 + 52 + 122 = 235.

Therefore, the final answers are 61 adults, 122 children, and 52 students.

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3 years ago
P(x) = x + 1x² – 34x + 343<br> d(x)= x + 9
Feliz [49]

Answer:

x=\frac{9}{d-1},\:P=\frac{-297d+378}{\left(d-1\right)^2}+343

Step-by-step explanation:

Let us start by isolating x for dx = x + 9.

dx - x = x + 9 - x > dx - x = 9.

Factor out the common term of x > x(d - 1) = 9.

Now divide both sides by d - 1 > \frac{x\left(d-1\right)}{d-1}=\frac{9}{d-1};\quad \:d\ne \:1. Go ahead and simplify.

x=\frac{9}{d-1};\quad \:d\ne \:1.

Now, \mathrm{For\:}P=x+1x^2-34x+343, \mathrm{Subsititute\:}x=\frac{9}{d-1}.

P=\frac{9}{d-1}+1\cdot \left(\frac{9}{d-1}\right)^2-34\cdot \frac{9}{d-1}+343.

Group the like terms... 1\cdot \left(\frac{9}{d-1}\right)^2+\frac{9}{d-1}-34\cdot \frac{9}{d-1}+343.

\mathrm{Add\:similar\:elements:}\:\frac{9}{d-1}-34\cdot \frac{9}{d-1}=-33\cdot \frac{9}{d-1} > 1\cdot \left(\frac{9}{d-1}\right)^2-33\cdot \frac{9}{d-1}+343.

Now for 1\cdot \left(\frac{9}{d-1}\right)^2 > \mathrm{Apply\:exponent\:rule}: \left(\frac{a}{b}\right)^c=\frac{a^c}{b^c} > \frac{9^2}{\left(d-1\right)^2} = 1\cdot \frac{9^2}{\left(d-1\right)^2}.

\mathrm{Multiply:}\:1\cdot \frac{9^2}{\left(d-1\right)^2}=\frac{9^2}{\left(d-1\right)^2}.

Now for 33\cdot \frac{9}{d-1} > \mathrm{Multiply\:fractions}: \:a\cdot \frac{b}{c}=\frac{a\:\cdot \:b}{c} > \frac{9\cdot \:33}{d-1} > \frac{297}{d-1}.

Thus we then get \frac{9^2}{\left(d-1\right)^2}-\frac{297}{d-1}+343.

Now we want to combine fractions. \frac{9^2}{\left(d-1\right)^2}-\frac{297}{d-1}.

\mathrm{Compute\:an\:expression\:comprised\:of\:factors\:that\:appear\:either\:in\:}\left(d-1\right)^2\mathrm{\:or\:}d-1 > This\: is \:the\:LCM > \left(d-1\right)^2

\mathrm{For}\:\frac{297}{d-1}:\:\mathrm{multiply\:the\:denominator\:and\:numerator\:by\:}\:d-1 > \frac{297}{d-1}=\frac{297\left(d-1\right)}{\left(d-1\right)\left(d-1\right)}=\frac{297\left(d-1\right)}{\left(d-1\right)^2}

\frac{9^2}{\left(d-1\right)^2}-\frac{297\left(d-1\right)}{\left(d-1\right)^2} > \mathrm{Since\:the\:denominators\:are\:equal,\:combine\:the\:fractions}> \frac{a}{c}\pm \frac{b}{c}=\frac{a\pm \:b}{c}

\frac{9^2-297\left(d-1\right)}{\left(d-1\right)^2} > 9^2=81 > \frac{81-297\left(d-1\right)}{\left(d-1\right)^2}.

Expand 81-297\left(d-1\right) > -297\left(d-1\right) > \mathrm{Apply\:the\:distributive\:law}: \:a\left(b-c\right)=ab-ac.

-297d-\left(-297\right)\cdot \:1 > \mathrm{Apply\:minus-plus\:rules} > -\left(-a\right)=a > -297d+297\cdot \:1.

\mathrm{Multiply\:the\:numbers:}\:297\cdot \:1=297 > -297d+297 > 81-297d+297 > \mathrm{Add\:the\:numbers:}\:81+297=378 > -297d+378 > \frac{-297d+378}{\left(d-1\right)^2}

Therefore P=\frac{-297d+378}{\left(d-1\right)^2}+343.

Hope this helps!

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3 years ago
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3 years ago
Directions: Find the indicated trigonometric ratio as a fraction in simplest form. 1. Sin L =
Rainbow [258]

Answer:

\sin L = 0.60

tan\ N = 1.33

\cos L = 0.80

\sin N = 0.80

Step-by-step explanation:

Given

See attachment

From the attachment, we have:

MN = 6

LN = 10

First, we need to calculate length LM,

Using Pythagoras theorem:

LN^2 = MN^2 + LM^2

10^2 = 6^2 + LM^2

100 = 36 + LM^2

Collect Like Terms

LM^2 = 100 - 36

LM^2 = 64

LM = 8

Solving (a): \sin L

\sin L = \frac{Opposite}{Hypotenuse}

\sin L = \frac{MN}{LN}

Substitute values for MN and LN

\sin L = \frac{6}{10}

\sin L = 0.60

Solving (b): tan\ N

tan\ N = \frac{Opposite}{Adjacent}

tan\ N = \frac{LM}{MN}

Substitute values for LM and MN

tan\ N = \frac{8}{6}

tan\ N = 1.33

Solving (c): \cos L

\cos L = \frac{Adjacent}{Hypotenuse}

\cos L = \frac{LM}{LN}

Substitute values for LN and LM

\cos L = \frac{8}{10}

\cos L = 0.80

Solving (d): \sin N

\sin N = \frac{Opposite}{Hypotenuse}

\sin N = \frac{LM}{LN}

Substitute values for LM and LN

\sin N = \frac{8}{10}

\sin N = 0.80

7 0
3 years ago
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