A movie theater has a seating capacity of 315. The theater charges $5.00 for children, $7.00 for students, and $12.00 of adults. There are half as many adults as there are children. If the total ticket sales was $ 2280, How many children, students, and adults attended?

84 students

Step-by-step explanation:

Let x be the number of adults.

Then the number of children is 2x, according to the condition,

and the number of students is the rest (315-x-2x) = (315-3x).

The "money" equation (the revenue equation) is

   12x + 5x(2x) + 7x(315-3x) = 2282.

   12x + 10x + 7x315 - 21x = 2282

   x = 2282 - 7x315 = 77  is the number of adults.

 77 adults :2 x 77 = 154 children and the rest (315-77-154) = 84 students.

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(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240