CubeyThePenguin

Supplementary angles add up to 180 degrees, so add the angles from each pair to see if they add to 180 degrees.

A) 59 + 121 = 180

B) 89 + 99 = 188

C) 62 + 138 = 200

D) 30 + 60 = 90

Only A is the right choice. Note that the angles in answer choice add up to 90 degrees, so they are called complements.

Translated to English:

A) How many natural numbers are there between the ends of the segments?

B) How many integers are there?

C) How many non-natural integers?

And why?

A) Natural numbers are positive, so you are only considering the numbers 1, 2, ..., 24.

There are 24 of them.

B) between -42 and 25: -41, -40, ..., 23, 24

There are 41 + 1 + 24 = 66 integers. (41 negative, 1 zero, 24 positive)

C) Non-natural numbers include the negatives and zero.

There are 41 + 1 = 42 of them.

Translate this solution into Spanish if necessary.

Traduzca esta solución al español si es necesario.

perpendicular bisector passes through midpoint: \((\frac{24-3}{2}, \frac{7+4}{2}) = (21/2, 11/2)\)

The midpoint is the "C" that the problem is talking about, so 2x - 4y = 21 - 22 = -1.

\(\frac{1}{x} + \frac{1}{y} = \frac{1}{18} + \frac{1}{3} = \frac{7}{18}\)

\(\frac{x+y}{xy} = \frac{7}{18}\)

\(18x + 18y = 7xy\)

\(0 = 7xy - 18x - 18y\)

\(324/7 = (7x - 18)(y-18/7)\)

\(324 = (7x - 18)(7y - 18)\)

Create a table with the factors of 324, which is what (7x - 18) and (7y - 18) will multiply to. By brute force, you can find the smallest value of x + y.

simplify the quadratic:

\(2x^2 - 3x + 27 + 3x^2 + 5 = 5x^2 - 3x + 32\)

Let the roots be x and y. From Vieta's formulas, we know that:

sum of roots = x + y = 3/5

product of roots = xy = 32/5

sum of squares of roots = x^2 + y^2 = (x + y)^2 - 2xy = (3/5)^2 - 2(32/5) = -12.44

\(f(x) + bg(x)\)

\(= (x^4 - 3x^2 + 2) + b(2x^8 - 2x^4 - 6x^2 + 2x - 1)\)

\(= 2bx^8 + (-2b + 1)x^4 + (-6b - 3)x^2 + 2xb + (2-b)\)

If b = 0, then the x^8 term is eliminated, and the smallest possible degree of the polynomial is 4.

g(x) = f(f(x)) - f(f(f(x)))

= f(3x+2) - f(f(3x+2))

= 3(3x+2) + 2 - f(3(3x+2) + 2)

= 9x + 8 - f(9x + 8)

 = 9x + 8 - (3(9x + 8) + 2)

= 9x + 8 - (27x + 26)

= -18x - 18

x is in [0, 2]

g(x) is in [-54, -18]

We are given two points: (x, y) = (12, 2.5) and (16, 3.5)

We can find the slope of this line:

\(\frac{3.5 - 2.5}{16 - 12} = \frac{1}{4} = 0.25\)

Now we can find the y-intercept. First, pick a random point:

y = mx + b

2.5 = (0.25)(12) + b

b = -0.5

The linear function is y = 0.25x - 0.5.

To find the missing y-value, plug in x = 8 to get y = 0.25(8) - 0.5 = 1.5

there are two options: 1) he did the calculations and got it wrong or 2) he guessed and got it wrong.

1) 1/4 * 1/5 = 1/20

2) 3/4 * 2/3 = 1/2

The probability we want is the answer for #1 divided by the sum of #1 and #2.

\(\frac{1/20}{1/20 + 1/2} = \frac{0.05}{0.55} = \boxed{\frac{1}{11}}\)

ratio of 3:2 -> Adriana's income is 3x and Belen's income is 2x

3x + 2x = 90000

5x = 90000

x = 18000

Adriana should receive 3x = $54000.