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Umnica [9.8K]
3 years ago
10

El administrador de una cafetería sabe que vende diariamente 50 litros de jugo mixto. Para elaborar este jugo se emplean dos pre

parados: uno de jugo de naranja, que contiene un 5% de jugo natural y otro preparado de jugo de toronja, que contiene 25% de jugo natural. Si se desea que el jugo mixto contenga un 10% de jugo natural ¿Cuántos litros de cada preparado se deben emplear?
Mathematics
1 answer:
Nina [5.8K]3 years ago
7 0

Answer:

Se necesitan 37.5 litros del preparado que contiene 5% de jugo natural, y 12.5 litros del que contiene 25%.

Step-by-step explanation:

Primero calculamos la cantidad total de jugo natural que se necesita, a partir del volumen de jugo mixto que se quiere vender:

  • 50 L * 10/100 = 5 L

Ahora, que X sean los litros del preparado con jugo de 5%, y que Y sean los litros del preparado con jugo de 25%:

La suma de ambos preparados tiene que ser 50 L:

1.    x + y = 50

La suma del jugo natural tiene que ser 5:

2.   x*5/100 + y*25/100 = 5

Por lo que tenemos <em>un sistema de dos ecuaciones y dos incognitas</em>.

Expresamos x en términos de y:

  • x + y = 50
  • x = 50 - y

Reemplazamos x en la segunda ecuación:

  • (50-y)*5/100 + y*25/100 = 5

Y despejamos <em>y</em>:

  • 2.5 - 0.05y + 0.25y = 5
  • 0.20y = 2.5
  • y = 12.5

Usando ese valor de y, calculamos x:

  • x + 12.5 = 50
  • x = 37.5

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The answer: The factors are (2x-9) and (x+6).

The Problem:

Factor 2x^2+3x-54

Step-by-step explanation:

So I'm going to do trial factors using the choices to aid me.

Factored form for this problem if it exist will be in the form:

(mx+n)(kx+p).

In general this is what it would look like if we factored any quadratic in terms of x (given the quadratic is not prime but technically you could factor even over the complex numbers).

Let's look at:

(mx+n)(kx+p)

We want to choose k \text{ and } m such that when you multiply them you get 2.  Well those would have to be 2 and 1.

(2x+n)(x+p)

we want to choose n \text{ and } p such that when you multiply them you get -54. Based on the choices we want to get with -9 and 6, or 9 and -6. We don't know the order we want to choose it in either.

For example which of these would work:

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We are going to consider only the outer and inner of FOIL since we already know the first times the first is 2x^2 and the last times the last is -54.

Let's test the first one:

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The first choice did not give us the middle term 3x.

Trying the second one would give us the opposite since they are in the same form as previous just the + and - are switched.

Let's look at the third one:

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Outer: 2x(6)=12x

Inner: -9(x)=-9x

--------------------------ADD!

12x-9x=3x

This is the winner.

The answer: The factors are (2x-9) and (x+6).

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