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Yakvenalex [24]

3 years ago

13

5. There are two kinds of products (called DA and GA) produced in your company every day. Alsothere are two kinds of materials (

denoted by A and B) in need One DA costs one unit of A, one unit of B: one GA costs one unit of A, two units of B. You get $3000 when you sell one DA and $5000 for one GA. The supply of the two materials every day is 12,22. a)What is the maximal profit $ b) How many DA and GA should be produced every day to obtain this maximal profit? Number of DA = Number of GA

1 answer:

chubhunter [2.5K]3 years ago

8 0

Answer:

a) $56000
b) DA = 2, GA = 10

Step-by-step explanation:

<u>Given</u>

DA = A + B ⇒ $3000
GA = A + 2B ⇒ $5000
Number of A = 12
Number of B = 22

<u>We can see from the equations that</u>

GA - DA = B ⇒ $2000 and
A = $1000 and B = $2000 in terms of profit

<u>So greater use of unit B brings greater profit. We don't want any unit is left over, so get this equation.</u>

A + B = 12
A + 2B = 22

Is the equation set to indicate use of the units A and B

<u>Solving we get </u>

A = 2 and
B = 10

<u>It this case all the units are used and profit is maximum</u>

2*$3000 + 10*$5000 = $56000

Number of DA = 2
Number of GA = 10

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Answer:

$a+b$ is divisible by $c$ since $d$ and $e$ are also whole numbers.

$b-a$ is divisible by $c$ since $d$ and $e$ are also whole numbers.

Step-by-step explanation:

Let $a$ , $b$ whole number that are divisible by $c$ . Then, we have that $d = \frac{a}{c}$ and $e = \frac{b}{c}$ , which are also whole numbers. We proceed to prove that $a + b$ is divisible by $c$ :

1) $d = \frac{a}{c}$ , $e = \frac{b}{c}$ , $a$ , $b$ , $c$ , $d$ , $e$ $\in \mathbb{Z}$ Given

2) $d + e = \frac{a}{c}+\frac{b}{c}$ Compatibility with addition

3) $d+e = a\cdot c^{-1}+b\cdot c^{-1}$ Definition of division

4) $d + e = c^{-1}\cdot (a+b)$ Commutative and distributive properties

5) $d+e = (a+b) \cdot c^{-1}$ Commutative properties

6) $d+e = \frac{a+b}{c}$ Definition of division/Result

Therefore, $a+b$ is divisible by $c$ since $d$ and $e$ are also whole numbers.

We proceed to prove that $b-a$ is divisible by $c$ :

1) $d = \frac{a}{c}$ , $e = \frac{b}{c}$ , $a$ , $b$ , $c$ , $d$ , $e$ $\in \mathbb{Z}$ Given

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5) $e + d\cdot (-1) = b\cdot c^{-1} + [a\cdot (-1)]\cdot c^{-1}$ Associative and commutative properties

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8) $e + d\cdot (-1) = \frac{b+a\cdot (-1)}{c}$ Definition of division

9) $e-d = \frac{b-a}{c}$ $(-1)\cdot a = -a$ /Definition of subtraction

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