On a number line, a number, b, is located the same distance from 0 as another number, a, but in the opposite direction. The

Mathematics

1 answer:

oksian1 [2.3K]2 years ago

3 0

Answer:

b=-a

Step-by-step explanation:

we know that b is located the same distance from 0 as another number, a, but in the opposite direction.

so the sum of the two numbers equal zero

a+b=0

a=-b or b=-a

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Keith_Richards [23]

Answer:

32.1 g

Step-by-step explanation:

In each 3 grams of C, there are 2 grams of A and 1 gram of B. So, for some amount C, the amount remaining of A is 40 -(2C/3), and the amount remaining of B is (50 -C/3). Since the reaction rate is proportional to the product of these amounts, we have ...

C' = k(40 -2C/3)(50 -C/3) = (2k/9)(60 -C)(150 -C) . . . for some constant k

This is separable differential equation with a solution of the form ...

ln((150 -C)/(60 -C)) = at + b

We know that C(0) = 0, so b=ln(150/60) = ln(2.5). And we know that C(10) = 20, so ln(130/40) = 10a +ln(2.5) ⇒ a = ln(1.3)/10

Then our equation for C is ...

ln((150 -C)/(60 -C)) = t·ln(1.3)/10 +ln(2.5)

For t=20, this is ...

ln((150 -C)/(60 -C)) = 2ln(1.3) +ln(2.5) = ln(2.5·1.3²) = ln(4.225)

Taking antilogs, we have ...

(150 -C)/(60 -C) = 4.225

1 +90/(60 -C) = 4.225

C = 60 -90/3.225 ≈ 32.093 . . . . . grams of product in 20 minutes