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Instructions: Attempt all questions $\newline$ $1$ . Two chemicals $A$ and $B$ are combined to form a chemical $C$ . The rate, or velocity, of the reaction is proportional to the product of the instantaneous amounts of $A$ and $B$ not converted to chemical $C$ . Initially, there are $100$ grams of $4$ and $50$ grams of $B$ , and for each gram of $B$ , $B$ $1$ grams of $A$ is used. It is observed that $B$ $3$ grams of $C$ is formed in $B$ $5$ minutes. $\newline$ a. Write formed an IVP for the problems. $\newline$ b. How much is formed at any time $B$ $6$ ? $\newline$ c. How much is formed in $B$ $7$ minutes? $\newline$ d. What is the limiting amount of $C$ after a long time? $\newline$ e. How much of chemicals $A$ and $B$ remains after a long time? $\newline$ $2$ . Solve the Cauchy-Euler equation $C$ $1$ ; $C$ $2$ , $C$ $3$

Full solution

Q. Instructions: Attempt all questions

\newline

1

. Two chemicals

A

and

B

are combined to form a chemical

C

. The rate, or velocity, of the reaction is proportional to the product of the instantaneous amounts of

A

and

B

not converted to chemical

C

. Initially, there are

100

grams of

4

and

50

grams of

B

, and for each gram of

B

B

1

grams of

A

is used. It is observed that

B

3

grams of

C

is formed in

B

5

minutes.

\newline

a. Write formed an IVP for the problems.

\newline

b. How much is formed at any time

B

6

\newline

c. How much is formed in

B

7

minutes?

\newline

d. What is the limiting amount of

C

after a long time?

\newline

e. How much of chemicals

A

and

B

remains after a long time?

\newline

2

. Solve the Cauchy-Euler equation

C

1

;

C

2

C

3

Given expression: The expression $6n-300$ is given to calculate the profit from the dance.
Revenue calculation: The term $6n$ represents the total revenue from selling $n$ tickets at $\$6$ each.
Calculation of $6n$ : To find out what $6n$ represents, we multiply the number of tickets, $n$ , by the price per ticket, which is $\$6$ .
Total revenue calculation: So, $6n = 6 \times n$ , which is the total amount of money collected from selling $n$ tickets.
Representation of $6n$ : Therefore, $6n$ represents the amount of money the school charges for all $n$ tickets.