In a fund raising activity, the Drama Club sells two types of ticket for a cultural show.

The club sells \(x\) tickets to members and \(y\) tickets to non-members.

The price of a ticket sold to members is \(\text{RM }20\) while the price of a ticket sold to non-members is \(\text{RM }30\).

The number of tickets sold depends on the following constraints.

i   : The number of tickets sold to non-members cannot exceed three times the number of tickets sold to members.

ii  : The hall can only accommodate a total of \(80\) people.

iii : The club needs to collect at least \(\text{RM }P\) for the show which is represented by the shaded region in the graph below.

Find the value of \(P\).

\(1\, 100\)

In a fund raising activity, the Drama Club sells two types of ticket for a cultural show.

The club sells \(x\) tickets to members and \(y\) tickets to non-members.

The price of a ticket sold to members is \(\text{RM }20\) while the price of a ticket sold to non-members is \(\text{RM }30\).

The number of tickets sold depends on the following constraints.

i   : The number of tickets sold to non-members cannot exceed three times the number of tickets sold to members.

ii  : The hall can only accommodate a total of \(80\) people.

iii : The club needs to collect at least \(\text{RM }P\) for the show which is represented by the shaded region in the graph below.

Construct and shade the region \(R\) which satisfies all the above constraints.

From the graph constraints, find the maximum total amount collected if the number of tickets sold to non-members is the same as the number of tickets sold to members.

\(\text{RM } 2\,000\)

A factory produces two types of computer chips, \(X\) and \(Y\), using two machines, \(A\) and \(B\).

In one week, the two machines produce \(x\) units of chip \(X\) and \(y\) units of chip \(Y\).

Machine \(A\) requires \(30\) minutes to produce a unit of \(X\) and \(25\) minutes to produce a unit of \(Y\).

Machine \(B\) needs \(25\) minutes to produce a unit of \(X\) and \(75\) minutes to produce a unit of \(Y\).

The factory production is limited by the following constraints.

i   : The total time machine \(A\) functions is not more than \(4\,800\) minutes.

ii  : The number of chip \(Y\) produced is at most twice the number of chip \(X\) produced. 

iii : The total time machine \(B\) functions is at least \(2 \, 625\) minutes.

Write all the inequalities satisfying the constraints above.

\(\begin{aligned} \text{I}&:30x+25y\ge 4\,800 \\\\ \text{II}&:y\ge2x\\\\ \text{III}&:25x+75y \le 2 \,625 \end{aligned}\)

A factory produces two types of computer chips, \(X\) and \(Y\), using two machines, \(A\) and \(B\).

In one week, the two machines produce \(x\) units of chip \(X\) and \(y\) units of chip \(Y\).

Machine \(A\) requires \(30\) minutes to produce a unit of \(X\) and \(25\) minutes to produce a unit of \(Y\).

Machine \(B\) needs \(25\) minutes to produce a unit of \(X\) and \(75\) minutes to produce a unit of \(Y\).

The factory production is limited by the following constraints.

i   : The total time machine \(A\) functions is not more than \(4\,800\) minutes.

ii  : The number of chip \(Y\) produced is at most twice the number of chip \(X\) produced. 

iii : The total time machine \(B\) functions is at least \(2 \, 625\) minutes.

Using a scale of \(2 \text{ cm}\) to \(20\) units of computer chips on both axes, construct and shade the region \(R\) which satisfies the constraints above.

Based on the graph, find the maximum units of \(X\) if the factory works to produce \(84\) units of chip \(Y\) in one week.

\(70 \text{ units}\)

A restaurant wants to buy two types of tables, \(M\) and \(N\), for its new outlet.

The price of a type \(M\) table is \(\text{RM }320\) and a type \(N\) table is \(\text{RM }240\).

The area of a tabletop of type \(M\) table and type \(N\) table are \(1 \text{ m}^2\) and \(2\text{ m}^2\) respectively.

The restaurant buys \(x\) tables of type \(M\) and \(y\) tables of type \(M\) and \(y\) tables of type \(N\).

The purchase of the table is based on the following constraints:

i   : The total area of all the tabletops is not less than \(60 \text{ m}^2\).

ii  : The total amount of money allocated is \(\text{RM } 19\,200\).

iii : The number of type \(N\) table is at most two times the number of type \(M\) table.

Write all the inequalities satisfying the constraints above.

\(\begin{aligned} \text{I}&:x+2y\gt 60 \\\\ \text{II}&:320x+240y\lt 19\,200\\\\ \text{III}&:y\lt2x\end{aligned}\)