A person burns 10 calories per minute while jogging and 5 calories per minute while cycling. He wants to burn at least 400 calories a day, but his doctor allows him at most 30 minutes of jogging a day. The least total time (jogging and cycling together) he needs each day is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 50 minutes
Let him jog for x minutes and cycle for y minutes. Then
Minimise \(Z = x + y\)
subject to
\(10x + 5y \ge 400\), that is, \(2x + y \ge 80\)
\(x \le 30\)
\(x \ge 0, \; y \ge 0\)
The line \(2x + y = 80\) meets \(x = 30\) at
\(y = 80 - 60 = 20\)
The region is unbounded (y can be as large as we like), with corner points (0, 80) and (30, 20). Values of Z:
\(Z(0, 80) = 80\)
\(Z(30, 20) = 50\)
For every feasible point, \(y \ge 80 - 2x\) and \(x \le 30\), so
\(x + y \ge 80 - x \ge 50\)
So the least time is 50 minutes: 30 minutes of jogging and 20 minutes of cycling.
Why the other options are incorrect
Option (1) 40 minutes — Incorrect. This needs 40 minutes of jogging alone, which breaks \(x \le 30\).
Option (3) 60 minutes — Incorrect. It can be done in 50 minutes, so 60 is not the least.
Option (4) 80 minutes — Incorrect. This is the time for cycling only, at (0, 80).