Ravi rides his motor cycle at 25 km per hour, he has to spend Rs. 4 per kilometer on petrol, if he rides at a faster speed of 40 km per hour, the petrol cost increases to Rs. 10 per kilometer. He can spend Rs. 200 on petrol and wishes to maximize distance he can travel within one hour. Which of the following is/are TRUE?
A. Maximum distance covered in one hour is 30 km
B. The distance covered at speed of 25km/hr is 50/3 km
C. The distance covered at speed of 25km/hr is \( \displaystyle\frac{25}{2} \) km
D. The distance covered at speed of 40km/hr is \( \displaystyle\frac{75}{4} \) km
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A and B only
Let him ride x km at 25 km/hr and y km at 40 km/hr. This is a linear programming problem: maximise Z = x + y subject to the time and cost limits.
Time (at most one hour):
\( \displaystyle \frac{x}{25}+\frac{y}{40}\le 1 \)
\( 8x+5y\le 200 \)
Cost (at most Rs. 200):
\( 4x+10y\le 200 \)
\( 2x+5y\le 100,\qquad x,y\ge 0 \)
The two boundary lines meet where
\( (8x+5y)-(2x+5y)=200-100 \)
\( 6x=100 \)
\( \displaystyle x=\frac{50}{3},\qquad y=\frac{40}{3} \)
Corner points:
At (25, 0): Z = 25
At (0, 20): Z = 20
At (50/3, 40/3): Z = 30
So the maximum distance is 30 km, with 50/3 km at 25 km/hr and 40/3 km at 40 km/hr.
Checking the statements
(A) Maximum distance covered in one hour is 30 km — Correct. The best corner gives 50/3 + 40/3 = 30 km.
(B) The distance covered at speed of 25km/hr is 50/3 km — Correct. At the best corner, x = 50/3 km.
(C) The distance covered at speed of 25km/hr is \( \displaystyle\frac{25}{2} \) km — Incorrect. The distance at 25 km/hr is 50/3 km, not 25/2 km.
(D) The distance covered at speed of 40km/hr is \( \displaystyle\frac{75}{4} \) km — Incorrect. The distance at 40 km/hr is 40/3 km, not 75/4 km.