The speed of a ship is increased to 18% above normal for 7.5 hours, then reduced to 9% below normal for 10 hours. The speed is then reduced for the remainder of the day so that the consumption for the day is the normal amount. Find the percentage difference between the distance travelled in that day and the normal distance travelled per day.
✓ Verified Model Answer (Text Solution)
Structured for DG Shipping MEO Class II examination scoring criteria.
Let normal speed be V and normal daily consumption be C. Consumption varies as the cube of speed, so C = k.V^3.
Step 1 - Find the speed for the remainder of the day.
For 7.5 hours the speed is 1.18V. Consumption in that period = k.(1.18V)^3 x 7.5/24 = k.V^3 x 1.643 x 0.3125 = 0.5134 k.V^3.
For 10 hours the speed is 0.91V. Consumption = k.(0.91V)^3 x 10/24 = k.V^3 x 0.7536 x 0.4167 = 0.3140 k.V^3.
Total consumption in first 17.5 hours = 0.5134 + 0.3140 = 0.8274 k.V^3.
Remaining time in the day = 24 - 17.5 = 6.5 hours.
For the day's total consumption to equal the normal amount k.V^3, the remaining consumption must be k.V^3 - 0.8274 k.V^3 = 0.1726 k.V^3.
If the reduced speed is Vr, then k.Vr^3 x 6.5/24 = 0.1726 k.V^3.
So Vr^3 = 0.1726 x 24/6.5 x V^3 = 0.6373 V^3.
Vr = (0.6373)^(1/3) V = 0.8606 V.
So the ship travels at 86.06% of normal speed for the last 6.5 hours.
Step 2 - Find the distance travelled that day.
Distance = speed x time.
Normal distance per day = V x 24 = 24V.
Actual distance = 1.18V x 7.5 + 0.91V x 10 + 0.8606V x 6.5
= 8.85V + 9.10V + 5.594V = 23.544V.
Step 3 - Percentage difference.
Difference = 24V - 23.544V = 0.456V.
Percentage difference = 0.456/24 x 100 = 1.9%.
Answer: The distance travelled that day is 1.9% less than the normal distance per day.