An airplane took off from the starting point 45 minutes later than the scheduled time. The destination was 2100 km away from the starting point. To reach on time, the pilot had to increase the speed by 40% of its usual speed. What was the increased speed (in km/h)?

Options

A. 1870
B. 2520
C. 2940
D. 1120

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Problem
The airplane took off 45 minutes late and needs to cover a distance of 2100 km. To reach on time, the pilot increased the speed by 40% of its usual speed.

### Step 2: Convert Time to Hours
Since the airplane took off 45 minutes late, we need to convert this time into hours:
\[ \text{Time in hours} = \frac{45 \text{ minutes}}{60} = 0.75 \text{ hours} \]

### Step 3: Define Variables
Let \( x \) be the usual speed of the airplane in km/h. Therefore, the increased speed of the airplane will be:
\[ \text{Increased Speed} = x + 0.4x = 1.4x \text{ km/h} \]

### Step 4: Calculate the Scheduled Time
Let \( t_s \) be the scheduled time to cover 2100 km at the usual speed \( x \):
\[ t_s = \frac{2100}{x} \text{ hours} \]

### Step 5: Calculate the Actual Time Taken
Since the airplane took off 45 minutes late, the actual time taken to reach the destination is:
\[ \text{Actual Time} = t_s - 0.75 = \frac{2100}{x} - 0.75 \]

### Step 6: Set Up the Equation
Using the increased speed, the time taken to cover 2100 km can also be expressed as:
\[ \text{Time at increased speed} = \frac{2100}{1.4x} \]

Since both expressions represent the time taken to cover the same distance, we can set them equal:
\[ \frac{2100}{x} - 0.75 = \frac{2100}{1.4x} \]

### Step 7: Solve the Equation
Multiply through by \( 1.4x \) to eliminate the denominators:
\[ 1.4 \cdot 2100 - 0.75 \cdot 1.4x = 2100 \]
\[ 2940 - 1.05x = 2100 \]
\[ 2940 - 2100 = 1.05x \]
\[ 840 = 1.05x \]
\[ x = \frac{840}{1.05} = 800 \text{ km/h} \]

### Step 8: Calculate the Increased Speed
Now that we have the usual speed, we can find the increased speed:
\[ \text{Increased Speed} = 1.4x = 1.4 \times 800 = 1120 \text{ km/h} \]

### Final Answer
The increased speed of the airplane is **1120 km/h**.

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