An aircraft travels at a speed of 400 km h⁻¹ in still air - Leaving Cert Applied Maths - Question 2 - 2018

To calculate the distance traveled by the aircraft after 20 minutes, we can use the formula:

$\text{Distance} = \text{Speed} \times \text{Time}$

Given the speed of the aircraft is 400 km h⁻¹, first convert 20 minutes into hours:

$20 \text{ minutes} = \frac{20}{60} = \frac{1}{3} \text{ hours}$

Now, substituting the values:

$\text{Distance} = 400 \times \frac{1}{3} = 133.33 \text{ km}$

Therefore, after 20 minutes, the aircraft is approximately 133.33 km away from point P.

The time taken to cross the river can be calculated using the formula:

$\text{time} = \frac{\text{distance}}{\text{speed}}$

Here, the distance is 60 m and the speed of the woman is 1 m s⁻¹. Therefore,

$\text{time} = \frac{60}{1} = 60 \text{ s}$

Thus, it will take the woman 60 seconds to cross from bank to bank.

To find the distance travelled by the boat, we use the geometry involved in crossing the river. The maximum angle for the boat, given the current speed, is given by the relationship:

$\sin(\beta) = \frac{1}{4}$

From this, we can deduce the angle

$\beta = 90°$

And applying it in the river crossing problem, we find:

Using the relation (\text{Using } 60 = 1 \cdot 4t), we can find (d):

$d = 240 \text{ m}$

Thus, the distance travelled by the boat when it crosses by the shortest path is 240 m.