A boat travels due north from A for 30 minutes at 20 km/h. It reaches B and then travels due east for 24 minutes at 10 km/h. It is then at C. (a) How many kilometers has the boat travelled? (b) On the diagram, draw a line segment that shows the shortest distance from C back to A. (c) Use Pythagoras’ theorem to calculate the shortest distance from C to A. Give your answer correct to the nearest metre.

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A boat travels due north from A for 30 minutes at 20 km/h - Junior Cycle Mathematics - Question 11 - 2012

Question 11

A boat travels due north from A for 30 minutes at 20 km/h. It reaches B and then travels due east for 24 minutes at 10 km/h. It is then at C. (a) How many kilometer... show full transcript

Worked Solution & Example Answer:A boat travels due north from A for 30 minutes at 20 km/h - Junior Cycle Mathematics - Question 11 - 2012

Step 1

How many kilometers has the boat travelled?

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Answer

To find the distance traveled by the boat, we calculate the distance for each segment of the journey:

Distance from A to B: The boat travels for 30 minutes at a speed of 20 km/h.

First, convert 30 minutes into hours:

$30 ext{ minutes} = \frac{30}{60} \text{ hours} = 0.5 \text{ hours}$

Now, calculate the distance:

$\text{Distance} = \text{Speed} \times \text{Time} = 20 \text{ km/h} \times 0.5 \text{ hours} = 10 \text{ km}$
Distance from B to C: The boat then travels for 24 minutes at a speed of 10 km/h.

Convert 24 minutes into hours:

$24 ext{ minutes} = \frac{24}{60} \text{ hours} = 0.4 \text{ hours}$

Calculate the distance:

$\text{Distance} = \text{Speed} \times \text{Time} = 10 \text{ km/h} \times 0.4 \text{ hours} = 4 \text{ km}$
Total distance travelled:

Therefore, the total distance travelled by the boat is:

$\text{Total Distance} = 10 \text{ km} + 4 \text{ km} = 14 \text{ km}$

Step 2

On the diagram, draw a line segment that shows the shortest distance from C back to A.

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Answer

To illustrate the shortest distance from C back to A, draw a straight line between points C and A on the diagram. This line segment should visually represent the direct route between the final destination C and the starting point A.

Step 3

Use Pythagoras’ theorem to calculate the shortest distance from C to A.

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Answer

To calculate the shortest distance from C back to A using Pythagoras' theorem, we note that the path forms a right triangle:

The length from A to B is 10 km (vertical leg). 2. The length from B to C is 4 km (horizontal leg).

Using Pythagoras' theorem:

$c^2 = a^2 + b^2$

Here, let $a = 10$ km and $b = 4$ km. Plugging in these values gives:

$c^2 = 10^2 + 4^2 = 100 + 16 = 116$

To find c, we take the square root:

$c = \sqrt{116} \approx 10.77$

Rounding to the nearest metre, the shortest distance from C to A is approximately 11 km.

A boat travels due north from A for 30 minutes at 20 km/h - Junior Cycle Mathematics - Question 11 - 2012

Worked Solution & Example Answer:A boat travels due north from A for 30 minutes at 20 km/h - Junior Cycle Mathematics - Question 11 - 2012

How many kilometers has the boat travelled?

On the diagram, draw a line segment that shows the shortest distance from C back to A.

Use Pythagoras’ theorem to calculate the shortest distance from C to A.

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