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(b) A driver traveled 108 km at an average speed of [tex]72 \text{ km/h}[/tex]. He then traveled at an average speed of [tex]76 \text{ km/h}[/tex] for the next [tex]2 \frac{1}{2}[/tex] hours. Calculate:

(i) the time taken for the first 108 km of the journey,

(ii) the average speed for the whole journey.

Sagot :

GinnyAnsweridn
Sure! Let's break down the problem step-by-step:

### Part (i)
Calculate the time taken for the first 108 km of the journey:

To find the time taken for a journey, you can use the formula:
[tex]\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \][/tex]

Given:
- Distance for the first part: [tex]\( 108 \)[/tex] km
- Speed for the first part: [tex]\( 72 \)[/tex] km/h

So, the time taken for the first part of the journey is:
[tex]\[ \text{Time}_1 = \frac{108 \text{ km}}{72 \text{ km/h}} = 1.5 \text{ hours} \][/tex]

Thus, the time taken for the first 108 km of the journey is [tex]\(1.5\)[/tex] hours.

### Part (ii)
Calculate the average speed for the whole journey:
First, we need to determine the total distance and total time for the entire journey.

1. The distance covered in the second part of the journey:

Given:
- Speed for the second part: [tex]\( 76 \)[/tex] km/h
- Time for the second part: [tex]\( 2.5 \)[/tex] hours

Distance can be calculated using the formula:
[tex]\[ \text{Distance} = \text{Speed} \times \text{Time} \][/tex]

So, the distance for the second part is:
[tex]\[ \text{Distance}_2 = 76 \text{ km/h} \times 2.5 \text{ hours} = 190 \text{ km} \][/tex]

2. Total distance:

Total distance = Distance of the first part + Distance of the second part
[tex]\[ \text{Total Distance} = 108 \text{ km} + 190 \text{ km} = 298 \text{ km} \][/tex]

3. Total time:

Total time = Time taken for the first part + Time taken for the second part
[tex]\[ \text{Total Time} = 1.5 \text{ hours} + 2.5 \text{ hours} = 4 \text{ hours} \][/tex]

4. Calculate the average speed:

Average speed is given by the formula:
[tex]\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \][/tex]

So, the average speed for the whole journey is:
[tex]\[ \text{Average Speed} = \frac{298 \text{ km}}{4 \text{ hours}} = 74.5 \text{ km/h} \][/tex]

Thus, the average speed for the whole journey is [tex]\(74.5\)[/tex] km/h.

In summary:
(i) The time taken for the first 108 km of the journey is [tex]\(1.5\)[/tex] hours.
(ii) The average speed for the whole journey is [tex]\(74.5\)[/tex] km/h.
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