Answer
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Hint: In this question, let us find the distance travelled with a speed of 70 km/hr in the first two hours. Then again increase speed by 10 km/hr and find the distance travelled in the next 2 hours. Now, subtract the total distance travelled in these 4 hours from 345 and then calculate the time required to travel that remaining distance with 90 km/hr. Then add this to the 4 hours to get the result.
Complete step-by-step solution -
Now, as we already know the relation between speed, time and distance we have
Let us assume speed as v, distance as d and time as t
\[v=\dfrac{d}{t}\]
Now, from the given conditions in the question let us assume speeds at the interval of 2 hours as v1, v2, v3
Now, from the conditions we have that
\[{{v}_{1}}=70,{{v}_{2}}=80,{{v}_{3}}=90\]
let us assume the distance travelled in first 2 hours as d1 and in next 2 hours as d2
Let us now calculate the distance travelled in first 2 hours
Here, we have that
\[{{v}_{1}}=70,{{t}_{1}}=2\]
Now, from the formula we get,
\[\Rightarrow {{v}_{1}}=\dfrac{{{d}_{1}}}{{{t}_{1}}}\]
Now, on substituting the respective values we get,
\[\Rightarrow 70=\dfrac{{{d}_{1}}}{2}\]
Now, on rearranging the terms we get,
\[\Rightarrow {{d}_{1}}=70\times 2\]
\[\therefore {{d}_{1}}=140km\]
Now, let us find the distance travelled in next 2 hours
Here, we have
\[{{v}_{2}}=80,{{t}_{2}}=2\]
Now, from the formula we get,
\[\Rightarrow {{v}_{2}}=\dfrac{{{d}_{2}}}{{{t}_{2}}}\]
Now, on substituting the values we get,
\[\Rightarrow 80=\dfrac{{{d}_{2}}}{2}\]
Now, on rearranging the terms we get,
\[\begin{align}
& \Rightarrow {{d}_{2}}=80\times 2 \\
& \therefore {{d}_{2}}=160km \\
\end{align}\]
Now, we need to subtract the distance travelled till now from the given total distance
\[\Rightarrow 345={{d}_{1}}+{{d}_{2}}+{{d}_{3}}\]
Now, on substituting the respective values we get,
\[\Rightarrow 345=140+160+{{d}_{3}}\]
Now, on rearranging and further simplifying we get,
\[\Rightarrow {{d}_{3}}=45km\]
Now, let us calculate the time taken to travel this distance with speed of 90 km/hr
Here, we have
\[{{v}_{3}}=90,{{d}_{3}}=45\]
Now, from the formula we get,
\[\Rightarrow {{v}_{3}}=\dfrac{{{d}_{3}}}{{{t}_{3}}}\]
Let us now substitute the respective terms
\[\Rightarrow 90=\dfrac{45}{{{t}_{3}}}\]
Now, on rearranging the terms we get,
\[\Rightarrow {{t}_{3}}=\dfrac{45}{90}\]
Now, on further simplification we get,
\[\therefore {{t}_{3}}=\dfrac{1}{2}hr\]
Let us now calculate the total time taken to travel 345 kms
\[t=2+2+\dfrac{1}{2}\]
\[\therefore t=4\dfrac{1}{2}hr\]
Hence, the correct option is (c).
Note: Instead of finding the distance traveled in the first two intervals and the time in the third interval we can directly find the time by calculating the average velocity and then dividing it with the 345 km. Both methods give the same result. It is important to note that to get the total time we need to find the time in the third interval using the distance and speed as they are known.
Complete step-by-step solution -
Now, as we already know the relation between speed, time and distance we have
Let us assume speed as v, distance as d and time as t
\[v=\dfrac{d}{t}\]
Now, from the given conditions in the question let us assume speeds at the interval of 2 hours as v1, v2, v3
Now, from the conditions we have that
\[{{v}_{1}}=70,{{v}_{2}}=80,{{v}_{3}}=90\]
let us assume the distance travelled in first 2 hours as d1 and in next 2 hours as d2
Let us now calculate the distance travelled in first 2 hours
Here, we have that
\[{{v}_{1}}=70,{{t}_{1}}=2\]
Now, from the formula we get,
\[\Rightarrow {{v}_{1}}=\dfrac{{{d}_{1}}}{{{t}_{1}}}\]
Now, on substituting the respective values we get,
\[\Rightarrow 70=\dfrac{{{d}_{1}}}{2}\]
Now, on rearranging the terms we get,
\[\Rightarrow {{d}_{1}}=70\times 2\]
\[\therefore {{d}_{1}}=140km\]
Now, let us find the distance travelled in next 2 hours
Here, we have
\[{{v}_{2}}=80,{{t}_{2}}=2\]
Now, from the formula we get,
\[\Rightarrow {{v}_{2}}=\dfrac{{{d}_{2}}}{{{t}_{2}}}\]
Now, on substituting the values we get,
\[\Rightarrow 80=\dfrac{{{d}_{2}}}{2}\]
Now, on rearranging the terms we get,
\[\begin{align}
& \Rightarrow {{d}_{2}}=80\times 2 \\
& \therefore {{d}_{2}}=160km \\
\end{align}\]
Now, we need to subtract the distance travelled till now from the given total distance
\[\Rightarrow 345={{d}_{1}}+{{d}_{2}}+{{d}_{3}}\]
Now, on substituting the respective values we get,
\[\Rightarrow 345=140+160+{{d}_{3}}\]
Now, on rearranging and further simplifying we get,
\[\Rightarrow {{d}_{3}}=45km\]
Now, let us calculate the time taken to travel this distance with speed of 90 km/hr
Here, we have
\[{{v}_{3}}=90,{{d}_{3}}=45\]
Now, from the formula we get,
\[\Rightarrow {{v}_{3}}=\dfrac{{{d}_{3}}}{{{t}_{3}}}\]
Let us now substitute the respective terms
\[\Rightarrow 90=\dfrac{45}{{{t}_{3}}}\]
Now, on rearranging the terms we get,
\[\Rightarrow {{t}_{3}}=\dfrac{45}{90}\]
Now, on further simplification we get,
\[\therefore {{t}_{3}}=\dfrac{1}{2}hr\]
Let us now calculate the total time taken to travel 345 kms
\[t=2+2+\dfrac{1}{2}\]
\[\therefore t=4\dfrac{1}{2}hr\]
Hence, the correct option is (c).
Note: Instead of finding the distance traveled in the first two intervals and the time in the third interval we can directly find the time by calculating the average velocity and then dividing it with the 345 km. Both methods give the same result. It is important to note that to get the total time we need to find the time in the third interval using the distance and speed as they are known.
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