Certainly! Let's break this problem down step-by-step.
Darpan runs a distance of 12 km and cycles a distance of 26 km. His running speed is [tex]\(x \, \text{km/h}\)[/tex], and his cycling speed is 10 km/h faster than his running speed, which is [tex]\((x + 10) \, \text{km/h}\)[/tex]. He takes a total time of 2 hours and 48 minutes. First, we need to express this total time purely in hours.
Since 48 minutes is [tex]\(\frac{48}{60}\)[/tex] or 0.8 hours, the total time in hours Darpan takes is:
[tex]\[ 2 + 0.8 = 2.8 \, \text{hours} \][/tex]
Part (a):
An expression for the time, in hours, Darpan takes to run the 12 km is [tex]\(\frac{12}{x}\)[/tex].
Part (b):
Now, let's write an expression for the time, in hours, he takes to cycle the 26 km. If his cycling speed is [tex]\((x + 10) \, \text{km/h}\)[/tex], the time taken to cycle 26 km is:
[tex]\[ \frac{26}{x + 10} \][/tex]
Equation:
The total time taken for both running and cycling is the sum of the time spent on running and the time spent on cycling. Since the total time is 2.8 hours, the equation for the total time in terms of [tex]\( x \)[/tex] becomes:
[tex]\[ \frac{12}{x} + \frac{26}{x + 10} = 2.8 \][/tex]
This is the required equation in terms of [tex]\(x\)[/tex] for the total time Darpan takes. To summarize:
1. The time taken to run 12 km is [tex]\( \frac{12}{x} \)[/tex].
2. The time taken to cycle 26 km is [tex]\( \frac{26}{x + 10} \)[/tex].
3. The total time equation is [tex]\( \frac{12}{x} + \frac{26}{x + 10} = 2.8 \)[/tex].
