QuestionA cyclist travels from A to B at 250 m/min, covering an angle of $1^{\circ} 5^{\prime}$ on Earth with radius $6440 \text{ km}$ . Find the time taken in hours.

Studdy Solution

STEP 1

Assumptions1. The speed of the cyclist is250 meters per minute. . The angle formed by the two colleges at the center of the earth is $1^{\circ}5^{\prime}$ .
3. The radius of the earth is6440 km.
4. We assume that the earth is a perfect sphere.
5. The cyclist travels along the surface of the earth.

STEP 2

First, we need to find the distance between A college and B college. We can do this by using the formula for the arc length of a circle, which is $\theta \times r$ , where $\theta$ is the angle in radians and $r$ is the radius of the circle.
$Distance = \theta \times r$

STEP 3

Now, convert the angle from degrees and minutes to radians, as the formula for arc length requires the angle to be in radians.1 degree is equal to $\frac{\pi}{180}$ radians and1 minute is equal to $\frac{\pi}{10800}$ radians.
$\theta =1^{\circ}5^{\prime} =1 \times \frac{\pi}{180} +5 \times \frac{\pi}{10800}$

STEP 4

Calculate the value of $\theta$ .
$\theta =1 \times \frac{\pi}{180} + \times \frac{\pi}{10800} =0.017 +0.000145 =0.017645 \, radians$

STEP 5

Now, plug in the given values for $\theta$ and $r$ to calculate the distance.
$Distance = \theta \times r =0.017645 \, radians \times6440 \, km$

STEP 6

Calculate the distance.
$Distance =0.017645 \, radians \times6440 \, km =113.6 \, km$

STEP 7

Now that we have the distance, we can find the time taken by the cyclist to travel from A college to B college. We can do this by dividing the distance by the speed of the cyclist.
$Time = \frac{Distance}{Speed}$

STEP 8

Convert the speed of the cyclist from meters per minute to kilometers per hour.
1 km =1000 meters and1 hour =60 minutes.
$Speed =250 \, meters/minute =250 \times \frac{1}{1000} \, km/minute =0.25 \, km/minute =0.25 \times60 \, km/hour =15 \, km/hour$

STEP 9

Now, plug in the values for the distance and the speed to calculate the time.
$Time = \frac{Distance}{Speed} = \frac{113.6 \, km}{15 \, km/hour}$

STEP 10

Calculate the time taken by the cyclist.
$Time = \frac{113.6 \, km}{15 \, km/hour} =7.57 \, hours$ The cyclist will take approximately7.57 hours to travel from A college to B college.

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