Question

Given below is a circular Park with centre A. Madhav walks at a uniform speed of 0.5 m/s from Gate P and reaches the centre of the park in 150 seconds .What is the straight line distance between the centre of the park and gate M?
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Answer 1

Answer:

Step-by-step explanation

speed = distance/time

0.5m/s = S / 150s

S = 150 × 0.5

S = 75m

Answer 2

The straight line distance between the center of the park and gate M is 129.1 m

To find,

The straight line distance between the center of the park and gate M.

Given,

Madhav walks at a uniform speed of 0.5 m/s from Gate P and

Reaches the center of the park in 150 seconds

Solution,

Since Madhav walks at a uniform speed and reaches the center of the park in 150 seconds, we can use the formula:

distance = speed x time

Let us assume that the radius of the circular park is r. Then, the distance Madhav walks from gate P to center A is equal to the length of the radius r. We can calculate the length of the radius using the formula:

length of the radius = speed x time

So, length of the radius = 0.5 m/s x 150 s = 75 m

Now, we need to find the straight line distance between the center of the park and gate M. To do this, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

In the given figure, we can see that the line segment PM is the hypotenuse of a right triangle PAM, where PA is the length of the radius of the park and PM is the straight line distance we want to find.

Let us assume that the length of PM is d. Then, we can apply the Pythagorean theorem as follows:

d^2 = PA^2 + AM^2

We know that the length of PA (the radius) is 75 m. To find the length of AM, we can use the fact that AM is equal to the radius minus the distance Madhav walks from gate M to center A. So,

AM = PA - length of the radius walked by Madhav from M to A

We can find the length of the radius walked by Madhav from M to A using the formula:

length of the radius walked = speed x time

The speed is the same as before (0.5 m/s), and the time is given by:

time = distance / speed = PM / speed

So, we can write:

length of the radius walked = 0.5 x (PM / 0.5) = PM

Substituting this in the formula for AM, we get:

AM = PA - PM = 75 - PM

Now, we can substitute the values of PA and AM in the Pythagorean theorem equation and solve for PM:

d^2 = PA^2 + AM^2

PM = sqrt(d^2 - PA^2) = sqrt((75 - PM)^2 - 75^2)

Simplifying this equation, we get:

PM^2 + 2 x 75 x PM - (2 x 75^2) = 0

Solving this quadratic equation, we get:

PM = 75 x (sqrt(5) - 1)

Therefore, the straight line distance between the center of the park and gate M is: PM = 75 x (sqrt(5) - 1) ≈ 129.1 m (rounded to one decimal place)

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