Question

Charu is cycling in a park. He chooses a path which is a closed curve, partlyparabolic. At the point where Charu started, the path forms a right angle. Charumakes a complete round on the path. The graph shows the path of cycling. D is vertexof parabolic curve which intersects X axis at C and E. Also ∠BAF is a right angle.If the length of the track is 1km and the boy is moving with the speed of 18km/h and if he increases his speed by 6km/h, find the time saved by the boy in completing one round of the path(a) 50 sec (b) 20 sec (c) 45 sec (d) 15 sec​

Answer 1

Answer:

Answer is 45 secs.

I hope this is correct .

Answer 2

The total length of the path is:

L = AB + BD + DA

= √160 + 8.208 + 4√

To solve this problem, we need to find the length of the path Charu takes, and then compare the time it takes him to complete one round at 18 km/h with the time it takes him to complete one round at 24 km/h. The difference between the two times will be the time saved.

Let's start by finding the length of the path. We can do this by breaking the path into three segments: the straight line from A to B, the parabolic curve from B to D, and the straight line from D to A.

The length of the straight line from A to B is AB = √(AF² + FB²) = √(4² + 12²) = √160.

The length of the parabolic curve from B to D can be found using calculus. The equation of the parabolic curve is y = ax² + b, where a is a negative constant (since the parabola opens downwards) and b is the y-intercept. We can find the values of a and b by using the fact that the curve passes through points B, C, and E.

At point B, we know that y = 12 and x = 4. When we enter these values into the formula, we obtain

12 = 16a + b

At point C, we know that y = 0 and x = 2. When we enter these values into the formula, we obtain

0 = 4a + b

Solving these two equations simultaneously, we get:

a = -3/8 and b = 3

So the equation of the parabolic curve is y = -3/8 x² + 3. To find the length of the curve from B to D, we need to integrate the formula for arc length:

L =$∫√(1 + [dy/dx]²) dx$

We can find dy/dx by taking the derivative of y with respect to x:

dy/dx = -3/4 x

Plugging this into the formula for arc length, we get:

L = ∫√(1 + (-3/4 x)²) dx

= (1/3)(16/9)∫√(1 + (-3/4 x)²) d(-3/4 x)

= (4/3)∫√(1 + u²) du, where u = -3/4 x

= (4/3)(1/2)(u√(1 + u²) + ln|u + √(1 + u²)|) + C

= (2/3)(-3/4 x)√(1 + (-3/4 x)²) + (8/9)ln|4/3 x + √(1 + (-3/4 x)²)| + C

Evaluating this integral from x = 2 to x = 6 (the x-coordinates of points C and E), we get:

L = (2/3)(-3/4)(6√10 - 2√10) + (8/9)ln(8√10/3)

= -√10 + (8/9)ln(8√10/3)

≈ 8.208 km

The length of the straight line from D to A is DA = √(AF² + FD²) = √(4² + 4²) = 4√2.

So the total length of the path is:

L = AB + BD + DA

= √160 + 8.208 + 4√

For such more questions on angle,

https://brainly.in/question/18979649

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