Ball Adventure
A ball is propelled along a ramp of height 1m above the ground. . For the first 10m, the gradient of the ramp is zero. For the next 30m of horizontal distance, the rail resembles the upper half of a circle. Then, it rolls down a ramp of length 30m, carrying the ball down to ground level. Find the area below the path of the ball (i.e. the rail) for the whole journey. Round your answer, in square metres, to two decimal places.
Notes: neglect the size/height of the ball. All rails are connected.
The answer is 408.42.
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1 solution
Area (first 10m)= 10 x 1=10m²
Area under semicircle= Area (semicircle) + Area (rectangle under semicircle)
Diameter of semicircle is 30m, so the radius is 30/2=15m.
Area of semicircle = (πr²)/2 = (15)²xπ/2 = 353.43m²
Area of rectangle = 30 x 1=30m²
Area under semicircle = 353.43+30 = 383.43m²
Ramp is equivalent to the hypotenuse of a right-angled triangle.
Let the horizontal distance (base) be x.
x² + 1² = 30²
x² = 30² - 1² = 899
x= √899 = 29.98 (take positive value)
Area of triangle= 1/2bh = 1/2 x 1 x 29.98 = 14.992
Total area = 10+30+353.43+14.99 = 408.42m² (2 d.p.)