find the distance traveled by a train on a circular track road with radius of 78.8 m which will cause the train to change direct
ion by 21.3 degrees. Round to desiml place.
1 answer:
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First, we have to convert our function (of x) into a function of y (we revolve the curve around the y-axis). So:
And the derivative of x:
Now, we can calculate the area of the surface:
We could calculate this integral (not very hard, but long), or use (1), (2) and (3) to get:
Calculate indefinite integral:
And the area:
![A=2\pi\int\limits_0^{10}x\sqrt{4x^2+1}\,dx=2\pi\cdot\dfrac{1}{12}\bigg[\left(4x^2+1\right)^\frac{3}{2}\bigg]_0^{10}=\\\\\\= \dfrac{\pi}{6}\left[\big(4\cdot10^2+1\big)^\frac{3}{2}-\big(4\cdot0^2+1\big)^\frac{3}{2}\right]=\dfrac{\pi}{6}\Big(\big401^\frac{3}{2}-1^\frac{3}{2}\Big)=\boxed{\dfrac{401^\frac{3}{2}-1}{6}\pi}](https://tex.z-dn.net/?f=A%3D2%5Cpi%5Cint%5Climits_0%5E%7B10%7Dx%5Csqrt%7B4x%5E2%2B1%7D%5C%2Cdx%3D2%5Cpi%5Ccdot%5Cdfrac%7B1%7D%7B12%7D%5Cbigg%5B%5Cleft%284x%5E2%2B1%5Cright%29%5E%5Cfrac%7B3%7D%7B2%7D%5Cbigg%5D_0%5E%7B10%7D%3D%5C%5C%5C%5C%5C%5C%3D%20%5Cdfrac%7B%5Cpi%7D%7B6%7D%5Cleft%5B%5Cbig%284%5Ccdot10%5E2%2B1%5Cbig%29%5E%5Cfrac%7B3%7D%7B2%7D-%5Cbig%284%5Ccdot0%5E2%2B1%5Cbig%29%5E%5Cfrac%7B3%7D%7B2%7D%5Cright%5D%3D%5Cdfrac%7B%5Cpi%7D%7B6%7D%5CBig%28%5Cbig401%5E%5Cfrac%7B3%7D%7B2%7D-1%5E%5Cfrac%7B3%7D%7B2%7D%5CBig%29%3D%5Cboxed%7B%5Cdfrac%7B401%5E%5Cfrac%7B3%7D%7B2%7D-1%7D%7B6%7D%5Cpi%7D)
Answer D.
And the derivative of x:
Now, we can calculate the area of the surface:
We could calculate this integral (not very hard, but long), or use (1), (2) and (3) to get:
Calculate indefinite integral:
And the area:
Answer D.
Answer:
Shown
Step-by-step explanation:
Given that twelve basketball players, whose uniforms are numbered 1 through 12, stand around the center ring on the court in an arbitrary arrangement.
Let us consider consecutive numbers in this set.
After this we find the totals are more than 20.
When 1 to 12 are arbitrarily arranged, there are chances that numbers from 6 and above are having consecutive numbers.
These totals are greater than 20
Hence shown that some three consecutive players have the sum of their numbers at least 20.
(i.e. starting from if we take)