Answer:
<em>Speed s= 19</em>
<u>Explanation:</u>
Take note of the following parameters:
Speed= s,
Number of cars= N,
Number N of cars that can pass a given spot per minute=
N(s)=88s/16+16(s19)2
The principle of differentiation is here:
We let N(s) = N
N = 86s / (17 + 17((s/19)^2))
17N = 86s /(1 + s²/19²)
17N = 361* 86s /(361 + s²)
17N = 31046s /(361 + s²)
Next step;
Differentiate with respect to s
17N = 31046s /(361 + s²)
Remember the quotient rule [u/v]’ = (vu’ - uv’) / v²
Therefore,
u = 31046s =====> du/ds = u’ = 31046
v = (361 + s²) ====>dv/ds = v’ = 2s
17N = 31046s /(361 + s²)
17 dN/ds = ( 31046(361 + s²) - 31046s(2s) ) / (361 + s²)²
The maximum when dN/ds = 0
17 dN/ds = ( 31046(361 + s²) - 31046s(2s) ) / (361 + s²)²
17 * 0 = ( 31046(361 + s²) - 31046s(2s) ) / (361 + s²)²
( 31046(361 + s²) - 31046s(2s) ) = 0
31046 [ (361 + s²) - 2s² ] = 0
31046 (361 - s²) = 0
(361 - s²) = 0
(19 - s)(19 + s) = 0
either s = -19 or s = 19, but s > 0
s = 19
