A cyclist rides down a long straight road at a velocity (in m/min) given by v (t) equals400minus20 t, for 0less than or equalstless than or equals10. a. How far does the cyclist travel in the first 6 min? b. How far does the cyclist travel in the first 10 min? c. How far has the cyclist traveled when his velocity is 290 m/min?

a. 2040 m

b. 3000 m

c. 1697.5 m

Explanation:

Given that:

v (t) = 400 - 20t

0 ≤ t ≤ 10

We can get the formula for distance traveled by integrating the velocity, v (t), with respect to time:

=> s (t) = 400t - 10t²

a. First 6 mins:

t = 6 mins

s (t) = 400 (6) - 10 (6) ²

s (t) = 2400 - 360

s (t) = 2040m

b. First 10 mins:

t = 10 mins

s (t) = 400 (10) - 10 (10) ²

s (t) = 4000 - 1000

s (t) = 3000 m

c. When v (t) = 290 m/mins

We insert this into the v (t) equation so we can find time t.

v (t) = 400 - 20t

290 = 400 - 20 (t)

20t = 400 - 290

20t = 110

t = 110/20 = 5.5 mins

Now we insert that into the s (t) equation:

s (t) = 400 (5.5) - 10 (5.5) ²

s (t) = 2200 - 302.5

s (t) = 1697.5 m