Calculus Related Rates

Question

Mario is driving his kart on a straight road towards the finish line of a race. However, due to a series of mishaps, Mario suddenly finds himself on a grassy field 1000 ft west of the road. the finish line is 1000ft east and 1500 ft north of Mario. The kart travels 30 ft/sec on the grass, and 90 ft/sec on the road. Mario will drive straight across the grass until he reaches a point on the road somewhere along the final 1500-foot stretch. Let T(x) be the time (in seconds) it takes for Mario to reach the finish line if he drives straight across the grass until he reaches the point on the road that is x feet south of the finish line and then drives due north of the road for the remaining x feet.

(a) find the formula for T(x).

(b) In the context of this problem, what is the domain of T(x)?

(c) What is the fastest time in which Mario can reach the finish line? Use calculus to find and justify your answer, and be sure to show enough evidence that you have indeed found the fastest time.

Answer 1

time = distance/rate

t=d/r

a) total time = T(x) = t_grass+t_road

The distance over grass = √(1000²+(1500-x)²)

The distance on the road = x

T(x) = √(1000²+(1500-x)²)/30 + x/90

b)

Domain of T(x): x= [0,1500]

c)

Need to find the value of x that minimizes T(x)

It is easier if you let y=1500-x and find the value of y that minimizes T(y)

T(y) = √(1000^2+y²)/30 + (1500-y)//90

T'(y) = y/√(10⁶+y²)-1/3

Setting T'(y) to 0 and solving for y

y = 353.553

x = 1500-y = 1146.4466

T(x) ≈ 48.09 seconds

Checking if minimum time for T(x)

T(1200) ≈ 48.13

T(1100) ≈ 48.12

So 48.09 is a minimum