Question

Two buses are driving along parallel freeways that are 5mi apart, one heading east and the other heading west. Assuming that each bus drives a constant 55 mph, find the rate at which the distance between the buses is changing when they are 13mi apart, heading toward each other.

Answer 1

Answer:

101.54m/h

Explanation:

Given that the buses are 5mi apart, and that they are both driving at the same speed of 55m/h, rate of change of distance can be determined using differentiation as;

Let l be the be the distance further away at which they will meet from the current points;

$l=\sqrt{13^2-5^2}=12m\\\\\frac{dl}{dt}=-(55m/h+55m/h})\\\\=-110m/h$#The speed toward each other.

$\frac{dh}{dt}=0, \ \ \ \ h=constant\\\\h^2+l^2=b^2\\\\2h\frac{dh}{dt}+2l\frac{dl}{dt}=2b\frac{db}{dt}\\\\2\times5\times0+2\times12\times(-110)=2\times13\frac{db}{dt}\\\\\frac{db}{dt}=-101.54m/h$

Hence, the rate at which the distance between the buses is changing when they are 13mi apart is 101.54m/h

Answer 2

The rate of change of distance between the two cars when they are 13 miles apart is 71.1 mph.

The given parameters;

• initial distance between the two cars,= 5 miles
• speed of each car, v = 55 mph
• final distance between the cars, = 13 miles

The time of motion of the car when they are 5 miles apart;

$t = \frac{5}{55} = \frac{1}{11} \ h$

The distance traveled by the cars when they are 13 miles apart

$a = \frac{1}{11} \times 55 = 5 \ m$

The displacement of the two cars when they are 13 miles apart;

$c^2 = a^2 + b^2\\\\c = \sqrt{a^2 + b^2} \\\\c = \sqrt{5^2 + 13^2} \\\\c = 13.93 \ m$

The change in the displacement;

$c^2 = a^2 + b^2\\\\2c \frac{dc}{dt} = 2a\frac{da}{dt} \ + \ 2b\frac{db}{dt} \\\\c \frac{dc}{dt} = a\frac{da}{dt} \ + \ b\frac{db}{dt} \\\\13.93 (\frac{dc}{dt} ) = 13(55) \ + \ 5(55)\\\\13.93 (\frac{dc}{dt} ) = 990\\\\\frac {dc}{dt} = \frac{990}{13.93 } \\\\\frac {dc}{dt} = 71.1 \ mph$

Thus, the rate of change of distance between the two cars when they are 13 miles apart is 71.1 mph.

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