Thread: calc help

The figure below shows a road running in the shape of a parabola from the bottom of a hill at A to point B. At B it changes to a line and continues on to C. The equation of the road is represented by R(x), also shown below.


B is 1000 feet horrizontally from A and 100 feet higher. Since the road is smooth, R'(x) is continuous. What is the value of b?


Multiple Choice:
A---> 0.2
B---> 0.02
C---> 0.002
D---> 0.0002
E---> 0.00002

i was thinking that the answer is (A). because (100/1000)*2 = 0.2.

thanks for the help guys

Let us imagine the line BC is straight, even if it ios not in the drawing.

So R(x) has two branches that are continuous at B.

The parabolic branch AB is
R(x) = ax^2
At B, it is (1000,100), so,
100 = a(1000)^2
a = 100 /(1000)^2
a = 10^(-4) = 0.0001
Hence,
R(x) = (0.0001)x^2

The tangent line at B has the slopes:
--R'(x) = 2ax for the parabola, and,
--R'(x) = b for the straight line.
Since R'(x) = R'(x) at B, thenn
b = 2ax
b = 2(0.0001)(1000)
b = 0.2 ----------------------answer.