Question

Problem A.1

The graph below is made of three line segments:

-1 1 2 3 4 5 6 7 8 9 10 11 12

1

2

3

4

y

x

f(x)

g(x)

h(x)

The segments correspond to the following three functions:

f(x) = x − 2, g(x) = p

4 − (x − 6)2 + 2, h(x) = x − 6

Find the total length L of the graph between x = 2 and x = 10.

Answer 1

(I) length of f(x)

Given

$f(x)=x-2, for \ 2\leq x\leq4$

the length would be the length of hypotenuse and right angled isosceles triangle of side length = 2 units

$\implies h^2=b^2+h^2\\\implies h^2=2^2+2^2\\\implies h^2=8\\h=2\sqrt{2}$

length of graph contributed by

$f(x)=2\sqrt{2}$

(ii) Length of g(x)

Given

$g(x)=2\sqrt{4-(x-6)^2+2}, \ for \ 4\leq x \leq8$

$\implies g(x)-2=\sqrt{4-(x-6)^2}\\\implies(g(x)-2)^2=4-(x-6)^2\\\implies(x-6)^2+(g(x)-2)^2=2^2$

so, g(x) is a circle with center at (6,2) and radius = 2 units

length contributed by g(x) would be circumference of semicircle

length contributed by g(x)=$\pi r=2\pi$

(III) Length of h(x)

Given,

$f(x)=x-6, for\ 8\leq x\leq10$

the length would be the length of hypotenuse of right angled triangle of base =height=2 units

$\implies h^2=b^2+h^2\\\implies h^2=2^2+2^2\\\implies h^2=8\\h=2\sqrt{2}$

length of graph contributed by

h(x)=$2\sqrt{2}$

Total length of the graph

$L=length\ contributed \ by \ f(x)+length\ contributed \ by \ g(x)+length\ contributed \ by \ h(x)$ $\implies L=2\sqrt{2}+2\pi+2\sqrt{2}\\\implies L=4\sqrt{2}+2\pi$