What is the minimum distance between C and D?
Hello Guest!
The sensor describes an arc of a circle:
\(f(x)=\sqrt{7^2-x^2}+9-4\)
The point of closest distance from the sensor to D is on the line (0,5) --> D:
\(g(x)= -\frac{5}{12}x+5\\ f(x)=g(x)\)
\(\sqrt{7^2-x^2}+5=-\frac{5}{12}x+5\\ 49-x^2=\frac{25x^2}{144}\\ \frac{169x^2}{144}=49\\ x=\frac{7\cdot 12}{13}\\ x=6.\overline{461538}\\ y=-\frac{5}{12}\cdot x+5\\ y=2.\overline{307692}\)
What is the minimum distance between C and D?
\(\overline{CD}_{min}=\sqrt{(12-x)^2+y^2}=\sqrt{(12-6.4615)^2+2.3077^2}\\ \color{blue}\overline{CD}_{min}=6\)
!