Question

Shyam, a fertilizer salesman, sells directly to farmers. He visits two villages A and B. Shyam starts from A, and travels 50 meters to the East, then 50 meters North-East at exactly \(45^\circ\) to his earlier direction, and then another 50 meters East to reach village B. If the shortest distance between villages A and B is in the form of \(a\sqrt{b}+\sqrt{c}\) meters, Find the value of \(a+b+c\).

• A. 52
• B. 54
• C. 58
• D. 59
• E. None of the above.

Hint:

- For multi-leg paths, add vectors component-wise and then take the magnitude.
- Expressions of the form \(\sqrt{m+n\sqrt{k}}\) equal \(\sqrt{u}+\sqrt{v}\) only when \(u+v=m\) and \(uv=\frac{n^2k}{4}\) admit integer solutions; if not, the “\(a\sqrt{b}+\sqrt{c}\)” form with integers is impossible.

Answer 1

Answer: (E)

Step 1: Resolve the three displacements into components.
Let East be \(+x\) and North be \(+y\).
First \(50\) m East: \((50,0)\).
Second \(50\) m at \(45^\circ\) NE: \((50\cos45^\circ,\,50\sin45^\circ)=(25\sqrt{2},\,25\sqrt{2})\).
Third \(50\) m East: \((50,0)\).
Net displacement \(\vec{AB}=(100+25\sqrt{2},\,25\sqrt{2})\).
Step 2: Compute the straight-line distance \(AB\).
\[ AB=\sqrt{(100+25\sqrt{2})^2+(25\sqrt{2})^2} =\sqrt{12500+5000\sqrt{2}}. \] Factor to simplify: \[ AB=\sqrt{25(500+200\sqrt{2})}=5\sqrt{500+200\sqrt{2}} =50\sqrt{\,5+2\sqrt{2}\,}. \] Step 3: Compare with the demanded form \(a\sqrt{b}+\sqrt{c}\).
Setting \(AB=a\sqrt{b}+\sqrt{c}\) and squaring would give \[ AB^2=a^2b+c+2a\sqrt{bc}=12500+5000\sqrt{2}. \] This requires \(2a\sqrt{bc}=5000\sqrt{2}\Rightarrow a\sqrt{bc}=2500\sqrt{2}\), with \(a^2b+c=12500\). There are \textit{no} integers \(a,b,c\) satisfying both simultaneously (one may check that taking \(b\) to absorb the factor \(2\) leads to a quadratic with non-integral solutions). Hence \(AB\) cannot be written in the stated form with integers \(a,b,c\). Therefore none of the numerical options for \(a+b+c\) applies.
\[ \boxed{AB=50\sqrt{\,5+2\sqrt{2}\,}\ \text{m}} \quad\Rightarrow\quad \boxed{\text{Answer: None of the above.}} \]