Question

At the foot of the mountain the elevation of its summit is ${45}^{\text{o}}$; after ascending $1000m$towards the mountain up a slope of ${30}^{\text{o}}$inclination, the elevation is found to be ${60}^{\text{o}}$.The height of the mountain is $\frac{\sqrt{a}+1}{b}$ km. Then find the value of $a+b$. (where $a,b$ are prime number.)

Answer 1

Let$P$ be the summit of the mountain and $Q$ be the foot. Let $A$ be the first position and $B$ the second position of observation. $BN$ and $BM$ are perpendicular 's from $B$ to $PQ$ and $AQ$, respectively. Then, $AB=1000m=1km,\angle MAB={30}^{\text{o}}$


$\angle MAP=45°,\angle NBP=60°Now,\angle BAP=\angle MAP–\angle MAB=45°–30°=15°\angle APB=\angle APN–\angle BPN=45°–30°=15°\therefore \angle ABP\text{is isosceles and}\therefore AB=BP\text{But}AB=1km,\therefore BP=1km\text{Now}PQ=PN+NQ=PN+BM=BPsin60°+ABsin30°=1\times \frac{\sqrt{3}}{2}+1\times \frac{1}{2}=\frac{\sqrt{3}+1}{2}km$