Question:

Step 1: Define the relationship between height, distance, and angle of elevation. If a pole at distance $x$ from point $O$ has height $y$, and subtends an angle of elevation $\alpha$ at $O$, then the relationship is given by: $$ \tan\alpha = \frac{y}{x} $$ This implies that the height of a pole is $y = x \tan\alpha$. Step 2: Identify the distances of the poles from $O$. Let $d$ be the distance between two consecutive poles. Since there are 10 poles, and the smallest pole is at a distance $a$ from $O$, the distances of the feet of the poles from $O$ are: $$ a, a+d, a+2d, \dots, a+9d $$ Step 3: Express the heights of the smallest and longest poles. The smallest pole is at distance $a$ from $O$. Its height is $y_1 = a \tan\alpha$. The longest pole is at distance $a+9d$ from $O$. Its height is given as $h$. Therefore, the height of the longest pole is: $$ h = (a+9d) \tan\alpha $$ Step 4: Expand the expression for the height of the longest pole. $$ h = a \tan\alpha + 9d \tan\alpha $$ Step 5: Isolate the term containing $d$. Subtract $a \tan\alpha$ from both sides of the equation: $$ h - a \tan\alpha = 9d \tan\alpha $$ Step 6: Solve for $d$. Divide both sides by $9 \tan\alpha$: $$ d = \frac{h - a \tan\alpha}{9 \tan\alpha} $$ Step 7: Rewrite the expression for $d$ using trigonometric identities. Substitute $\tan\alpha = \frac{\sin\alpha}{\cos\alpha}$ into the expression for $d$: $$ d = \frac{h - a \frac{\sin\alpha}{\cos\alpha}}{9 \frac{\sin\alpha}{\cos\alpha}} $$ To simplify the complex fraction, multiply the numerator and the denominator by $\cos\alpha$: $$ d = \frac{\left(h - a \frac{\sin\alpha}{\cos\alpha}\right) \cos\alpha}{\left(9 \frac{\sin\alpha}{\cos\alpha}\right) \cos\alpha} $$ $$ d = \frac{h\cos\alpha - a\sin\alpha}{9\sin\alpha} $$