Multiple choice

The angle of elevation of the top of a tower standing on a horizontal plane from point A is $\alpha$. After walking a distance d towards the foot of the tower, the angle of elevation is found to be $\beta$. The height of the tower is

  1. $\dfrac{d \,sin\, \alpha\, sin\, \beta}{sin\, (\beta - \alpha)}$
  2. $\dfrac{d \,sin\, (\beta - \alpha)}{sin\, \alpha\,sin\, \beta}$
  3. $\dfrac{d \,sin\,\alpha\, sin\, \beta}{sin\, (\alpha\,- \beta)}$
  4. $\dfrac{d\, sin\, (\alpha\, - \beta)}{sin\, \alpha\,sin \, \beta}$
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A Correct answer
Explanation

Let h be the height. From the geometry, h = x * tan(alpha) and h = (x-d) * tan(beta). Solving for x and then h gives h = (d * tan(alpha) * tan(beta)) / (tan(alpha) - tan(beta)). Using sine and cosine identities, this simplifies to (d * sin(alpha) * sin(beta)) / sin(beta - alpha).

AI explanation

Let the tower height be h, making the initial distance h cot alpha and the final distance h cot beta. The walked distance d equals h cot alpha minus h cot beta, which factors into h multiplied by cos alpha divided by sin alpha minus cos beta divided by sin beta. Finding a common denominator gives h multiplied by the quantity sin beta cos alpha minus cos beta sin alpha all divided by sin alpha sin beta. Using the sine difference identity, the numerator is sin (beta minus alpha), so h equals d sin alpha sin beta divided by sin (beta minus alpha).

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