The angle of elevation of the top of a building from a point on the ground is 30 degrees. On walking 20 m towards the building the angle becomes 60 degrees. Find the height of the building, and draw a diagram.

Step 1: Define variables and set up equations

Let $AB$ be the height of the building, denoted by $h$. Let $BC$ be the distance from the second observation point to the building, denoted by $x$. The first observation point is $D$, so $BD = BC + CD = x + 20$. We use the tangent function, which relates the opposite side (height) to the adjacent side (distance from the building) in a right-angled triangle.

Step 2: Express height in terms of distance

From $\triangle ABC$, we have $h = x \tan 60^\circ$. Since $\tan 60^\circ = \sqrt{3}$, we get $h = x\sqrt{3}$. From $\triangle ABD$, we have $h = (x+20) \tan 30^\circ$. Since $\tan 30^\circ = \frac{1}{\sqrt{3}}$, we get $h = (x+20)\frac{1}{\sqrt{3}}$.

Step 3: Solve for x

Now we equate the two expressions for $h$ from equations (1) and (2). This allows us to solve for $x$. Multiplying both sides by $\sqrt{3}$ eliminates the fraction. Then, we rearrange the terms to isolate $x$ and find its value.

Step 4: Calculate the height of the building

Substitute the value of $x$ back into equation (1) to find the height $h$ of the building. The height is $10\sqrt{3}$ meters.

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