A lamp is on a 6 m pole. A 1.5 m woman casts a 3 m shadow. How far is she from the pole?
\(9\,\text{m}\)
Step 1: Draw the situation in your mind (or on paper). - The lamp is at the top of a vertical pole of height \(6\,\text{m}\). - A woman of height \(1.5\,\text{m}\) is standing on the ground. - She casts a shadow that is \(3\,\text{m}\) long on the ground.
Step 2: Notice that two right-angled triangles are formed: - The big triangle: from the top of the pole to the tip of the shadow. - The small triangle: from the top of the woman to the tip of her shadow. These two triangles are similar (same shape), so the ratios of their corresponding sides will be equal.
Step 3: Let the distance of the woman from the pole be \(d\,\text{m}\). Then, the total distance from the pole to the end of the shadow is \(d + 3\,\text{m}\).
Step 4: Write the similarity ratio: \[ \dfrac{\text{Height of pole}}{\text{Distance from pole to shadow end}} \,=\, \dfrac{\text{Height of woman}}{\text{Length of shadow}} \]
That means: \[ \dfrac{6}{d+3} = \dfrac{1.5}{3} \]
Step 5: Simplify the right-hand side: \(\tfrac{1.5}{3} = 0.5 = \tfrac{1}{2}\).
So, \[ \dfrac{6}{d+3} = \dfrac{1}{2} \]
Step 6: Cross multiply: \[ 6 \times 2 = d + 3 \]
\[ 12 = d + 3 \]
Step 7: Solve for \(d\): \[ d = 12 - 3 = 9 \]
Final Answer: The woman is 9 m away from the pole.