A 30 meter deep well, with a diameter of 7 meters is dug and the earth from digging is evenly spread out to form a platform of size 22 meters by 14 meters. The height of the platform is:

The Key Insight: When earth is dug from the well, it has to go somewhere! All that earth gets spread out to form the platform, so the volumes must be equal.

Volume of earth from well = Volume of platform

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The well is shaped like a cylinder, so we use the cylinder volume formula:

Volume $= \pi \times r^2 \times h$

Given information:

• Depth (height) $= 30$ meters
• Diameter $= 7$ meters, so radius $= \dfrac{7}{2} = 3.5$ meters

Volume of well $= \pi \times (3.5)^2 \times 30$

$= \pi \times 12.25 \times 30$

$= 367.5\pi$ cubic meters

Why use exact form with $\pi$? It makes our final calculation cleaner!

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The platform is rectangular, so:

Volume $= \text{length} \times \text{width} \times \text{height}$

Given information:

• Length $= 22$ meters
• Width $= 14$ meters
• Height $= h$ meters (this is what we need to find!)

Volume of platform $= 22 \times 14 \times h = 308h$ cubic meters

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Since all the earth from the well forms the platform:

$367.5\pi = 308h$

Solving for $h$:

$h = \dfrac{367.5\pi}{308}$

Pro tip: Let's simplify this fraction first!

$\dfrac{367.5 \times 3.14159}{308} = \dfrac{1154.07}{308} = 3.75$ meters

ANSWER: The height of the platform is $3.75$ meters

Quick Check: Does this make sense? We took earth from a deep, narrow well ($7$m diameter) and spread it over a much larger area ($22$m $\times$ $14$m), so the height should be much less than the well's depth. ✓