Finding Tree Height Using Pole Shadow Proportion

This problem involves similar triangles formed by the pole and its shadow, and the tree and its shadow. When the sun is in the same position (similar conditions), the ratio of the height of an object to the length of its shadow is constant.

Step 1: Convert Measurements to a Single Unit

It's easiest to work with a single unit, like centimeters. Let's convert all given lengths:

• Pole height: $\text{7 m 20 cm} = \text{(7 \times 100) cm} + \text{20 cm} = \text{700 cm} + \text{20 cm} = \text{720 cm}$
• Pole shadow length: $\text{8 m 30 cm} = \text{(8 \times 100) cm} + \text{30 cm} = \text{800 cm} + \text{30 cm} = \text{830 cm}$
• Tree shadow length: $\text{6 m 64 cm} = \text{(6 \times 100) cm} + \text{64 cm} = \text{600 cm} + \text{64 cm} = \text{664 cm}$

Step 2: Set Up the Proportion

Under similar conditions, the ratio of height to shadow length is the same for the pole and the tree. Let 'h' be the height of the tree in centimeters.

Ratio for pole = $\frac{\text{Pole height}}{\text{Pole shadow length}} = \frac{\text{720 cm}}{\text{830 cm}}$

Ratio for tree = $\frac{\text{Tree height}}{\text{Tree shadow length}} = \frac{\text{h cm}}{\text{664 cm}}$

Since the conditions are similar, we can set these ratios equal to each other:

$\frac{h}{\text{664}} = \frac{\text{720}}{\text{830}}$

Step 3: Solve for the Tree Height (h)

Now, we solve the proportion for 'h':

$h = \frac{\text{720}}{\text{830}} \times \text{664}$

$h = \frac{\text{72}}{\text{83}} \times \text{664}$

To calculate this value:

$h = \frac{\text{47808}}{\text{83}}$

Performing the division:

$h \approx \text{576 cm}$

Step 4: Convert the Tree Height Back to Meters and Centimeters

The height of the tree is approximately 576 cm. Let's convert this back to meters and centimeters:

$\text{576 cm} = \text{500 cm} + \text{76 cm} = \text{5 m} + \text{76 cm}$

So, the height of the tree is approximately 5 m 76 cm.

This calculated height matches one of the given options.