Question

Create a sketch of the object and its shadow and another sketch of you and your shadow. Collect the measurements of your height, the length of your shadow, and the length of the shadow of the object. Label the measures on your sketches. Propose a method to calculate the height of the object based on the measurements you have collected. Perform your proposed calculation. Does your solution make sense? Explain your method and solution. Why does your method work? Explain in terms of similar triangles.

Answer 1

Certainly! While I can't create graphical sketches directly, I can help you visualize and describe how to create them, along with the method to calculate the height of the object based on measurements and the principles behind it.

### Sketches Description

1. **Sketch of the Object and Its Shadow**:
- Draw a vertical line representing the object (let's say a tree) and label its height as “h_object”.
- Draw a horizontal line extending from the base of the object, representing the shadow. Label the length of the shadow as “s_object”.

2. **Sketch of You and Your Shadow**:
- Draw a vertical line representing your height and label it as “h_yours”.
- Draw a horizontal line from your base extending outward, representing your shadow. Label this length as “s_yours”.

### Measurements
- Let's say you measure:
- Your height (h_yours): 1.7 meters
- The length of your shadow (s_yours): 1.5 meters
- The length of the object's shadow (s_object): 3.0 meters

### Propose a Method to Calculate the Height of the Object

To calculate the height of the object, we can use the properties of similar triangles. When the sun's rays cast shadows, the triangles formed by the object and its shadow and by you and your shadow are similar. This means that the ratios of corresponding sides are equal.

### Formula
Using the ratios of similar triangles:

\[
\frac{h_yours}{s_yours} = \frac{h_object}{s_object}
\]

Rearranging this gives:

\[
h_object = h_yours \times \frac{s_object}{s_yours}
\]

### Perform the Calculation

Substituting in our values:

\[
h_object = 1.7 \, \text{meters} \times \frac{3.0 \, \text{meters}}{1.5 \, \text{meters}}
\]

Calculating this gives:

\[
h_object = 1.7 \times 2 = 3.4 \, \text{meters}
\]

### Does Your Solution Make Sense?

Yes, the calculated height of the object, 3.4 meters, makes sense in relation to the measurements of your height and the lengths of the shadows.

### Explanation of Method and Solution

This method works because of the concept of similar triangles. When light rays from the sun strike both you and the object, they create right triangles where:
- The height of you or the object acts as one side,
- The length of the shadow acts as another side.

In similar triangles:
- The angles at the tip (where the light is coming in) are the same,
- The ratio of corresponding sides remains constant.

Thus, we can set up a proportion with the heights and shadow lengths to derive the height of the object based on your known height and the lengths of both shadows.

In summary, using the property of similar triangles allows us to easily and accurately calculate the height of the object based on our measurements.