One way you can measure the height of a tree, lamppost, or building is with a very tall ladder and a very large tape measure. Another way is by using geometry. Devise a plan to measure something tall in your neighborhood by using your shadow and what you have learned about similar triangles.

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.

Answer 1

Sketch 1: Object and its shadow
- Object: Tall building
- Shadow: Length of shadow of the building

Sketch 2: You and your shadow
- You: Standing next to the building
- Shadow: Length of your shadow

Measurements:
- Your height: 5 feet
- Length of your shadow: 3 feet
- Length of the building's shadow: 30 feet

Plan:
- Use the concept of similar triangles to calculate the height of the building.
- Similar triangles have proportional sides, so the ratio of corresponding sides will be equal.

Calculations:
- Let x be the height of the building
- (x + 5) / 30 = 5 / 3
- Cross multiply: 3(x + 5) = 150
- 3x + 15 = 150
- 3x = 135
- x = 45 feet

Solution:
- The height of the building is calculated to be 45 feet.
- This solution makes sense because the proportions of the similar triangles match up correctly and the calculated height is reasonable for a tall building.

Explanation:
- By using the measurements of your height, your shadow, and the building's shadow, you can apply the concept of similar triangles to calculate the height of the building. This method allows you to measure tall objects in your neighborhood without the need for a ladder or tape measure.