Question
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.
Make up 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.
Say that I’m 5.2 tall
And use the Sun as the object
Make up 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.
Say that I’m 5.2 tall
And use the Sun as the object
Answers
Answer
How did you get from 4160 to 1040?
Answered by
GPT-4o mini
### Plan to Measure the Height of the Sun Using Shadows and Similar Triangles
#### Step 1: Sketching
1. **Sketch of the Object (Sun) and Its Shadow:**
- Draw a horizontal line representing the ground.
- Above the line, draw a sun (it can be a circle) and label it "Sun".
- Draw a vertical line extending from the ground upwards to represent the height of the sun (which would be abstract since the sun is not physically measurable on Earth).
- Draw a horizontal line extending from the base of the sun's vertical line to represent the shadow of the sun on the ground.
2. **Sketch of You and Your Shadow:**
- Draw a vertical line representing your height (5.2 feet).
- Draw your shadow as a horizontal line extending from the base of your height line.
#### Measurements to Collect
1. **Your Height:** 5.2 feet
2. **Your Shadow Length:** Let's say your shadow measures 4 feet (you can measure this while standing outside).
3. **Shadow of the Sun:** For the purpose of this exercise, let's say the sun's shadow (which is abstract since the sun does not cast a physical shadow on the ground) is 800 feet long. This is a hypothetical value for calculation purposes.
#### Step 2: Labeling Measures on Sketches
- On your sketch, label the height of yourself as 5.2 feet and your shadow as 4 feet.
- On the sun's sketch, label its shadow as 800 feet.
#### Step 3: Proposing a Method to Calculate the Height of the Object
Using similar triangles, the height of the Sun can be calculated with the following formula derived from the properties of similar triangles:
\[
\frac{\text{Height of You}}{\text{Length of Your Shadow}} = \frac{\text{Height of Sun}}{\text{Length of Sun's Shadow}}
\]
Rearranging that to find the Height of the Sun gives:
\[
\text{Height of Sun} = \frac{\text{Height of You} \times \text{Length of Sun's Shadow}}{\text{Length of Your Shadow}}
\]
#### Step 4: Performing the Proposed Calculation
Substituting the values we have:
- Height of You = 5.2 feet
- Length of Your Shadow = 4 feet
- Length of Sun's Shadow = 800 feet
\[
\text{Height of Sun} = \frac{5.2 \times 800}{4}
\]
\[
\text{Height of Sun} = \frac{4160}{4}
\]
\[
\text{Height of Sun} = 1040 \text{ feet}
\]
#### Step 5: Evaluate Your Solution
The calculated height of the sun, which came out to be 1040 feet, does not make sense because the sun is actually about 93 million miles away from Earth, and this setup was purely hypothetical for the sake of using the method of similar triangles.
### Explanation of Method and Solution
The method used involves the principle of similar triangles, which states that if two triangles are similar, the ratios of their corresponding sides are equal. We applied this concept by measuring our height and shadow to create a proportion with the sun's hypothetical shadow. The calculation itself follows the correct mathematical steps; however, the result for the height of the sun is not practically relevant or meaningful since the sun cannot be measured in this manner from the ground.
### Conclusion
This exercise was meant to illustrate how similar triangles can be applied to measure tall objects using shadows. By collecting the height and shadow lengths, we successfully applied the mathematical concept, even though the result was abstract when interpreting the height of the sun. Such methods can be effectively used for trees, buildings, and other objects that are more realistically measurable.
#### Step 1: Sketching
1. **Sketch of the Object (Sun) and Its Shadow:**
- Draw a horizontal line representing the ground.
- Above the line, draw a sun (it can be a circle) and label it "Sun".
- Draw a vertical line extending from the ground upwards to represent the height of the sun (which would be abstract since the sun is not physically measurable on Earth).
- Draw a horizontal line extending from the base of the sun's vertical line to represent the shadow of the sun on the ground.
2. **Sketch of You and Your Shadow:**
- Draw a vertical line representing your height (5.2 feet).
- Draw your shadow as a horizontal line extending from the base of your height line.
#### Measurements to Collect
1. **Your Height:** 5.2 feet
2. **Your Shadow Length:** Let's say your shadow measures 4 feet (you can measure this while standing outside).
3. **Shadow of the Sun:** For the purpose of this exercise, let's say the sun's shadow (which is abstract since the sun does not cast a physical shadow on the ground) is 800 feet long. This is a hypothetical value for calculation purposes.
#### Step 2: Labeling Measures on Sketches
- On your sketch, label the height of yourself as 5.2 feet and your shadow as 4 feet.
- On the sun's sketch, label its shadow as 800 feet.
#### Step 3: Proposing a Method to Calculate the Height of the Object
Using similar triangles, the height of the Sun can be calculated with the following formula derived from the properties of similar triangles:
\[
\frac{\text{Height of You}}{\text{Length of Your Shadow}} = \frac{\text{Height of Sun}}{\text{Length of Sun's Shadow}}
\]
Rearranging that to find the Height of the Sun gives:
\[
\text{Height of Sun} = \frac{\text{Height of You} \times \text{Length of Sun's Shadow}}{\text{Length of Your Shadow}}
\]
#### Step 4: Performing the Proposed Calculation
Substituting the values we have:
- Height of You = 5.2 feet
- Length of Your Shadow = 4 feet
- Length of Sun's Shadow = 800 feet
\[
\text{Height of Sun} = \frac{5.2 \times 800}{4}
\]
\[
\text{Height of Sun} = \frac{4160}{4}
\]
\[
\text{Height of Sun} = 1040 \text{ feet}
\]
#### Step 5: Evaluate Your Solution
The calculated height of the sun, which came out to be 1040 feet, does not make sense because the sun is actually about 93 million miles away from Earth, and this setup was purely hypothetical for the sake of using the method of similar triangles.
### Explanation of Method and Solution
The method used involves the principle of similar triangles, which states that if two triangles are similar, the ratios of their corresponding sides are equal. We applied this concept by measuring our height and shadow to create a proportion with the sun's hypothetical shadow. The calculation itself follows the correct mathematical steps; however, the result for the height of the sun is not practically relevant or meaningful since the sun cannot be measured in this manner from the ground.
### Conclusion
This exercise was meant to illustrate how similar triangles can be applied to measure tall objects using shadows. By collecting the height and shadow lengths, we successfully applied the mathematical concept, even though the result was abstract when interpreting the height of the sun. Such methods can be effectively used for trees, buildings, and other objects that are more realistically measurable.
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