Jared wants an estimate of the height of some trees near his lake house. Use ideas of size transformationto design a plan for him to use his height to estimate accurately a tree's height without directly measuring........
I understand the concept of size transformation, I have used it a lot in paint to shrink and blow up pictures. I understand that ABCD in one onject will still equal ABCD after size transformation, however I am unsure where to begin when comparing it to a set height. Any help would be appriciated. Thank you.
To help Jared estimate the height of the trees near his lake house using size transformation, we can apply the concept of similar triangles. Similar triangles have corresponding angles that are equal and their sides are proportional to each other.
Here's a step-by-step plan for Jared to estimate the height of a tree using his own height without directly measuring:
1. Find a reference object: Jared needs to find a reference object of known height near the tree, such as a lamppost, a flagpole, or any other tall object with a known height. Let's call the height of this reference object "Hr."
2. Measure Jared's height: Jared needs to measure his own height accurately. Let's call Jared's height "Hj."
3. Determine the distance: Jared needs to stand at a distance from the tree that allows him to see both the top and base of the tree clearly. Let's call this distance "D."
4. Measure the length of Jared's shadow: When Jared stands at distance D from the tree, he needs to measure the length of his own shadow. Let's call this length "Sj."
5. Measure the length of the reference object shadow: Jared also needs to measure the length of the reference object's shadow when it is cast by the same light source at the same time. Let's call this length "Sr."
6. Set up the similar triangles: Let's consider two similar triangles - one formed by Jared's height, his shadow, and his distance from the tree, and the other formed by the reference object's height, its shadow, and its distance from the tree.
7. Set up ratios: Since the triangles are similar, the ratios of corresponding sides will be equal. Therefore, we have the following ratios:
- Hj (Jared's height) / Sj (Jared's shadow) = Hr (Reference object height) / Sr (Reference object shadow)
8. Solve for the tree's height: Now, we know Jared's height (Hj), the length of his shadow (Sj), the distance between him and the tree (D), and the ratio of the reference object's height (Hr) to its shadow (Sr). We can set up a proportion using the ratio of the reference object's height to its shadow and the ratio of Jared's height to his shadow:
- Hj / Sj = Hr / Sr
Rearrange the proportion to solve for the tree's height (Ht):
- Ht = (Sj * Hr) / Sr
9. Calculate the tree's height: Using the value of Ht obtained from the above equation, Jared can calculate the estimated height of the tree based on his own height and the measurements he obtained.
Remember, this method provides an estimated height and may not be perfectly accurate. Make sure to choose a reference object with a known height for better accuracy.