https://file.aiquickdraw.com/mathbot/file/1734248192126058hbbrt.jpg

Let me extract the information from the image and provide a solution to the given mathematical problem. One moment, please.### Problem Analysis
The problem involves a geometric setup where points \( A \) and \( B \) are projected as shadows \( A' \) and \( B' \) on a vertical wall, formed by a lamp \( L \) positioned on a straight line. The questions involve calculating distances, given certain relationships and coordinates.

---

### **For Question 72**
Given:
- \( t = 2 \)
- \( x = 3 \)
- The distance \( A' \) from the wall is \( 6 \) meters.

We need to find the distance of the lamp \( L \) from the wall.

---

### Solution:
The setup forms two similar triangles:
1. Triangle \( \triangle LA'A \)
2. Triangle \( \triangle LB'B \)

Using the **similar triangles property**:
\[
\frac{\text{Height of shadow}}{\text{Height of object}} = \frac{\text{Distance of lamp from shadow}}{\text{Distance of lamp from object}}
\]

Let \( d \) be the distance of the lamp \( L \) from the wall. Then:
\[
\frac{6}{2} = \frac{d}{t}
\]
\[
d = 2 \cdot t = 2 \cdot 2 = 4 \, \text{meters}
\]

Thus, the answer is **4 meters**.

---

### **For Question 73**
Given:
- \( t = 2 \)
- \( x = 2 \)
- Length of \( AB' \) (shadow) = 3 meters.

We need to calculate the distance \( d \) of the lamp \( L \) from the wall.

---

### Solution:
Using the Pythagorean theorem in triangle \( \triangle LAB' \):
\[
\text{Distance of lamp from wall}^2 = \text{Shadow length}^2 + \text{Object height}^2
\]
\[
d^2 = AB'^2 + t^2
\]
\[
d^2 = 3^2 + 2^2 = 9 + 4 = 13
\]
\[
d = \sqrt{13}
\]

Thus, the answer is \( \sqrt{13} \) meters.

---

### Final Answers:
- **72**: \( \mathbf{4} \) meters.
- **73**: \( \mathbf{\sqrt{13}} \) meters.

Would you like detailed diagrams or further clarifications? Here are 5 follow-up questions to enhance understanding:

1. How are similar triangles used in real-world applications?
2. Why does the Pythagorean theorem apply to this problem?
3. Can you generalize the formula for the lamp’s distance?
4. What is the relationship between shadow length and object height?
5. How would the answer change if \( t \) or \( x \) were different?

**Tip:** Always visualize problems involving geometry for clarity and to reduce calculation errors!

Given a geometric problem involving a lamp L, shadows A' and B' on a wall, and specific distances. Calculate the distance of the lamp L to the wall based on given parameters t = 2, x = 3, and shadow lengths.

Solve a geometric problem involving a lamp and the projection of shadows onto a wall. Learn how to calculate the lamp's distance to the wall using triangle similarity and the Pythagorean theorem. Parameters include t = 2, x = 3, and given shadow lengths. Suitable for high school math students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation