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The figure represents a boy who is 1.6 m tall standing in front of a lamppost that is 8 m tall. As shown, the boy casts a shadow that is 1.5 m long when he is 6 m from the lamppost. Note that \(\frac{8}{1.6} = \frac{6+1.5}{1.5}\). As the boy walks in a straight line away from the lamppost, his distance from the lamppost increases and his shadow gets longer. When the boy is 7 m from the lamppost, he casts a shadow that is 1.75 m long.
From each drop-down menu, select the option that creates the most accurate statement based on the information provided.
| Text Component | Literal Content | Simple Interpretation |
|---|---|---|
| Boy's height | "1.6 m tall" | The boy is 1.6 meters tall |
| Lamppost height | "8 m tall" | The lamppost is 8 meters tall |
| Position 1 | "6 m from the lamppost", "shadow that is 1.5 m long" | At 6 meters from lamppost, the boy's shadow is 1.5 meters long |
| Proportionality check | "\(\frac{8}{1.6} = \frac{6+1.5}{1.5}\)" | The ratio of heights equals the ratio of distances (verifying similarity) |
| Shadow trend | "As the boy...walks away..., his shadow gets longer" | Farther from lamppost, shadow grows longer |
| Position 2 | "7 m from the lamppost", "shadow that is 1.75 m long" | At 7 meters, shadow is 1.75 meters |
| Chart Component | What It Shows | Interpretation |
|---|---|---|
| Triangle Type | Right triangle diagram | Geometric layout of lamppost, boy, and shadow |
| Vertical side | 8 m | Lamppost height (matches text) |
| Base (horizontal segment) | Two pieces: 6 m (boy's distance), 1.5 m (shadow), total 7.5 m | Boy's distance + shadow length |
| Perpendicular segment | 1.6 m | The boy's height |
| Hypotenuse | Slant from lamppost top through boy to shadow tip | Represents the path of the light |
When the boy is 8 m from the lamppost, he casts a shadow that is ___ m long.
Statement Breakdown 1:
Statement Breakdown 2:
What is needed: The length of the boy's shadow when he stands 8 meters from the lamppost.
Condensed Solution Implementation:
Use the proportional pattern found in the given examples: shadow length = distance from lamppost divided by 4.
Necessary Data points:
At 6m, shadow = 1.5m (since \(6 \div 4 = 1.5\)). At 7m, shadow = 1.75m (\(7 \div 4 = 1.75\)).
Calculations Estimations:
For 8m: shadow length = \(8 \div 4 = 2\)m.
Comparison to Answer Choices:
Possible answers are 2, 2.25, 2.50, 2.75, 3. The correct value 2m matches the first option.
FINAL ANSWER Blank 1: 2
When the boy casts a shadow that is 3 m long, he is ___ m from the lamppost.
Statement Breakdown 1:
Statement Breakdown 2:
What is needed: The boy's distance from the lamppost when the shadow is 3 meters long.
Condensed Solution Implementation:
Rearrange the proportional rule to solve for distance: distance = shadow length × 4.
Necessary Data points:
Shadow length is given as 3 meters. This follows the same proportional pattern as above.
Calculations Estimations:
Distance = \(3 \times 4 = 12\) meters.
Comparison to Answer Choices:
Possible answers are 9, 10, 11, 12, 13. The correct value 12m matches the fourth option.
FINAL ANSWER Blank 2: 12
Both blanks are solved using the proportional pattern: shadow length equals distance from lamppost divided by 4. For 8m the shadow is 2m, and for a 3m shadow, the distance is 12m. This predictable relationship allows direct calculation for each scenario.
The two blanks are independent. The first requires determining the shadow length for a given distance, and the second requires finding the distance for a given shadow length. While both use the same proportional relationship, neither blank's answer depends on the other's result.