Question

1. What relationships do you notice between an object's height and the length of its shadow?

2. Make a conjecture about the height of the lamppost and explain your thinking.

Answer 1

1) The object's height is approximately 1.5 times the length of its shadow.

2) 171 inches

Explanation:

Question 1: Relationships between Height and Shadow Length

There appears to be a direct relationship between an object's height and its shadow length. As the height of the object increases, the length of its shadow also increases. This relationship is likely to be proportional.

To determine if the relationship is proportional, we can calculate the ratio of height to shadow length for each person:

`\begin{aligned}\text{Younger Boy:} & \quad (43)/(29) \approx 1.48 \\\\\text{Man:} & \quad (72)/(48) = 1.5 \\\\\text{Older Boy:} & \quad (51)/(34) =1.5\end{aligned}`

Therefore, the ratio of height to shadow length is approximately 1.5 : 1. This means that the height of an object is approximately 1.5 times the length of its shadow.

Question 2: Conjecture about the Lamppost's Height

To make a conjecture about the height of the lamppost, we can use the observed relationship between height and shadow length.

We have determined that the height of an object is approximately 1.5 times the length of its shadow. So, we can apply this ratio to calculate the height of the lamppost:

`\frac{\text{Lamppost height}}{\text{Lamppost shadow length}}=1.5`

`\frac{\text{Lamppost height}}{114}=1.5`

`\text{Lamppost height}=1.5* 114`

`\text{Lamppost height}=171\; \text{inches}`

Based on this estimation, the conjecture is that the lamppost's height is approximately 171 inches.

It is important to note that this is a simple estimation based on the observed ratios and assumes that the relationship between height and shadow length is consistent across different objects and distances from the light source. The accuracy of this estimate may depend on factors such as the angle of the light source and the evenness of the ground.