The sun rises at 6 AM and is directly overhead at noon, estimate the time (hour and minutes) when a 34-foot tree will have a 14-foot shadow.
To estimate the time when a 34-foot tree will have a 14-foot shadow, we can use similar triangles and the relationship between the height of an object, its shadow, and the angle of the sun.
First, let's establish the relationship between the height of the tree, its shadow, and the angle of the sun. When the sun is directly overhead (at noon in this case), the angle of the sun with the ground is 90 degrees. This means that the height of the tree and its shadow form a right triangle, with the height of the tree being the vertical (opposite) side and the shadow being the horizontal (adjacent) side. Using the tangent function, we can express this relationship:
tan(angle) = opposite/adjacent
In this case, the angle is 90 degrees and the opposite side is the height of the tree (34 feet), while the adjacent side is the length of the shadow (unknown). Therefore, we have:
tan(90 degrees) = 34 feet/adjacent
Since the tangent of 90 degrees is undefined, we can't solve this equation directly. However, we know that the height of the tree and the length of its shadow are proportional as the sun moves across the sky. This means that the ratio of the height of the tree to the length of its shadow remains constant throughout the day.
So let's find this constant ratio. At noon, the tree's shadow is 0 feet (directly underneath the tree), so we have:
34 feet/0 feet = k
Here, k represents the constant ratio. Dividing by 0 is undefined, but we observe that the ratio k is approaching infinity. This means that when the sun is directly overhead, the ratio of the tree's height to its shadow is very large.
Now, let's estimate the time when the tree will have a 14-foot shadow. Since the ratio remains constant, we can set up another proportion to find this time. Let's call this time t hours after 6 AM. At t hours after 6 AM, the height of the tree will remain the same (34 feet), but the length of the shadow will be 14 feet. Therefore:
34 feet/14 feet = k
Solving for k, we find:
k = 34/14
k ≈ 2.4286
Now, we can determine the time t when the tree will have a 14-foot shadow:
k = t hours after 6 AM/14 feet
2.4286 = t/14
Cross-multiplying, we get:
t ≈ 2.4286 * 14
t ≈ 34.0
Therefore, the estimated time (hour and minutes) when a 34-foot tree will have a 14-foot shadow is approximately 6:34 AM.