If the sun rises at 6 AM and is directly overhead at 12 noon, estimate the time (hour and minutes) when a 34-foot tree will have a 24-foot shadow.
Let T be the tree top
Let B be the tree base
Let S be the tip of the shadow
tan(angle BST) = 34/24
angle BST = 54.78°
This is also the angle of the sun from the horizon
54.78/90 = .61
so, the sun has gone .61* 6 hours = 3.66 hrs = 3 hr 40 min, so the time is 9:40 am.
Of course, the shadow is the same length at 2:20 pm, assuming no cloud cover...
To estimate the time when a 34-foot tree will have a 24-foot shadow, we need to understand the relationship between the length of the shadow and the position of the sun. The length of the shadow is determined by the angle at which the sun's rays hit the tree.
Given that the sun is directly overhead at 12 noon, we can assume that the angle between the sun and the tree is 90 degrees, resulting in no shadow. As the sun moves, the angle decreases, causing the shadow to become longer.
To estimate the time, we can analyze the rate at which the shadow length changes. From 6 AM to 12 noon, the sun moves from the horizon to directly overhead, which is a 90-degree change in the angle. This 90-degree change occurs over 6 hours, so the rate of change is 90 degrees / 6 hours = 15 degrees per hour.
Now, let's calculate how much time it takes for the shadow to change by 10 feet (34 feet - 24 feet). Since we know that the rate of change is 15 degrees per hour, we need to find the angle such that sin(angle) = 10 feet / 34 feet.
Using the inverse sine function (sin^(-1)) on a calculator, we can find that the angle is approximately 17.1 degrees.
Next, we divide this angle by the rate of change (15 degrees per hour) to get the estimated time it takes for the shadow to change by 10 feet: 17.1 degrees / 15 degrees per hour = 1.14 hours.
Finally, we add this estimated time to 12 noon (when the sun is directly overhead) to find the expected time for a 34-foot tree to have a 24-foot shadow: 12 noon + 1.14 hours.
Thus, the estimated time is approximately 1:08 PM (12 noon + 1 hour and 8 minutes).