A clock tower has been constructed such that the center of the clock is 40 feet above ground. The minute hand on the clock is 3 feet long. How far above the ground the end of the minute hand at 12:05? 1:00?
12:05 >>> 42,598 feet
1:00 >>> 43 feet
To find how far above the ground the end of the minute hand is at different times, we need to consider the position of the minute hand relative to the center of the clock.
Let's assume that the center of the clock is at point C, and the end of the minute hand is at point M.
At 12:00, the minute hand points straight up, pointing directly at the 12th hour mark on the clock face. In this position, the distance from the center of the clock to the end of the minute hand is equal to the length of the minute hand, which is 3 feet.
At 12:05, the minute hand moves 1/12th of its total rotation, which is equivalent to moving 5 minutes out of 60 minutes. To calculate the distance from the ground to the end of the minute hand at 12:05, we can use trigonometry.
Since the minute hand forms a right triangle with the vertical line passing through point C, we can use the sine function to find the height of point M above the center of the clock:
sine(angle) = opposite/hypotenuse
In this case, the angle is 5 minutes out of 60 minutes, or 5/60, and the hypotenuse is the length of the minute hand, which is 3 feet.
So, sine(5/60) = height of M/3
Rearranging the equation, we get:
height of M = 3 * sine(5/60)
Using a calculator, we can find that sine(5/60) is approximately 0.0823.
Therefore, the height of point M above the ground at 12:05 is:
height of M = 3 * 0.0823 = 0.247 feet (rounded to the nearest thousandth)
At 1:00, the minute hand points straight down, pointing at the 6th hour mark on the clock face. In this position, the distance from the center of the clock to the end of the minute hand is equal to the sum of the length of the minute hand and the height of the center of the clock above the ground.
So, the height of point M above the ground at 1:00 is:
height of M = 3 + 40 = 43 feet
Therefore, the end of the minute hand is 0.247 feet above the ground at 12:05 and 43 feet above the ground at 1:00.