An electric hoist is used to lift a piece of equipment y = 3 feet. (see figure in the link below). The diameter of the drum on the hoist is x = 12 inches. Find the number of degrees through which the drum must rotate. (Round to the nearest integer.)
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To find the number of degrees through which the drum must rotate, we need to calculate the circumference of the drum.
The circumference of a circle (C) can be calculated using the formula:
C = π * diameter
Here, the diameter of the drum (x) is given as 12 inches. So, the circumference of the drum is:
C = π * 12 = 12π inches
Now, we need to convert the distance the hoist lifts (y) from feet to inches because the circumference of the drum is in inches.
To convert inches to feet, we know that 1 foot is equal to 12 inches. So, we can calculate:
Distance lifted (y) = 3 feet * 12 inches/foot = 36 inches
Since the circumference of the drum represents 360 degrees, we can set up a proportion to find the number of degrees the drum must rotate for a given distance lifted:
Circumference (degrees) / Circumference (inches) = Distance lifted (inches) / Total circumference (inches)
Let's substitute the values we found:
360 degrees / 12π inches = 36 inches / 12π inches
Now, we can solve this proportion to find the number of degrees through which the drum must rotate:
360 degrees = (36 inches / 12π inches) * 360 degrees
360 degrees = 1080 / π degrees
To round the answer to the nearest integer, we can use the value of π ≈ 3.14:
360 degrees ≈ 1080 / 3.14 degrees
360 degrees ≈ 344.62 degrees
Rounding to the nearest integer, the drum must rotate approximately 345 degrees.