The number of oscillations made by a pendulum in a given time is inversely proportional to the length of the pendulum. A certain clock with a 20-in.-long pendulum is losing 11.00 min/day. Should the pendulum be lengthened or shortened, and by how much?

n/t = k/L

when t = 24 * 60 = 1440
let's say n should be 1440
then
k = 20
now if n = 1440-11 = 1429
we want
1440 = 1429 (20)/L
L = 19.85

To determine whether the pendulum should be lengthened or shortened, we need to analyze the given relationship between the number of oscillations and the length of the pendulum.

The statement "The number of oscillations made by a pendulum in a given time is inversely proportional to the length of the pendulum" can be mathematically represented as:

Number of oscillations ∝ 1/Length of pendulum

Now let's analyze the given situation. The clock with a 20-in.-long pendulum is losing 11.00 min/day. This means that the clock is running slower than it should.

To determine if the pendulum should be lengthened or shortened, we need to consider the inverse relationship. If the clock is running slower than it should, we need to increase the number of oscillations, which means we need to increase the length of the pendulum.

To find out by how much the pendulum should be lengthened, we can set up a proportion. Let's assume the clock should be running at the correct rate, which is 24 hours/day.

Original number of oscillations / Original length of pendulum = New number of oscillations / New length of pendulum

The original number of oscillations can be calculated by dividing 24 hours (or 1440 minutes) by the time it currently takes for one oscillation. The original length of the pendulum is given as 20 inches.

Using this information, we can set up the proportion:

(1440 min) / (Current time for one oscillation) = New number of oscillations / (20 + x) inches

Since the clock is losing 11.00 min/day or (11.00 min / 24 hours), we can adjust the proportion:

(1440 min - 11.00 min) / (Current time for one oscillation) = New number of oscillations / (20 + x) inches

Now we can solve for x to find out by how much the pendulum should be lengthened.