The length of time, t (in seconds), it takes the pendulum of a clock to swing through one complete cycle is a function of the length of the pendulum in feet. Defined by:

t = f(L) = 2pi sqrt L/32

A.Rewrite the formula using fractional exponents.

B.Determine the length of the cycle in time if the pendulum is 4 feet long.

good grief, g = 32 ft/s^2, old text :)

T = 2 pi (L/32)^(1/2)

T = 2 pi (4/32)^.5

= 2.22 seconds

A. To rewrite the formula using fractional exponents, we can express the square root as an exponent of 1/2. Therefore, the formula can be rewritten as:

t = f(L) = 2π(L/32)^(1/2)

B. To determine the length of the cycle in time if the pendulum is 4 feet long, we can substitute L = 4 into the formula:

t = 2π(4/32)^(1/2)

Simplifying the expression inside the square root:

t = 2π(1/8)^(1/2)

Taking the square root of 1/8:

t = 2π(1/√8)

Simplifying the denominator:

t = 2π(1/√(4*2))

t = 2π(1/(2√2))

Since √2 is irrational, we cannot simplify it any further. Therefore, the length of the cycle in time for a 4-foot long pendulum is:

t = 2π/(2√2)