Question

unit radical expressions and quadratic equations
radical expressions and quadratic equations
the period ( t ) (in seconds) of a pendulum is given by ( t = 2pi sqrt{\frac{l}{32}} ), where ( l ) stands for the length (in feet) of the pendulum. if ( pi = 3.14 ), and the period is 15.7 seconds, what is the length?
the length of the pendulum is ____ feet.
the solution is (\boxed{})

Explanation:

Step1: Identify given values

We know \( T = 15.7 \) seconds, \( \pi = 3.14 \), and the formula \( T = 2\pi\sqrt{\frac{L}{32}} \).

Step2: Substitute \( T \) and \( \pi \) into the formula

Substitute \( T = 15.7 \) and \( \pi = 3.14 \) into \( T = 2\pi\sqrt{\frac{L}{32}} \):

$$ 15.7 = 2\times3.14\times\sqrt{\frac{L}{32}} $$

Step3: Simplify the left - hand side multiplication

First, calculate \( 2\times3.14 = 6.28 \), so the equation becomes:

$$ 15.7 = 6.28\times\sqrt{\frac{L}{32}} $$

Step4: Solve for \( \sqrt{\frac{L}{32}} \)

Divide both sides of the equation by \( 6.28 \):

$$ \sqrt{\frac{L}{32}}=\frac{15.7}{6.28} $$

Since \( \frac{15.7}{6.28}=2.5 \), we have \( \sqrt{\frac{L}{32}} = 2.5 \)

Step5: Eliminate the square root

Square both sides of the equation \( \sqrt{\frac{L}{32}} = 2.5 \):

$$ \frac{L}{32}=(2.5)^2 $$

Since \( (2.5)^2 = 6.25 \), the equation is \( \frac{L}{32}=6.25 \)

Step6: Solve for \( L \)

Multiply both sides of the equation by \( 32 \):

$$ L = 6.25\times32 $$

Calculate \( 6.25\times32 = 200 \)

Answer:

The length of the pendulum is \( 200 \) feet.