The time it takes for a pendulum
to swing back and forth can be represented by the function,
S of x equals two pi times the square-root of x
divided by 32, also as shown here below.
Where S of x is the time in seconds,
and x is the length of the pendulum in feet.
We want to use our function
to answer the following two questions.
We first want to determine how many seconds it will take
for a seven foot pendulum to swing back and forth one time,
and we're asked to round our answer to one decimal place.
Well, we're told the pendulum is seven feet long,
which means, x equals seven,
and therefore, the time it takes to swing back and forth
would be S of seven.
So, we evaluate S of seven, we substitute seven for x,
which would give us two pi, times the square-root
of seven divided by 32,
and now, we'll go to the calculator
to get a decimal approximation.
So, we first want to enter two pi,
so here's two.
Pi is second,
and then, the exponent key, or the caret,
and then, times the square-root,
so second x squared, for the square-root,
and then, seven divided by 32.
Notice how numerator and denominator are both
underneath the square-root, which is correct,
and press enter.
So, the time would be approximately 2.9 seconds,
rounded to one decimal place.
So, the pendulum will swing back and forth one time
in approximately 2.9 seconds.
And now, for the second part,
we're told it takes eight seconds
for the pendulum to swing back and forth one time.
We're asked to determine the length of the pendulum,
and again, round to one decimal place.
So notice, here, we're told the time it takes
to swing back and forth, which means, now we know S of x,
the function value is equal to eight.
We want to find the input x,
which would be the length of the pendulum.
So, we'll substitute eight for S of x,
which will give us the equation, eight
equals two pi
times the square-root of x divided by 32,
so to solve this for x, we'll first isolate the square root,
and then, square both sides of the equation.
So, to isolate the square root,
we would divide both sides by two pi.
So, this simplifies to one, and notice,
eight divided by two equals four,
so this simplifies to one, this simplifies to four,
so now, we have the equation, four divided by pi,
equals the square root of x divided by 32.
And now, to undo the square root,
we'll now square both sides of the equation.
Squaring this fraction, we just square the numerator,
and square the denominator,
so we'd have 16 divided by pi squared equals,
on the right side, when we square
the square root of x divided by 32,
we just get the radicand, x divided by 32.
So now, to solve for x, we multiply both sides by 32.
So, we have 32 times 16 divided by pi-squared equals x.
Notice how this simplifies to one.
Now, let's go to the calculator,
and get a decimal approximation for this length.
So, in parentheses, we have 32 times 16,
and then, divided by pi-squared,
so second, caret, or second, exponent key,
and then, to square it, we press x-squared, and enter.
So, the pendulum would be approximately 51.9,
so again, if it takes eight seconds
for the pendulum to swing back and forth one time,
the length of the pendulum is approximately 51.9 feet.
I hope this was helpful.