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------------------------------------------------ The formula [tex] s = 16t^2 [/tex] is used to approximate the distance [tex] s [/tex], in feet, that an object falls freely (from rest) in [tex] t [/tex] seconds.

A branch of a tree hangs 90 ft above the soil. If a twig breaks off, how long will it take, falling freely, to reach the soil? Round to the nearest tenth.

A. About 1,440 sec
B. About 37.9 sec
C. About 5.6 sec
D. About 2.4 sec

We start with the equation that gives the distance an object falls:
[tex]$$
s = 16t^2.
$$[/tex]

Since the branch is 90 ft above the soil, we set
[tex]$$
s = 90.
$$[/tex]

Plugging this into the equation gives:
[tex]$$
16t^2 = 90.
$$[/tex]

Next, solve for [tex]$t^2$[/tex]:
[tex]$$
t^2 = \frac{90}{16} = 5.625.
$$[/tex]

To find [tex]$t$[/tex], take the square root of both sides:
[tex]$$
t = \sqrt{5.625} \approx 2.3717.
$$[/tex]

Rounding to the nearest tenth of a second, we have:
[tex]$$
t \approx 2.4 \text{ seconds}.
$$[/tex]

Thus, it takes about [tex]$2.4$[/tex] seconds for the twig to reach the soil. This corresponds to option D.