Video Transcript
A flashlight emits a light beam
whose width 𝑤 at a distance of 𝑑 is given by 𝑤 equals 1.2 times the square root
of 𝑑. Determine, to the nearest tenth,
the width of the beam 26 feet away from the flashlight.
Let us begin by analyzing the
information given to us. We need to determine the width of
the beam, which is denoted by 𝑤, given that the distance away, denoted by 𝑑, is 26
feet. We also have a formula that relates
𝑤 and 𝑑. Therefore, our strategy will be to
use this formula to find the width. So, let us first write out this
formula. And, we will substitute in 𝑑
equals 26.
Now, we could just evaluate this
with a calculator to get 𝑤. But let us think about how we can
find the answer by approximating root 26. If we consider that the square root
of 25 is five, then we can see that the number that squares to make 26 must be
slightly greater than five. So, let us consider 5.1 squared as
an approximation. We can make this calculation easier
by forming a multiplication table and summing the values.
First, we have five times five,
which is 25. We then have five times 0.1 is 0.5,
and 0.1 times 0.1 which is 0.01. And the sum of these values is
26.01. Now, since 26.01 is greater than
26, this tells us that root 26 is between five and 5.1, although clearly it is much
closer to 5.1.
Now, recall that our final answer
needs to be correct to the nearest tenth. While we have found that root 26 is
5.1 to the nearest tenth, in order to make sure our final answer is completely
accurate, we should find root 26 to the nearest hundredth. To be sure that our approximation
is correct to the nearest hundredth, let us calculate 5.09 squared to determine
whether it is a better approximation than 5.1 squared.
So, let us draw another
multiplication table. The products of these values are
25, 0.45, and 0.0081. And If we add up these values, we
can add up the two 0.45s to get 0.9. So, this is 25 plus 0.9 plus
0.0081, which is 25.9081. So, if we now compare these two
values, we can clearly see that 5.1 squared is much closer to 26 than 5.09
squared. So, to two decimal places, root 26
is 5.10.
Let us finally finish this
calculation, substituting in 5.1. So, we have 1.2 times 5.1. And using a multiplication table or
otherwise, this is 6.12. Therefore, to one decimal place,
and recalling the correct units, the width of the beam 26 feet away from the
flashlight will be 6.1 feet.
