Question

The tip of a 15-inch wiper blade wipes a path that is 36 inches long. What is the angle of rotation of the blade in radians to the nearest tenth?

2.4 radians

1.2 radians

2.8 radians

0.4 radians

Answer 1

\(\textsf{s = length of arc, which is = 36}\)
\(\textsf{rotation (r) = 15}\)
so \(\textsf{s=rθ}\)
Say \(\textsf{36=15θ}\)
\(θ=\frac{36}{15}\)

Answer 2

Refresh the page if you see "��" :)

Answer 3

I don't get it

Answer 4

There are (at least) 2 ways to do this. one way is to *memorize* the formula

arc length s = radius * angle_in_radians
written as
\[ s = r \ \theta\]
for your problem, "solve for θ"
\[ \theta= \frac{s}{r} \]
and remember this *only works for theta in radians* (not degrees)

Answer 5

the other way is to use ratios. we know all the way around the circle takes 2 pi radians (i.e. 360 degrees)
the circumference (all the way round) is 2 pi r
now write a ratio of angle/distance = angle/distance
where the first ratio is the whole circle:

\[ \frac{2 \pi}{2 \pi r} = \frac{\theta}{s} \]
the left side simplifies to
\[ \frac{1}{r} = \frac{\theta}{s} \]


if we cross multiply, we get
\[ s = r \ \theta\]
which seems vaguely familiar

Answer 6

using a radius r=15 inches,
all the way round your circle i.e. its circumference is 2 pi * 15 = 30 pi inches
(about 94.25 inches)

36 inches will only be part way round
as a ratio:
\[ \frac{2 \pi \ radians}{30 \pi \ inches} = \frac{\theta}{36 \ inches} \]
or, after simplifying
\[ \frac{1 \ radian}{15} = \frac{\theta}{36 } \]
multiply both sides by 36 to get
\[ \frac{36}{15} \ radians= \theta \]
so theta is 12/5 radians. as a decimal, it is 2.4 radians