This question is asking for the length of an arc corresponding to $\displaystyle \frac{58}{60}$ of a circle with radius eight feet. The question can be answered by evaluating for $\displaystyle r=8$:

$\displaystyle \frac{58}{60} \cdot 2 \pi r=\frac{58}{60} \cdot 2 \pi \cdot 8 \approx 48.6$

To compare $\displaystyle m \widehat{ABC}$ and $\displaystyle m \widehat{CAB}$, we note that

$\displaystyle m \widehat{ABC} = m \widehat{AB} + m \widehat{BC}$

and 

$\displaystyle m \widehat{CAB} = m \widehat{AB} + m \widehat{AC}$

We need to be able to compare $\displaystyle m \widehat{BC}$ and $\displaystyle m \widehat{AC}$. If we know which of the intercepting angles is the greater, then we know which of the arcs is greater. The intercepting angles are $\displaystyle \angle A , \angle B$, respectively. However, we are not given this relationship. 

Every hour, the tip of the minute hand travels the circumference of a circle with radius six feet, which is

$\displaystyle C = 2 \pi r= 2 \pi \cdot 6 = 12 \pi$ feet.

Since it is now 3:50 PM, the minute hand made three complete revolutions since noon, plus $\displaystyle \frac{50}{60 } = \frac{5}{6}$ of a fourth, so its tip has traveled this circumference $\displaystyle 3\frac{5}{6}$ times. 

This is

$\displaystyle 3\frac{5}{6} \times 12 \pi = \frac{23}{6} \times 12 \pi = 46 \pi$ feet. This is 

$\displaystyle 46 \pi \times 12 = 552 \pi$ inches.

The tip of a minute hand travels a circle whose radius is equal to the length of that minute hand, which, in this question, is seven feet long. The circumference of this circle is $\displaystyle 2 \pi$ times the radius, or $\displaystyle 2 \pi \times 7 = 14\pi$ feet; over the course of thrity minutes (or one-half of an hour) the tip of the minute hand covers half this distance, or $\displaystyle \frac{1}{2} \times 14 \pi = 7 \pi$ feet.

The circumference of a circle seven feet in diameter is $\displaystyle \pi$ times this diameter, or $\displaystyle 7 \pi$ feet.

The quantities are equal.

Every hour, the tip of the minute hand travels the circumference of a circle, which here is 

$\displaystyle C = 2 \pi r = 2 \pi \times 4 \frac{1}{2} = 9 \pi$ feet.

The minute hand has traveled $\displaystyle 110 \pi$ feet since noon, so it has traveled the circumference of the circle

$\displaystyle \frac{110 \pi}{9 \pi} = \frac{110 }{9 } =12 \frac{2}{9}$ times.

Since $\displaystyle 12 \leq 12 \frac{2}{9} \leq 12 \frac{1}{2}$, between 12 and $\displaystyle 12 \frac{1}{2}$ hours have elapsed since noon, and the time is between 12:00 midnight and 12:30 AM.

Examine the figure below, which shows $\displaystyle \bigtriangleup ABC$ inscribed in a circle.

By the Arc Addition Principle,

$\displaystyle m \overarc {ABC} = m \overarc {AB}+ m \overarc {BC}$

and

$\displaystyle m \overarc {ACB} = m \overarc {AC}+ m \overarc {BC}$

Consequently,

$\displaystyle m \overarc {ABC} - m \overarc {ACB}$

$\displaystyle = (m \overarc {AB}+ m \overarc {BC} ) - ( m \overarc {AC}+ m \overarc {BC})$

$\displaystyle = m \overarc {AB}+ m \overarc {BC} - m \overarc {AC}- m \overarc {BC}$

$\displaystyle = m \overarc {A B} - m \overarc {A C}$

The two quantities are equal.