Answer
The person is $1377$ ft from the building when she makes her second observation.
Work Step by Step
Find the angles of the larger triangle that includes both triangles
$\angle A = 90˚$
$\angle B = 28˚$
$\angle C = 180 - (90+28)$
$= 62˚$
2. Find the angles inside the smaller triangle on the left side
$\angle A = 180 - 38$ $= 142˚$
$\angle B = 28˚$
$\angle C = 180 - (142 + 28$ $= 10˚$
3. Find the angles inside the large triangle on the right side
$\angle A = 90˚$ $\angle B = 38˚$
$\angle C = 62 - 10 = 52˚$
4. Use the sine law to solve for the missing length of the small triangle on the left
$\frac{x}{sin(28)} = \frac{440}{sin(10)}$
$x = \frac{440sin(28)}{sin(10)}$
by GDC / calculator
$x = 1189.574.... $ft
5. Use the answer in #4 to solve for the missing angle in the large triangle $\frac{y}{sin(52)} = \frac{1189.574...}{sin(90)}$
$y = \frac{1189..sin(52)}{1}$ $y = 937.39...$ ft
6. Add up the previous length with the known length of $440$ ft
$= 937.39... + 440$
$=1377$ ft
Therefore the person is $1377$ ft from the building when she makes her second observation.
