Aforest ranger in the west observation tower spots a fire 41° north of east. fifteen miles directly east, the forest ranger in the east tower spots the same fire at 56° north of west. how far away is the ranger who is closest to the fire? approximate the distance by rounding to the nearest hundredth of a mile. 9.91 mi 11.87 mi 12.53 mi 18.95 mi

9.91 miles

Step-by-step explanation:

Refer the attached figure

Forest Ranger at point A observes the fire at angle of 41° north of east i.e.∠CAB = 41°

The distance between the two rangers is 15 miles i.e. AB = 15 miles

Forest Ranger at point B observes the fire at at 56° north of west. i.e.∠CBA= 56°

Now we are supposed to find who is closest to the fire

So, we are supposed to find the length of AC and BC

So, first calculate ∠ACB

We will use angle sum property of triangle

Angle  sum property of triangle  : Sum of all angles of triangle is 180°

So, ∠CBA+∠ACB+∠CAB =180°

56°+∠ACB+41° =180°

97°+∠ACB =180°

∠ACB =180°-97°

∠ACB =83°

Now to find  AC and BC we will use law of sines

Refer the attached figure

So,

So, BC< AC

So, the ranger who is closest to fire is at a distance of 9.91 miles .

So, Option 1 is true

(d)

step-by-step explanation:

given: the line in the graph passes through points (5,30) and (6,26).

then, the equation of line is:

now, put y=a(n) and x=n,

hence, option d is correct.

option 2 is correct.

step-by-step explanation:

we are given two inequality equation

x+y> 5 and -3x+2y< -2

first we draw the graph of both function using t-table method

equation: x+y> 5

for x=0, y=5

for x=-5, y=0

table:

x   :   0     5

y   :   5     0

test point (0,0) , 0> 5

false

equation: -3x+2y< -2

for x=0, y=-1

for x=2, y=4

table:

x   :   0     2

y   : -1       4

test point (0,0) , 0< -2

true

now we draw the graph. see the attachment for graph.

common region would be solution set.

option 2 is correct.