Width of the crater in meter

Question

Points A and B are in opposite sides of a lunar crater. Point C is 45 meters from point A. The measure if angle BAC is 114 degrees and the measure of angle ACB is 45 degrees. What is the width of the crater. round to the nearest hundredth 

Answer 1

Hi Lauren,

 

This one is a bit hard to explain without drawing a diagram, but the law of sines states that, for a triangle, the sine of one angle divided by the length of the side opposite that angle is equal to the sine of any other angle divided by the length of the side opposite that angle.

For this problem, you must first solve the measure of angle ABC by the formula:

180 degrees (always the sum of the three angles of a triangle)-114 degrees (angle BAC)-45 degrees (angle ACB)= 21 degrees

(it may be helpful to draw yourself a diagram here)...

 

So the solution then becomes sin(21^o)/45=sin(45^o)/x

 

Doing some basic algebra and rounding to the nearest hundredth, we arrive at

 

x=88.79 meters

 

Let me know if any of this is not clear; I hope it helps!

 

Jason

 

 

EDIT: Here is a rough sketch of the problem to help you visualize it better (sorry the formatting is a bit off!):

 

                 C                       

               /    \                     

             / 45^o \           

  45m /            \                  

         /                \               

      / 114^o     21^o \             

   /________________\           

A             X                 B

Answer 2

Lauren

 

It would be much easier to describe the triangle expressed in the problem statement, if it could be drawn. 

 

In any case, the triangle discussed has 3 angles of 114 degrees (BAC), 45 degrees (ACB), and 21 degrees (ABC).  (The 3rd angle of 21 degrees (for ABC) is computed as 180-114-45, since the sum of the angles in a triangle equals 180).

 

Also from the problem statement, the side AC (which is the side opposite to angle ABC) has a length of 45 meters.

 

The width of the crater being asked for is the distance from point A to point B or side AB of the triangle.

 

Using the law of sines:

(Sin of angle ABC)/(distance AC) = (Sin of angle ACB)/(AB)

 

Substituting, we have:

(Sin 21 degrees)/(45) = (Sine 45 degrees)/(AB)

 

Solving for AB:

AB = ((Sin 45)*(45))/(Sin 21)

AB = (.707*45)/(.358)

AB = 88.87 meters

 

I realize this is difficult to visualize without a picture.  You should try drawing one out for yourself from the problem statement.  Draw the crater as a circle.  Then pick two points on the edge that are opposite each other and label them as A and B.  Then pick a point C outside the circle which is 45 meters away from A.  Draw the triangle ABC.  Hopefully, you can follow the steps explained above from this picture.

 

Hope this helps!

 

George T

 

Answer 3

Find angle ABC first: ABC = 180-114-45=21°

 

Use Law of Sines to find AB: AB/sin(45) = 45/sin(21)

 

AB = 45 sin(45)/sin(21) = 88.79 meters

Answer 4

angle ABC = 180-45-114 = 21 degrees

Using law of sines,

x/sin45 = 45/sin21, where x is the width of the crater.

Solve for x,

x = 88.8 m <==Answer