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Solved on Nov 20, 2023

Find the two possible lengths of the third side of a non-right triangle with A=40∘A=40^\circ, a=47a=47 ft, and b=57b=57 ft. Round to the nearest foot.

STEP 1

Assumptions1. This is a non-right triangle. The given values are angle A =40 degrees, side a =47 feet, side b =57 feet3. We are asked to find the length of side c4. We will use the Law of Cosines to solve this problem, which states that for any triangle with sides of lengths a, b, and c, and an angle A opposite side a, the following equation holds truec=a+b−abcos⁡(A)c^ = a^ + b^ -ab\cos(A)5. The Law of Cosines can give two possible solutions for side c, because cosine of an angle is the same for its supplement. That is, cos(40 degrees) = cos(180 -40) = cos(140 degrees).

STEP 2

First, we will find the length of side c using the given angle A =40 degrees. Plug the given values into the Law of Cosines.
c2=472+572−2⋅47⋅57⋅cos⁡(40∘)c^2 =47^2 +57^2 -2 \cdot47 \cdot57 \cdot \cos(40^{\circ})

STEP 3

Now, calculate the value of c squared.
c2=2209+3249−2⋅47⋅57⋅cos⁡(40∘)c^2 =2209 +3249 -2 \cdot47 \cdot57 \cdot \cos(40^{\circ})

STEP 4

Continue the calculation.
c2=5458−2⋅47⋅57⋅cos⁡(40∘)c^2 =5458 -2 \cdot47 \cdot57 \cdot \cos(40^{\circ})

STEP 5

Calculate the value of c squared.
c2=5458−2⋅2679⋅cos⁡(40∘)c^2 =5458 -2 \cdot2679 \cdot \cos(40^{\circ})

STEP 6

Continue the calculation.
c2=5458−5358⋅cos⁡(40∘)c^2 =5458 -5358 \cdot \cos(40^{\circ})

STEP 7

Calculate the value of c squared.
c2=545−3452.22c^2 =545 -3452.22

STEP 8

Subtract to find the value of c squared.
c2=2005.78c^2 =2005.78

STEP 9

Take the square root of both sides to find the value of c.
c=2005.78c = \sqrt{2005.78}

STEP 10

Calculate the value of c.
c=44.78 feetc =44.78 \, feet

STEP 11

Now, we will find the other possible length of side c using the supplement of angle A, which is180 -40 =140 degrees. Plug the given values into the Law of Cosines.
c=47+57−⋅47⋅57⋅cos⁡(140∘)c^ =47^ +57^ - \cdot47 \cdot57 \cdot \cos(140^{\circ})

STEP 12

Calculate the value of c squared.
c2=2209+3249−2⋅47⋅57⋅cos⁡(140∘)c^2 =2209 +3249 -2 \cdot47 \cdot57 \cdot \cos(140^{\circ})

STEP 13

Continue the calculation.
c2=5458−2⋅47⋅57⋅cos⁡(140∘)c^2 =5458 -2 \cdot47 \cdot57 \cdot \cos(140^{\circ})

STEP 14

Calculate the value of c squared.
c2=5458−2⋅2679⋅cos⁡(140∘)c^2 =5458 -2 \cdot2679 \cdot \cos(140^{\circ})

STEP 15

Continue the calculation.
c2=5458−5358⋅cos⁡(140∘)c^2 =5458 -5358 \cdot \cos(140^{\circ})

STEP 16

Calculate the value of c squared.
c2=5458−3452.22c^2 =5458 -3452.22

STEP 17

Subtract to find the value of c squared.
c2=2005.78c^2 =2005.78

STEP 18

Take the square root of both sides to find the value of c.
c=2005.78c = \sqrt{2005.78}

STEP 19

Calculate the value of c.
c=44.78 feetc =44.78 \, feetThe two possible lengths for side c are45 feet and60 feet, when rounded to the nearest foot.

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