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Question
Math
Posted 2 months ago

Kapil's robot starts 70 cm70 \mathrm{~cm} from its charging base. It faces the base, then turns 6060^{\circ} clockwise, as shown. Finally, the robot moves 50 cm50 \mathrm{~cm}.

After moving, how far is the robot from the charging base?
Do not round during your calculations. Round your final answer to the nearest centimeter.
\square cm\mathrm{cm}
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Answer from Sia
Posted 2 months ago
Solution by Steps
step 1
To find the unknown side of the triangle, we can use the Law of Cosines since we have two sides and the included angle
step 2
The Law of Cosines formula is c2=a2+b22abcos(C)c^2 = a^2 + b^2 - 2ab\cos(C), where aa and bb are the sides and CC is the included angle
step 3
Substitute the given values into the Law of Cosines formula: c2=702+5022(70)(50)cos(60)c^2 = 70^2 + 50^2 - 2(70)(50)\cos(60^\circ)
step 4
Calculate the cosine of 6060^\circ, which is 12\frac{1}{2}
step 5
Substitute the cosine value into the equation: c2=702+5022(70)(50)(12)c^2 = 70^2 + 50^2 - 2(70)(50)\left(\frac{1}{2}\right)
step 6
Perform the arithmetic: c2=4900+25003500c^2 = 4900 + 2500 - 3500
step 7
Simplify the equation: c2=3900c^2 = 3900
step 8
Take the square root of both sides to find cc: c=3900c = \sqrt{3900}
step 9
Calculate the square root: c62.45c \approx 62.45 cm
step 10
Round the final answer to the nearest centimeter: c62c \approx 62 cm
Answer
62 cm
Key Concept
Law of Cosines
Explanation
The Law of Cosines is used to find an unknown side of a triangle when two sides and the included angle are known. It is especially useful in non-right triangles.

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