Kapil's robot starts $70 \mathrm{~cm}$ from its charging base. It faces the base, then turns $60^{\circ}$ clockwise, as shown. Finally, the robot moves $50 \mathrm{~cm}$.

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.
$\square$ $\mathrm{cm}$

Question

Kapil's robot starts $70 \mathrm{~cm}$ from its charging base. It faces the base, then turns $60^{\circ}$ clockwise, as shown. Finally, the robot moves $50 \mathrm{~cm}$.

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.
$\square$ $\mathrm{cm}$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Use the Law of Cosines to find the length of the unknown side opposite the 60° angle. ‖ ‖ step 2 ⋮ The Law of Cosines formula is $c^2 = a^2 + b^2 - 2ab\cos(\gamma)$, where $c$ is the length of the side opposite the angle $\gamma$, and $a$ and $b$ are the lengths of the other two sides. ‖ ‖ step 3 ⋮ Substitute the given values into the Law of Cosines formula: $c^2 = 70^2 + 50^2 - 2(70)(50)\cos(60°)$. ‖ ‖ step 4 ⋮ Calculate the cosine of 60°, which is $\frac{1}{2}$. ‖ ‖ step 5 ⋮ Substitute the cosine value into the equation: $c^2 = 70^2 + 50^2 - 2(70)(50)\left(\frac{1}{2}\right)$. ‖ ‖ step 6 ⋮ Simplify the equation: $c^2 = 4900 + 2500 - 3500$. ‖ ‖ step 7 ⋮ Calculate the value of $c^2$: $c^2 = 3900$. ‖ ‖ step 8 ⋮ Take the square root of both sides to find $c$: $c = \sqrt{3900}$. ‖ ‖ step 9 ⋮ Round the final answer to the nearest centimeter: $c \approx 62$ cm. ‖ ∻Answer∻ ⚹ The robot is approximately 62 cm from the charging base. ⚹ ∻Key Concept∻ ⚹Law of Cosines⚹ ∻Explanation∻ ⚹The Law of Cosines is used to find the length of a side of a triangle when two sides and the included angle are known.⚹◊What is the formula to calculate the distance of the robot from the charging base in the given geometric problem?⍭ Generate me a similar question◊

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