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 ⋮ 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 $c^2 = a^2 + b^2 - 2ab\cos(C)$, where $a$ and $b$ are the sides and $C$ is the included angle. ‖ ‖ step 3 ⋮ Substitute the given values into the Law of Cosines formula: $c^2 = 70^2 + 50^2 - 2(70)(50)\cos(60^\circ)$. ‖ ‖ step 4 ⋮ Calculate the cosine of $60^\circ$, 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 ⋮ Perform the arithmetic: $c^2 = 4900 + 2500 - 3500$. ‖ ‖ step 7 ⋮ Simplify the equation: $c^2 = 3900$. ‖ ‖ step 8 ⋮ Take the square root of both sides to find $c$: $c = \sqrt{3900}$. ‖ ‖ step 9 ⋮ Calculate the square root: $c \approx 62.45$ cm. ‖ ‖ step 10 ⋮ Round the final answer to the nearest centimeter: $c \approx 62$ 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.⚹◊What is the formula for calculating the unknown side of a triangle given two sides and the included angle⍭ Generate me a similar question◊

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