Question

Math

Posted 4 months ago

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

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Answer from Sia

Posted 4 months ago

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.

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