Question

Math

Posted 2 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 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 $c^2 = a^2 + b^2 - 2ab\cos(C)$, 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.

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