Question

kapil’s robot starts 70 cm from its charging base. it faces the base, then turns 60° clockwise, as shown. finally, the robot moves 50 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.

Explanation:

Step1: Identify the triangle sides and angle

We have a triangle with two sides \( a = 50 \) cm, \( b = 70 \) cm, and the included angle \( C = 60^\circ \). We need to find the third side \( c \) (distance from robot to base) using the Law of Cosines. The Law of Cosines formula is \( c^{2}=a^{2}+b^{2}-2ab\cos(C) \).

Step2: Substitute the values into the formula

Substitute \( a = 50 \), \( b = 70 \), and \( \cos(60^\circ)=\frac{1}{2} \) into the formula:
\[

$$\begin{align*} c^{2}&=50^{2}+70^{2}-2\times50\times70\times\cos(60^\circ)\\ &= 2500 + 4900-2\times50\times70\times\frac{1}{2}\\ &=2500 + 4900 - 3500\\ &=2500+1400\\ &= 3900 \end{align*}$$

\]

Step3: Find the value of \( c \)

Take the square root of \( c^{2} \) to find \( c \):
\( c=\sqrt{3900}\approx62.45 \approx 62 \) (rounded to the nearest centimeter)

Answer:

62