Scientists are making an aerial study of a volcano. Their helicopter is circling at

an 8 km radius around the volcano's crater, and one of the scientists notices a

new vent that is 45° east of due north from the crater. What is the position of the

new vent relative to the crater?

(x + 5.66, y + 5.66) where (x, y) is the location of the crater

Step-by-step explanation:

Locating volcano's crater at (0, 0), the new vent is at angle equal to 45° respect x-axis, and at distance of 8 km from the crater. In polar coordinates that is radius (r) = 8, angle (α) = 45°. We are asked to transform these polar coordinates into rectangular coordinates. To do it, we have to use the following formulas:

x = r*cos α = x = 8*cos 45 = 5.66

y = r*sin α = 8*sin 45 = 5.66

If the crater is at position (x, y) then the vent would be translated at position (x + 5.66, y + 5.66)