Due to construction, traffic's getting detoured from main street by turning slightly right onto oak avenue and continuing straight for 4 mi. to get back onto main stree, you need to make a 90° left turn onto lilac lane and continue straight for x miles. lilac lane intersects main street at 30° for the end of the detour. approximately how many miles should you travel on lilac lane to reach main street? i don't know how to get the answer, but they gave the answer, and it's 6.9 mi.

If you actually draw the diagram described in the text (Use something like Microsoft Paint if you're using Microsoft Windows), you'll quickly realize that the detour is a 30/60/90 right triangle. The 90 and 30 degree values are given in the text. And since all triangles have a total of 180 degrees, the 3rd angle is 180-90-30 = 60. Now for a 30/60/90 right triangle, if you consider the hypotenuse to be of length 1, the short leg will be of length 0.5 and the long leg will be of length sqrt(3)/2. Those are values that you should have memorized. And if you haven't memorized them, you can easily derive them from realizing that if you take the triangle and copy it, then rotate the copy around the long leg, you'll get an equilateral triangle with one of the 60 degree angles bisected. So you'll quickly realize that the hypotenuse has a length of 1, the short leg will be half of that, and the long leg you can calculate using the Pythagorean theorem. And for this problem, the short leg is of length 4. So: 4/0.5 = X/(sqrt(3)/2) 8 = X/(sqrt(3)/2) (sqrt(3)/2)*8 = X sqrt(3)*4 = X 1.732050808*4 = X 6.92820323 = X And after rounding to 1 decimal place, you get 6.9 miles.

Answer: 10pi

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.