The path traveled by a bottlenose dolphin as it jumps out of water is modeled by the equation y = −0.4x2 + 3x, where y is the height above water and x is the horizontal distance in feet. if a beam of light is shone upward at an angle modeled by the equation x +  y = 10, at what height from the water's surface will the beam of light hit the dolphin

The beam of light hit the dolphin at the common point that is the solution of a system:

.

From the second equation express y: y=10-x and substitute it into the first equation:

.

Therefore, the horizontal distance in feet is x=5 ft and the height above water is y=10-5=5 ft.

the beam of light hit the dolphin at 5 ft above water

x is variable or an unknown value to find

example:

2x = 5 which means x = 3

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.