40.7 miles.
For this problem, we want to know the length of the chord created by the line and the circle. So let's first create the equations needed.
The slope intercept equation for a line is:
y = ax + b
the value for a will be the the difference in y divided by the difference in x. We're going from y=61 to y=0 for a chance of -61 and from x=0 to x=62 for a change of 62. So the value of a is
-61/62, giving us the formula
y = -(61/62)x + b
Substituting x = 0, we can calculate b
61 = -(61/62)0 + b
61 = b
So the equation for the line is:
y = -(61/62)x + 61
Now for the equation for the circle. Since the circle is centered at the origin, the equation is:
x^2 + y^2 = 48^2
Now we to calculate the intersections.
y = -(61/62)x + 61
x^2 + y^2 = 48^2
x^2 + (-(61/62)x + 61)^2 = 48^2
x^2 + (3721/3844)x^2 - (3721/31)x + 3721 = 2304
(7565/3844)x^2 - (3721/31)x + 3721 = 2304
(7565/3844)x^2 - (3721/31)x + 1417 = 0
1.968002081x^2 - 120.0322581x + 1417 = 0
And we have a rather ugly quadratic equation which we can solve using the quadratic formula, giving the solutions x = 16.00512574 and x = 44.98681081
Now we need to calculate the y values for those 2 x values.
y = -(61/62)x + 61
y = -(61/62)16.00512574 + 61
y = 45.25302145
y = -(61/62)x + 61
y = -(61/62)44.98681081 + 61
y = 16.73878292
So the 2 endpoints are
(16.00512574, 45.25302145) and (44.98681081, 16.73878292)
The distance between those points can be calculated using the Pythagorean theorem.
sqrt((16.00512574 - 44.98681081)^2 + (45.25302145 - 16.73878292)^2)
= sqrt(-28.98168506^2 + 28.51423853^2)
=
sqrt(839.938069 + 813.0617988)
=
sqrt(1652.999868)
= 40.65710107
And finally, we have the solution of 40.7 miles.
