Final answer:
The zeros of the function F(r) = (r-14)²-56.25 are smaller r = 6.5 and larger r = 21.5.
Step-by-step explanation:
To find the zeros of the function F(r) = (r-14)²-56.25, we need to set the function equal to zero and solve for r.
(r-14)²-56.25 = 0
Expanding and simplifying the equation, we get:
r² - 28r + 196 - 56.25 = 0
r² - 28r + 139.75 = 0
Since this is a quadratic equation, we can use the quadratic formula to find the solutions:
r = (-b ± √(b²-4ac)) / (2a)
Here, a = 1, b = -28, and c = 139.75.
Substituting these values into the quadratic formula, we get:
r = (-(-28) ± √((-28)²-4(1)(139.75))) / (2(1))
r = (28 ± √(784-559)) / 2
r = (28 ± √(225)) / 2
r = (28 ± 15) / 2
So the two solutions are:
Smaller r = (28 - 15) / 2 = 13 / 2 = 6.5
Larger r = (28 + 15) / 2 = 43 / 2 = 21.5