Angelo the alligator is laying out next to a perfectly circular pond. he is 25 feet from the edge of the pond and 4 feet from the point of tangency. Determine the radius of the pond

19.5 feet

Step-by-step explanation:

I assume you mean 40 feet, not 4 feet.

Draw a picture of the pond and Angelo the alligator.  Angelo is 25 feet from the edge of the pond.  If we draw a tangent line from Angelo to the pond, the length of the tangent line is 40 feet.

Now draw a radius line from the center of the circle to the point of tangency; this is perpendicular to the tangent line.  Draw another radius line from the center of the circle towards Angelo.

The result is a right triangle, where the legs are r and 40 feet, and the hypotenuse is r+25 feet.

Using Pythagorean theorem:

(r + 25)² = r² + 40²

r² + 50r + 25² = r² + 40²

50r + 25² = 40²

50r + 625 = 1600

50r = 975

r = 19.5

The radius of the pond is 19.5 feet.

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.