Afarmer has 1,800 feet of fencing and wants to fence off a rectangular field that borders a straight river. he needs no fence along the river. write the function that will produce the largest area if x is the short side of the rectangle.

There are 2 short sides x and 1 long side y.

x+x+y=1,800
2x+y=1,800
y=1,800-2x

The area of the rectangle is A=x(1,800-x).

A is a function with variable x. Moreover it is a quadratic (second degree) function.

The graph of the function is a parabola which opens downwards, because the signs of the leading coefficients of the factors, multiply to minus:

(x)(-x)=-x^2.

This means that the highest point of the parabola, is its vertex V(h,k), so the function takes its largest value for x=h.

The roots of A=x(1,800-x) are x=0 and x=1,800, thus the x coordinate of the vertex, must be the midpoint of 0 and 1,800, that is 900.

A=x(1,800-x)

largest area is when x=900

45x is the correct answer

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12