An outdoor track consists of a rectangular region with a semi-circle on each end. if the perimeter of the track must be 200 meters, find the dimensions that will make the area of the rectangular region as large as possible.

Length = 50m

Width = 31.84m

Step-by-step explanation:

Perimeter of a track = 200m

The perimeter of a track with two semi-circular end and two rectangular regions = P

Perimeter of a semi-circle = ½(2πr)

Length of a rectangle = x

Perimeter of the rectangle = x+x

P = ½(2πr) + ½(2πr)+x + x

P = 2πr + 2x

200 =2πr + 2x

2πr = 200 – 2x

2πr = 2(100 – x)

r = 2(100 – x) / 2π

r = (100-x)/π

total area of the rectangular region =A

A= x(2r)

= (x)2[(100-x)/π]

= 2x[(100-x)/π]

A = (200x – 2x^2)/π

Differentiate A with respect to x

dA/dx = (200 -4x)/π

at critical point, first derivative vanishes(dA/dx = 0)

(200 -4x)/π = 0

-4x =-200

x =-200/-4

x = 50

this means length (x) = 50m

put x=50 into r = (100-x)/π

r= (100-50)/π

r =50/π

width = 2r

width = 2(50/π)

= 2(19.62)

= 31.84m

guess at question:

a vertical line intersects the graph of g(x,y) = 0 at two points. what conclusion follows about the function f(x) implicitly defined by g(x,f(x)) = 0?

the domain of f(x) cannot include the x coordinate of any such vertical line.

informally, f(x) is not a function, and

g(x,y)=0 cannot be solved for y uniquely.

the relation r = {(x,y) such that g(x,y) = 0} is a function if

for all (a,b) ∊ r and (c,d) ∊ r, a=c implies b=d.

step-by-step explanation:

what? that isnt a question

step-by-step explanation: