Optimization Question

Question

With the upcoming football season, we need to keep Mator the Gator safe from

the coronavirus! Caden and Rebecca have decided to build him a pen in the shape of

an isosceles triangle with the vertex at the origin and base parallel to the x-axis. (This

means it will look like an upside-down triangle with the point at (0,0)). The triangle

has to fit in the curve at the end of the track that is modeled by

y = 36 − x^2

Find the height and base of the triangle that will maximize the area of the triangle.

Answer 1

Area = A =2xy = 2x(36-x^2)^1/2

take the derivative and set = 0, solve for x^2, then for x

That's probably the usual method

But just let x=y = sqr(36-x^2)

square both sides

x^2 = 36- x^2

2x^2 = 36

x^2 = 18

x = sqr18 = about 4.24 = y

dimensions of the triangle are

8.48 by 4.24

base = 2x= 8.48 = 2sqr18

height = y = 4.24= sqr18

max Area = 8.48 by 4.24 divided by 2 or exactly 18

graph it and estimate what looks like the biggest area, it's somewhere between 3 and 5 height. Height of the parabola is 6. h= 6 or h=0 gives zero Area. A rough guess would have been about 4