Bamboo Forest

Suppose the point nearest to the origin is $(a,b)$ . Then $f(a,b) = b^2 + a^3 - 2ab + 7 = 12$ and $f(a, b+3) = (b+3)^2 + a^3 -2a(b+3)+7 = 27$

$f(a, b+3) - f(a, b) = 15 = 6b+9-6a$ ; $b=a+1$ .

$f(a,b) = (a+1)^2 + a^3 - 2a(a+1) + 7 = 12$

$a^3- a^2 -4 = 0$

$(a-2)(a^2+a+2) = 0$

Thus, $a=2$ and $b=3$ .

Now we can set up a region of $[2,4]\times[3,6]$ for $f(x,y)$ , and the average height of the bamboo can be calculated as the ratio total volume over the base area. The volume in this case = $\displaystyle \int_{2}^{4}\int_{3}^{6} y^2 + x^3 - 2xy +7 \,dy \,dx$

Taking the first integral, $\displaystyle \int_{3}^{6} y^2 + x^3 - 2xy +7 \ dy = \left[\left.\dfrac{y^3}{3} + yx^3 -xy^2 + 7y\right|_3^6\right] = 3x^3 - 27x + 84$

Then taking the second integral, $\displaystyle \int_{2}^{4} 3x^3 - 27x + 84 \ dx = \left[\left.\dfrac{3x^4}{4} - \dfrac{27x^2}{2} + 84x \right|_2^4\right] = 168+ 180-162 = 186$ .

Therefore, the average height = $\dfrac{186}{2\times3} = \boxed{31}$ .