Đáp án: $890$ m 

Giải thích các bước giải:

Ta có:

$f(x)$ tiếp xúc với $Ox$ tại $(0,0)$ và cắt $Ox$ tại $(3,0)$

$\to  y=f(x)=ax^2(x-3)=ax^3-3ax^2$

$\to y'=3x^2a-6ax=3ax(x-2)$

Giải $y'=0$

$\to 3ax(x-2)=0$

$\to x\in\{0, 2\}$

$\to x=2$ là cực trị của $f(x)$

$\to x=2\to y=a\cdot 2^2(2-3)=-4a$

Ta có:

$S_2-S_1=\dfrac{11}2$

$\to \displaystyle\int ^3_0ax^2(x-3)dx-\displaystyle\int ^2_0 (-4a-ax^2(x-3))dx=\dfrac{11}2$

$\to \displaystyle\int ^3_0a(x^3-3x^2)dx+\displaystyle\int ^2_0(4a+ax^3-3ax^2)dx=\dfrac{11}2$

$\to a(\dfrac{x^4}4-x^3)\bigg|^3_0+(4ax+\dfrac14ax^4-ax^3)\bigg|^2_0=\dfrac{11}2$

$\to a(\dfrac{3^4}4-3^3)+(8a+a\cdot \dfrac{2^4}4-a\cdot 2^3)=\dfrac{11}2$

$\to -\dfrac{11a}{4}=\dfrac{11}2$

$\to a=-2$

$\to f(x)=-2x^2(x-3)=-2x^3+6x^2$

$\to g(x)=f(x)+2=-2x^2(x-3)+2=-2x^3+6x^2+2$

Lấy $C\in f(x),D\in g(x)$

$\to C(x_c, -2x_c^3+6x_c^2); D(x_d; -2x_d^3+6x_d^2+2)$

$\to CD=\sqrt{(x_d-x_c)^2+(-2x_d^3+6x_d^2+2+2x_c^3-6x_c^2)^2}$

$\to \begin{cases}f'_{x_c}=\dfrac{-2(x_d-x_c)+2(6x_c^2-12x_c)(-2x_d^3+6x_d^2+2+2x_c^2-6x_c^2)}{2\sqrt{(x_d-x_c)^2+(-2x_d^3+6x_d^2+2+2x_c^2-6x_c^2)^2}}\\ f'_{x_d}=\dfrac{2(x_d-x_c)+2(-6x_d^2+12x_d)(-2x_d^3+6x_d^2+2+2x_c^3-6x_c^2)}{\sqrt{(x_d-x_c)^2+(-2x_d^3+6x_d^2+2+2x_c^2-6x_c^2)^2}}\end{cases}$

Giải hệ $f'_{x_c}=0, f'_{x_d}=0$

$\to \begin{cases} \dfrac{-2(x_d-x_c)+2(6x_c^2-12x_c)(-2x_d^3+6x_d^2+2+2x_c^2-6x_c^2)}{2\sqrt{(x_d-x_c)^2+(-2x_d^3+6x_d^2+2+2x_c^2-6x_c^2)^2}}=0\\\dfrac{2(x_d-x_c)+2(-6x_d^2+12x_d)(-2x_d^3+6x_d^2+2+2x_c^3-6x_c^2)}{\sqrt{(x_d-x_c)^2+(-2x_d^3+6x_d^2+2+2x_c^2-6x_c^2)^2}}=0\end{cases}$

$\to \begin{cases} -2(x_d-x_c)+2(6x_c^2-12x_c)(-2x_d^3+6x_d^2+2+2x_c^2-6x_c^2) =0\\2(x_d-x_c)+2(-6x_d^2+12x_d)(-2x_d^3+6x_d^2+2+2x_c^3-6x_c^2)=0\end{cases}$

$\to \begin{cases} 2(x_d-x_c)=2(6x_c^2-12x_c)(-2x_d^3+6x_d^2+2+2x_c^2-6x_c^2)\\2(x_d-x_c)=-2(-6x_d^2+12x_d)(-2x_d^3+6x_d^2+2+2x_c^3-6x_c^2)\end{cases}$

Chia vế cho vế

$\to \dfrac{6x_c^2-12x_c}{6x_d^2-12x_d}=1$

$\to \dfrac{x_c^2-2x_c}{x_d^2-2x_d}=1$

$\to x_c^2-2x_c=x_d^2-2x_d$

$\to (x_c^2-x_d^2)-2(x_c-x_d)=0$

$\to (x_c+x_d-2)(x_c-x_d-2)=)$

$\to x_c+x_d=2$

$\to x_d=2-x_c$

$\to  2(2-x_c-x_c)=2(6x_c^2-12x_c)(-2(2-x_c)^3+6(2-x_c)^2+2+2(2-x_c)^2-6(2-x_c)^2)=0$

$\to x_c\approx 0.33, x_d\approx 1.67$

$\to CD=\sqrt{(1.67-0.33)^2+(-2\cdot 1.67^3+6\cdot 1.67^2+2+2\cdot 0.33^3-6\cdot 0.33^2)^2}\cdot 100\approx 890(m)$