The figure shows part of the curve $y=2x^2+3$. The point $B(x,y)$ is a variable point that moves

Question

The figure shows part of the curve $y=2x^2+3$. The point $B(x,y)$ is a variable point that moves along the curve for $0<x<6$. $C$ is a point on the $x$-axis such that $BC$ is parallel to the $y$-axis and $A(6,0)$ lies on the $x$-axis. Express the area of triangle $ABC$, $T$units$^2$, in terms of $x$, and find an expression for $\frac{dT}{\mathrm{d}x}$. Given that when $x=2$, $T$ is increasing at the rate of $0.8$ units$^2$/s, find the corresponding rate of change of $x$ at this instant.

Accepted Answer

$T=\frac12(6-x)(2x^2+3)$, $\frac{\mathrm dT}{\mathrm dx}=12x-3x^2-\frac32$, $\frac{\mathrm dx}{\mathrm dt}=\frac8{105}\text{ unit s}^{-1}$

Connected Rate of Change involving Area

Connected Rate of Change involving Area

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