A curve passes through the point $( 8,2 )$ and has the property that the slope of the curve at every point $P$ is 3 times the y-coordinate $P$. What is the equation of the curve?
A) $y = 2 e ^ { 3 x + 24 }$
B) $y = \frac { e ^ { 3 x - 24 } } { 2 }$
C) $y = 2 e ^ { 3 x - 8 }$
D) $y = 2 e ^ { 3 x - 24 }$
E) $y = 2 e ^ { x - 24 }$

Question

Quizplus · Accepted Answer

Let the equation of the curve be $y=f(x)$. We know that $f(8)=2$ and $f'(x)=3y=3f(x)$. Solving the differential equation $f'(x)=3f(x)$ gives $f(x)=Ce^{3x}$, where $C$ is a constant. To find $C$, we use the initial condition $f(8)=2$, letting $x=8$ and $y=2$ in the equation $f(x)=Ce^{3x}$: $$2=Ce^{3\cdot 8}.$$ Solving for $C$ gives $C=\frac{2}{e^{24}}$, so the equation of the curve is: $$y=f(x)=\frac{2}{e^{24}}e^{3x}=2e^{3x-24}.$$ Therefore, the answer is $\boxed{	extbf{(D)}}$.

A Curve Passes Through the Point (8,2)( 8,2 )(8,2) And Has the Property That the Slope of the Curve

A Curve Passes Through the Point $( 8,2 )$ And Has the Property That the Slope of the Curve