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). Then we are given that:

f'(x) = 3y = 3f(x)

This is a separable differential equation, which we can solve as follows:

f'(x)/f(x) = 3
ln|f(x)| = 3x + C
|f(x)| = e^(3x+C) = e^3x * e^C

Since we know that the curve passes through the point (8,2), we can substitute these values into the equation to solve for e^C:

|f(8)| = 2 = e^3(8) * e^C
e^C = 2 / e^24

Now we can write the full equation of the curve:

f(x) = ±e^3x * 2/e^24

Since we want the curve to pass through the point (8,2), we choose the positive sign:

f(x) = 2/e^24 * e^3x = 2e^(3x-24)

Therefore, the equation of the curve is y = 2e^(3x-24), which is choice (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