The tangent to the curve $y = ax^3$ at the point $(5, b)$ has a gradient of $30$. Find the values of the constants $a$ and $b$.

Question

The tangent to the curve $y = ax^3$ at the point $(5,b)$ has a gradient of $30$. Find the values of the constants $a$ and $b$.

My working so far:

$$\frac{dy}{dx} = 3ax^2$$

tangent: $y = mx + c$

substituting $3ax^2$ (derivative) as gradient we get $75a=30, a=0.4$

How do you solve for $b$ from there?

Answer 1

What you did is fine. Then, $b=\frac25\times5^3=50$.