Find the slope of the tangent to the curve x=4 cos3⁡3θ and y=5 sin3⁡⁡3θ at θ=π/4

The correct option is 5/4.

The given parametric equations of the curve are:
$x=4{\cos }^3\left(\theta \right)$
and
$y=5{\sin }^3\left(\theta \right)$

To find the slope of the tangent to the curve, we calculate the derivative $\frac{dy}{dx}$ using parametric differentiation:
$\frac{dy}{dx}=\frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}$

First, we differentiate $x$ with respect to $\theta$:
$\frac{dx}{d\theta }=\frac{d}{d\theta }\left[4{\cos }^3\right(\theta \left)\right]=12{\cos }^2\left(\theta \right)\sin \left(\theta \right)$

Next, we differentiate $y$ with respect to $\theta$:
$\frac{dy}{d\theta }=\frac{d}{d\theta }\left[5{\sin }^3\right(\theta \left)\right]=15{\sin }^2\left(\theta \right)\cos \left(\theta \right)$

Now, we divide $\frac{dy}{d\theta }$ by $\frac{dx}{d\theta }$:
$\frac{dy}{dx}=\frac{15{\sin }^2\left(\theta \right)\cos \left(\theta \right)}{12{\cos }^2\left(\theta \right)\sin \left(\theta \right)}$

Simplifying the trigonometric expressions yields:
$\frac{dy}{dx}=\frac{5}{4}·\frac{\sin \left(\theta \right)}{\cos \left(\theta \right)}=\frac{5}{4}\tan \left(\theta \right)$

To find the slope of the tangent at $\theta =\frac{\pi }{4}$, we substitute this value into our equation:
${\left[\frac{dy}{dx}\right]}_{\theta =\frac{\pi }{4}}=\frac{5}{4}\tan \left(\frac{\pi }{4}\right)$

Since $\tan \left(\frac{\pi }{4}\right)=1$:
$\frac{dy}{dx}=\frac{5}{4}·1=\frac{5}{4}$