Question

Find the slope of the tangent to the curve x=4 cos^3⁡3θ and y=5 sin^3⁡⁡3θ at θ=π/4.

(a) –\(\frac{3}{4}\)

(b) –\(\frac{1}{4}\)

(c) \(\frac{5}{4}\)

(d) –\(\frac{5}{4}\)

This question was addressed to me in an online quiz.

Enquiry is from Derivatives Application in portion Application of Derivatives of Mathematics – Class 12

Answer 1

The correct choice is (c) \(\frac{5}{4}\)

Easy explanation: Given that, x=4 cos^3⁡3θ and y=5 sin^3⁡3θ

\(\frac{dx}{dθ}\)=4(3)(3 cos^2⁡3θ)(-sin⁡3θ)

\(\frac{dy}{dθ}\)=5(3)(3 sin^2⁡3θ)(cos⁡3θ)

\(\frac{dy}{dx}\)=\(\frac{dy}{dθ}.\frac{dθ}{dx}=\frac{5(3)(3 sin^2⁡3θ)(cos⁡3θ)}{4(3)(3 cos^23θ)(-sin⁡3θ)}\)

\(\frac{dy}{dx}\)=-\(\frac{5 tan⁡3θ}{4}\)

\(\frac{dy}{dx}\)]θ=π/4=-\(\frac{5}{4} tan\frac{⁡3 \pi}{4}\)=-\(\frac{5}{4} (-1)=\frac{5}{4}\).