# Question 5. Find the maximum height above the $$x$$-axis of the polar curve $$r=2+2 \cos (\theta)$$.

Transcribed Image Text: 5. Find the maximum height above the $$x$$-axis of the polar curve $$r=2+2 \cos (\theta)$$.
Transcribed Image Text: 5. Find the maximum height above the $$x$$-axis of the polar curve $$r=2+2 \cos (\theta)$$.
&#12304;General guidance&#12305;The answer provided below has been developed in a clear step by step manner.Step1/2To find the maximum height above the x-axis of the polar curve r = 2 + 2cos(&#952;), we need to first convert the polar equation to rectangular coordinates:$$\mathrm{{x}={r}{\cos{{\left(θ\right)}}}={\left({2}+{2}{\cos{{\left(θ\right)}}}\right)}{\cos{{\left(θ\right)}}}={2}{\cos{{\left(θ\right)}}}+{2}{{\cos}^{{2}}{\left(θ\right)}}}$$$$\mathrm{{y}={r}{\sin{{\left(θ\right)}}}={\left({2}+{2}{\cos{{\left(θ\right)}}}\right)}{\sin{{\left(θ\right)}}}={2}{\sin{{\left(θ\right)}}}+{2}{\cos{{\left(θ\right)}}}{\sin{{\left(θ\right)}}}}$$Then we can eliminate &#952; by using the identity $$\mathrm{{{\cos}^{{2}}{\left(θ\right)}}+{{\sin}^{{2}}{\left(θ\right)}}={1}}$$ to get:$$\mathrm{{x}={2}{\cos{{\left(θ\right)}}}+{2}{{\cos}^{{2}}{\left(θ\right)}}={2}{\left({\cos{{\left(θ\right)}}}+{{\cos}^{{2}}{\left(θ\right)}}\right)}}$$$$\mathrm{{y}={2}{\sin{{\left(θ\right)}}}+{2}{\cos{{\left(θ\right)}}}{\sin{{\left(θ\right)}}}={2}{\sin{{\left(θ\right)}}}+{\sin{{\left({2}θ\right)}}}}$$To find the maximum height above the x-axis, we need to find the maximum value of y. To do this, we can take the derivative of y concerning &#952; and set it equal to 0:\( \mathrm{\frac{{\left.{d}{y}\right.}}{{{d}θ}}={2}{\cos{{\ ... See the full answer