Question 5. Find the maximum height above the \( x \)-axis of the polar curve \( r=2+2 \cos (\theta) \).
【General guidance】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(θ), 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 θ 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 θ and set it equal to 0:\( \mathrm{\frac{{\left.{d}{y}\right.}}{{{d}θ}}={2}{\cos{{\ ... See the full answer