Question

find an equation in rectangular coordinates for the surface represented by the cylindrical equation. r = 5 cos(θ) sketch its graph.

Explanation:

Step1: Recall conversion formulas

We know that $x = r\cos(\theta)$ and $r^{2}=x^{2}+y^{2}$. Given $r = 5\cos(\theta)$, multiply both sides by $r$.
$r\times r=r\times5\cos(\theta)$

Step2: Substitute conversion - formulas

Since $r^{2}=x^{2}+y^{2}$ and $x = r\cos(\theta)$, we have $x^{2}+y^{2}=5x$.

Step3: Rearrange the equation

$x^{2}-5x + y^{2}=0$. Completing the square for the $x$ - terms: $(x-\frac{5}{2})^{2}-\frac{25}{4}+y^{2}=0$.
$(x - \frac{5}{2})^{2}+y^{2}=\frac{25}{4}$

Answer:

$(x - \frac{5}{2})^{2}+y^{2}=\frac{25}{4}$