Q 53. - Calculus

Question

Annie is using wood framing and a fabric shell to design a kayak. For simplicity, she considers making the cross-section of the kayak follows the curve $r = A(1 - 0.5 \sin \theta )$ where $A = 1$ at the middle of the kayak and becomes smaller as the cross-section is taken farther from the middle of the boat. Sketch the cross-section of the boat when $A = 1$

Step-by-Step Solution

Answer

The graph is 

Step 1: Given information

$r = A(1 - 0.5 \sin \theta )$

Step 2: Concept

A collection of quantities is defined as a function of one or more independent variables called parameters in a parametric equation.

Step 3: Calculation

Consider the curve $r=\mathrm{A}(1-0.5\sin \theta )$

When $A=1$ the goal is to draw the cross-section of the curve.

The curve $r=(1-0.5\sin \theta )$ when $A=1$

Assume $\theta =0,\frac{\pi }{2},\pi ,\frac{3\pi }{2},2\pi$

We obtain the values of the equation $r=(1-0.5\sin \theta )$ for various $\theta$ values.

When $\theta =0$

$$\begin{gathered} r=(1-0.5\sin 0)[\text{ since }r=(1-0.5\sin \theta \left)\right] \\ r=(1-0)[\text{ since }\sin 0=0] \\ r=1 \end{gathered}$$

Then the coordinate $(r,\theta )=(1,0)$

When $\theta =\frac{\pi }{2}$

$$\begin{gathered} r=\left(1-0.5\sin \frac{\pi }{2}\right)[\text{ since }r=(1-0.5\sin \theta \left)\right] \\ r=(1-(0.5)·1)\left[\text{ since }\sin \frac{\pi }{2}=1\right] \\ r=0.5 \end{gathered}$$

Then the coordinate $(r,\theta )=\left(0.5,\frac{\pi }{2}\right)$

When $\theta =\pi$

$$\begin{gathered} r=(1-0.5\sin \pi )[\text{ since }r=(1-0.5\sin \theta \left)\right] \\ r=(1-(0.5)·0)\left[\text{ since }\sin \frac{\pi }{2}=1\right] \\ r=1 \end{gathered}$$

Then the coordinate $(r,\theta )=(1,\pi )$

When $\theta =\frac{3\pi }{2}$

$$\begin{gathered} r=\left(1-0.5\sin \frac{3\pi }{2}\right)[sincer=(1-0.5\sin \theta \left)\right] \\ r=(1-(0.5)·-1)\left[since\sin \frac{\pi }{2}=1\right] \\ r=1.5 \end{gathered}$$

When $\theta =2\pi$

$$\begin{gathered} r=(1-0.5\sin 2\pi )[\text{ since }r=(1-0.5\sin \theta \left)\right] \\ r=(1-(0.5)·0)\left[\text{ since }\sin \frac{\pi }{2}=1\right] \\ r=1 \end{gathered}$$

Then the coordinate $(r,\theta )=(1,2\pi )$

Step 4: Calculation

Draw the graph by putting all of the following points on it.

$(1,0)\left(0.5,\frac{\pi }{2}\right)(1,\pi )\left(1.5,\frac{3\pi }{2}\right)(1,2\pi )$

This is the graphical representation.