Q 52. - Calculus

Question

Ian sometimes sews his own outdoor gear. He wants to make a body-hugging climbing pack. The bottom of the pack is the area outside the circle r = 14, but inside r = 14√2 sin θ, where r is measured in inches. Ian requires the pack to carry 2500 cubic inches of gear. If the pack has vertical sides, how tall does Ian need to make it?

Step-by-Step Solution

Answer

5.4 units

Step 1. Given information

The bottom of the pack is the area outside the circle r = 14, but inside r = 14√2 sin θ, where r is measured in inches. 

Ian requires the pack to carry 2500 cubic inches of gear..

Step 2. Explanation

Ian sometimes sews his own outdoor gear such that the bottom of the pack is the area outside the circle r = 14, but inside r = 14√2 sin θ, where r is measured in inches.

Area of circle with radius 14 units $=\frac{22}{7}{\left(14\right)}^2$

                                                          $=616$ square units.

Area of circle with radius $14\sqrt{2}\sin \theta$ units $=\frac{22}{7}{\left(14\sqrt{2} \sin \theta \right)}^2$

                                                                $=1232 {\sin }^2\theta$ square units

So, area between these circles $=1232{\sin }^2\theta -616$

Ian requires the pack to carry 2500 cubic inches of gear.

Ian need to make pack of $\frac{1232{\sin }^2\theta -616}{2500}=\frac{1232\left(2\right)-616}{2500}=5.4$ units.