20-21 Width of the second corridor for a pole to pass horizontally

Tags:

Problem 20
A pole 27 feet long is carried horizontally along a corridor 8 feet wide and into a second corridor at right angles to the first. How wide must the second corridor be?

Solution 20

Click here to show or hide the solution

Horizontal pole passing through 90 degree corridors

$a = 8 \sec \theta$

$b = w \csc \theta$

$a + b = 24$

$8 \sec \theta + w \csc \theta = 24$

$w = \dfrac{24 - 8\sec \theta}{\csc \theta}$

$w = 24\dfrac{1}{\csc \theta} - 8\dfrac{\sec \theta}{\csc \theta}$

$w = 24\sin \theta - 8\dfrac{\sin \theta}{\cos \theta}$

$w = 24\sin \theta - 8\tan \theta$

$\dfrac{dw}{\theta} = 24 \cos \theta - 8 \sec^2 \theta = 0$

$24\cos \theta = 8 \sec^2 \theta$

$24\cos \theta = \dfrac{8}{\cos^2 \theta}$

$\cos^3 \theta = \dfrac{8}{24}$

$\cos^3 \theta = \dfrac{1}{3}$

$\theta = 46.10^\circ$

$w = 24 \sin \theta - 8 \tan \theta$

$w = 24 \sin 46.10^\circ - 8 \tan 46.10^\circ$

$w = 8.98 \, \text{ ft}$ answer

Problem 21
Solve Problem 20 if the pole is of length $L$ and the first corridor is of width $C$ .

Solution 21

Click here to show or hide the solution

$a = C \sec \theta$

$b = W \csc \theta$

$a + b = L$

$C \sec \theta + W \csc \theta = L$

$W = \dfrac{L - C \sec \theta}{\csc \theta}$

$W = \dfrac{L}{\csc \theta} - \dfrac{C \sec \theta}{\csc \theta}$

$W = L\sin \theta - \dfrac{C \sin \theta}{\cos \theta}$

$W = L\sin \theta - C \tan \theta$

$\dfrac{dW}{d\theta} = L\cos \theta - C \sec^2 \theta$

$L \cos \theta = C \sec^2 \theta$

$L \cos \theta = \dfrac{C}{\cos^2 \theta}$

$\cos^3 \theta = \dfrac{C}{L}$

$\cos \theta = \dfrac{C^{1/3}}{L^{1/3}}$

$W = L \sin \theta - C \tan \theta$

$W = L \left( \dfrac{\sqrt{L^{2/3} - C^{2/3}}}{L^{1/3}} \right) - C \left( \dfrac{\sqrt{L^{2/3} - C^{2/3}}}{C^{1/3}} \right)$

$W = L^{2/3}\sqrt{L^{2/3} - C^{2/3}} - C^{2/3}\sqrt{L^{2/3} - C^{2/3}}$

$W = \left(L^{2/3} - C^{2/3}\right)\sqrt{L^{2/3} - C^{2/3}}$

$W = \left(L^{2/3} - C^{2/3}\right)\left(L^{2/3} - C^{2/3}\right)^{1/2}$

$W = \left(L^{2/3} - C^{2/3}\right)^{3/2}$ answer

(You may check the answer of Problem 20 above by using this formula)

‹ 19 Direction of the man to reach his destination as soon as possible up 22 - Smallest cone that may circumscribe a sphere ›

Submitted by Romel Verterra on August 20, 2010 - 2:41pm

9266 reads

Recent Updates

022 Review Problem - Tin required to create a funnel
021 Review Problem - Finding the volume of a stone
020 Review Problem - Volume of a factory chimney
019 Review Problem - Which buy is more economical?
018 Review Problem - Volume of Stone Bridge
017 Review Problem - Mammoth Dredges Used in Panama Canal
016 Review Problem - Mammoth Dredges Used in Panama Canal
015 Review Problem - Volume of spherical shell
014 Review Problem - Weight of concrete surge chamber when full of water
013 Review Problem - Volume of water inside the Venturi meter
012 Review Problem - Lake for household water supply
011 Review Problem - Cutting the strongest square timber
010 Review Problem - Volume of a stick
006 Review Problem - Area illuminated by light placed in front of the globe
007 Review Problem - Length of wire
008 Review Problem - Pint tin can
009 Review Problem - Volume of anchor ring
005 Review Problem - Weight of gravity dam
004 Review Problem - Size of a column in Fingal's Cave
003 Review Problem - Sphere melted into cylinder

MATHalino.com

Subjects

20-21 Width of the second corridor for a pole to pass horizontally

Recent Updates

Differential Calculus

Our Online Reviewers

MATHalino.com

Subjects

You are here

20-21 Width of the second corridor for a pole to pass horizontally

Recent Updates

Differential Calculus

Our Online Reviewers