# A steel girder is taken to a 15ft wide corridor. At the end of the corridor there is a 90° turn, to a 9ft wide corridor. How long is the longest girder than can be turned in this corner?

Maximum girder length is

Let us set up the following variables:

# {(x, "Partial distance along bottom",x+15=OA), (y, "total length along side",y=OB), (l, "Line Segment AB",l=AB),(alpha, angle " between AB and bottom",) :} #

All of

At a max/min we have

Either

# 15+x = 0 = > x=-15 # . But#x gt 0#

Or# \ \ \ \ \ \ 2 -3000/x^3 = 0 => 1500/x^3=1 => x=1500^(1/3)#

Thus we have:

# x = 11.44714 ... #

# L = 1233.2326... #

# l \ = 35.11741 ... #

I won't show that this is minimum (as required) but a plot or the second derivative test will show this is the case.

METHOD 2

We can also use a max/min method using trigonometry, considering the angle

From the large

# cos alpha = (15+x)/l #

# :. l = (15+x)/cos alpha #

# :. l = (15+x)sec alpha #

From the small

# tan alpha = 10/x #

# :. x = 10/tan alpha #

# :. x = 10cot alpha #

Combining these results we get:

# l = (15 + 10 cot alpha)sec alpha #

# \ = 15sec alpha + 10 cot alpha sec alpha #

# \ = 15sec alpha + 10 cos alpha /sin alpha 1/cos alpha #

# \ = 15sec alpha + 10 /sin alpha #

# \ = 15sec alpha + 10 csc alpha #

Differentiating

# (dl)/(d alpha) = 15sec alpha tan alpha - 10 csc alpha cot alpha #

At a max/min we have (dl)/(d alpha) #

# :. 15sec alpha tan alpha - 10 csc alpha cot alpha = 0 #

# :. 3/(cos alpha) (sin alpha)/(cos alpha) = 2/(sin alpha) (cos alpha)/(sin alpha) #

# :. 3sin^3alpha=2cos^3alpha #

# :. tan^3alpha=2/3 #

From which we get:

# alpha = 0.71802 ... ^c#

# alpha = 41.1398 ... ^o#

# l \ = 35.1174 ... # , as with Method 1

By signing up, you agree to our Terms of Service and Privacy Policy

The longest girder that can be turned in the corner is the diagonal of the rectangle formed by the two corridors. Using the Pythagorean theorem, the length of the diagonal can be calculated as √(15^2 + 9^2) = √(225 + 81) = √306 ≈ 17.49 feet. Therefore, the longest girder that can be turned in the corner is approximately 17.49 feet long.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you minimize and maximize #f(x,y)=x^2+y^3# constrained to #0<x+3y<2#?
- How do you find the linearization at a=0 of #f(x) = e^(5 x)#?
- At what approximate rate (in cubic meters per minute) is the volume of a sphere changing at the instant when the surface area is 5 square meters and the radius is increasing at the rate of 1/3 meters per minute?
- How do you approximate #sqrt(17.02)#?
- What is the maximum possible area of the rectangle that is to be inscribed in a semicircle of radius 8?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7