An airline requires that the total outside dimensions (length + width + height)

Question

An airline requires that the total outside dimensions (length + width + height) of a checked bag not exceed 62 inches. Suppose you want to check a bag whose height equals its width. What is the largest volume bag of this shape that you can check on a flight? (Round your answers to two decimal places.) length width height Need Help? Fly-by-Night Airlines has a peculiar rule about luggage: The length I and width w of a bag must add up to at most 60 inches, and the width w and height h must also add up to at most 60 inches. What are the dimensions of the bag with the largest volume that Fly-by-Night will accept? w=160/3 in Need Help? ReadWaich 14. 0/1 points ! ng that the

Explanation / Answer

Given dimention length, width and height should not exceed 62 inches. Also, you want to check a baggage whose height=width

Let height=width=x and length=y -----------------------------------------(1)

so

y + x + x <=62

2x+y <=62

y <= 62 - 2x -------------------------------------------------------------(2)

We know that the volume of cuboid:

Volume = Length * Height * Width

V = y * x * x from (1)

V=x2y = x2 (62-2x) from (2)

V = 62x2 - 2x3

Let us now find the first derivative of V(x)

dV/dx = 124 x - 6x2

Let us now find all values of x that makes dV / dx = 0 by solving the quadratic equation

124 x - 6x2 =0

x(124-6x) =

x=20.67

then y = 62 - 2x = 62 - 2 (20.67)= 62 - 41.34=20.66

Length = 20.66 in

width=height = 20.67 in

2)

length l and width w must add up to at most 60

l+w <=60 or l <= 60 -w -----------------------------(3)

height h and width w must add up to at most 60

h+w <=60 or h <=60-l------------------------ (4)

We know that the volume of cuboid:

Volume = Length * Height * Width = l * w * h

V = (60-w) * w * (60 - w)

V = 3600w -120w2 +w3

Let us now find the first derivative of V(w)

dV/dw = 3600-240w+3w2

Let us now find all values of w that makes dV / dw = 0 by solving the quadratic equation

3600-240w+3w2 =0

1200 - 80w+w2=0

w2 - 60w-20w+1200 = 0

w(w-60)-20(w-60) = 0

w=20 and 60

w=60 not possible as length and height will be 0 in this case

so lets take w=20 and put the value in (3) and (4)

l = 60 - w = 60 - 20 = 40

h = 60-w = 60-20 = 40

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