If the surface area of the rectangular box is 112 square feet, find dy/dx when x=4 feet and y=5 feet.

Question

If the surface area of the rectangular box is 112 square feet, find dy/dx when x=4 feet and y=5 feet.

Answer 1

surface area of a rectangular box = 112 square feet = 2hw +2wL + 2hL

if x= w=4 feet wide

y= L=5 feet Long

then 112 = 8h +40 + 10h

18h = 112-40 = 72

h = 72/18 =4 feet= box height

the dimensions of the box are 4 by 4 by 5 feet

find dy/dx = dL/dw

112= 2(4)x +2(xy) + 2(4)y

112 = 8x+2xy+8y

(2x+8)y=112-8x

y=(112-8x)/(2x+8)

= (56-4x)/(x+4)

dy/dx= y'=(56-4x)(-1)/(x+4)^2 -4/(x+4)

y'(4)=-40/64 -4/8

= -5/8-4/8= (-5-4)/8

= -9/8

with fixed surface area=112 and fixed height=4, when x=4 and y=5

Answer 2

So here we're thinking of a box with a fixed surface area of 112 sq ft. Meanwhile, we're wondering: if I make the x side length a little bit longer but keep the surface area at 112 sq ft., how much smaller does that let y be?

First, let the height of our box be called "z". We're not really aiming to change that height, because the question asks just about dy/dx (nothing about z). Recall that the surface of the box is made up of 6 sides. Two of the sides have area xy, and two of them have area yz, and two of them have area xz. So in other words,

2xy+2yz+2xz = 112.

We can plug in our known values of x and y here to find z:

2*4*5 + 2*5*z+2*4*z=112 yields z=4.

Plugging in z=4, this makes our equation with just x and y as variables this new thing here:

2xy+8y+8x = 112, or, dividing all by 2:

xy+4y+4x=56

Our next step is implicit differentiation, that is, we differentiate everything with respect to x while thinking of z as a constant but considering y a function of x. This will involve the [product rule], so let me know if you have questions on how that works. Since the dx/dx = 1, we get:

[x (dy/dx) + y (dx/dx)] + 4 dy/dx + 4 (dx/dx) = 0

x (dy/dx) + y + 4 (dy/dx) + 4 = 0

Now that we're done with the differentiation step, we can plug in our known values of x and y:

4 (dy/dx) + 5 + 4 (dy/dx) + 4 = 0, so

8 (dy/dx) + 9 = 0.

Solving for (dy/dx) as if it's just a regular variable, we get (dy/dx) = -9/8.

Quick sanity check: why is it negative? Well, it's negative because when you make one dimension (x) bigger but you're trying to keep the same surface area, the other dimension (y) can get smaller. (The specific 9/8 amount makes sense if you think about it getting closer to a perfect cube, so it's more "efficient", but that's a bit subtle.)