A box with a square base and open top must have a volume of
y. Since the volume of a box is equal to the length times the width times the height,
the volume of this box would be x times x times y, or x squared y. So we know that V
must equal x squared y, which in this case must be forty-two thousand five hundred ninety-two cubic
centimeters. Because this equation must be true, we call this the constraint.
Now we also need an equation for the surface area so that we can minimize that equation. The
surface area would be the sum of the area of each face, and since our box has an open top,
our box has five faces: the bottom and the four sides. The area of the bottom would be x times x,
or x squared. Then notice that the area of each face would be x times y, and since
there are four of them, we have plus four xy. This is the equation that we want to minimize.
But to do this, we need to write S in terms of one variable. Right now, it's written in terms
of x and y, so if we solve the constraint for y, we can perform a substitution for y in the surface
area formula, and then we'll have S in terms of x. Looking at our volume formula or our constraint,
if we solve this equation for y, we would divide both sides by x squared. Therefore,
y is equal to forty-two thousand five hundred ninety-two divided by x squared.
Now, if we take a look at the x's, one factor of x will simplify out.
This would simplify to one. This would simplify to x to the first power, and four
times forty-two thousand five hundred ninety-two would give us
one hundred seventy thousand three hundred sixty-eight divided by x.
We'll use calculus techniques to find the first derivative and determine where it's
undefined or equal to zero to find our critical numbers. So S prime is going to be equal to two
x minus one hundred seventy thousand three hundred sixty-eight divided by x squared. We want to know
when this is equal to zero or undefined. Let's go ahead and write this as two x minus one hundred
seventy thousand three hundred sixty-eight divided by x squared equals zero. Notice how this would be
undefined when x equals zero, but x can't equal zero; otherwise, this length here would be zero.
So let's go ahead and solve this equation for x. On the next slide, let's go ahead and clear
the fraction by multiplying both sides of the equation by x squared. So we'd have two x cubed
equals one hundred seventy thousand three hundred sixty-eight. Notice here the x's would simplify
out, so we have two x cubed equals one hundred seventy thousand three hundred sixty-eight.
Let's go ahead and isolate x cubed. We'll add this constant to both sides and divide both sides
by two. So we have x cubed equals eighty-five thousand six hundred eighty-four, and now we'll
cube root both sides in order to solve for x. This comes out nicely to x equals forty-four.
So we only have one critical number, so we can probably assume this x value will
minimize the surface area formula. To be completely sure, let's go ahead and use
the second derivative test. Remember, the first derivative was equal to two
x minus one hundred seventy thousand three hundred sixty-eight divided by x squared.
So to find the second derivative, we'll find the derivative of the first derivative. The derivative
of two x would be two. Now for the next term, we're going to multiply by negative two. So
that'll be plus three hundred forty thousand seven hundred thirty-six times x to the negative three.
Let's go ahead and write this using a positive exponent. So S double prime is equal to two
plus three hundred forty thousand seven hundred thirty-six divided by x cubed.
Notice how this sum is going to be positive or greater than zero,
which means our surface area function S of x is concave up at x equals forty-four.
So if the function is concave up at x equals forty-four, it would look something like this,
and therefore at x equals forty-four, we know we have a minimum function value.
Now, if we go back and take a look at our figure for a moment,
our goal is to find the dimensions of the box that will minimize the surface area. So
we know our box must be forty-four centimeters by forty-four centimeters by y centimeters. So
we still have to find the value of y, and we can find y using this equation here.
Y is equal to forty-two thousand five hundred ninety-two divided by forty-four squared, which we
already found this value. Y is equal to twenty-two. So to answer
the question, the dimensions of the box that would minimize the surface
area with a volume of forty-two thousand five hundred ninety-two cubic centimeters would be
forty-four centimeters by forty-four centimeters by twenty-two centimeters.
To verify this, let's take a look at the graph of the surface area formula once we
had it in terms of x. Here's the surface area formula that we used. Notice if we graph this,
here's the location of a minimum value for the surface area. Notice the x value does appear
to be forty-four. The y value would actually be what that minimum surface area would be.
So let's go ahead and substitute forty-four for x to verify that this surface area is less than
six hundred thousand cubic centimeters. We weren't actually asked to find this,
but the surface area of that box would be equal to forty-four squared plus one hundred seventy
thousand three hundred sixty-eight divided by forty-four. So if x is equal to forty-four, this
value comes out to five hundred eighty-eight cubic centimeters, which again is verified by our graph.
I hope you found this explanation helpful.