A box with a square base and open top must have a volume of 97556 cm^3. We wish

Question

A box with a square base and open top must have a volume of 97556 cm^3. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base. Simplify your formula as much as possible. A(x)= Next, find the derivative, A' (x). A'(x)= Now calculate when the derivative equals zero, that is when A'(x)=0. A'(x)=0 when x= We next have to make sure that this value of x gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x). A"(x) = Evaluate A"(x)at the x-value you gave above.

Explanation / Answer

Let the height of the box = h.
Then the volume of the box, V = x²h.

The are of the base = x².
There are 4 sides, each side has an area = x.h.
So total area of 4 sides = 4xh.
So, total area, 4 sides + base = x² + 4xh.
Let A = x² + 4xh.

So, we have: V = 4000 = x²h, and A = x² + 4xh.
Therefore, h = 4000/x²; so A = x² + 4x.(4000/s²) = x² + 16,000/x,
i.e.: A = x² + 97556/x.

We want to find x for A = a minimum.
That will be when dA/dx = 0.
Since: A = x² + 97556/x
dA/dx = 2x - 97556/x².
For dA/dx = 0, 2x = 97556/x²,
so: 2x^3 = 97556;

=> x = 36.5cm.

Now, A"(x);

A"(x) = 2+195112/x3 ;

A”(36.5) = 2 + 195112/(36.5)3

A”(36.5) = 6.0

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.