Plz help me with this question
1 answer:
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Answer:
x=46 gives us the minimum value
Step-by-step explanation:
Let us have the height of the box to be y. Since the box has a square base, let us call the side of the base as x. So, the volume of the box is given by
. From here, we know that
Now, we will find the area of the box. Since the box has an open top, then we only have the 4 sides of the box and the base. The base is a square of side x, so its area is . Recall that each side is a rectangle of base x and height y. So, the are of one side is
. Then, the area of the box is given by the function
. Since we can relate y to x through the volume, this can be a function of one variable. Namely
If we want to find the mininum value, we should derive the function A and find the value of x for which A'(x) is zero. REcall that the derivative of a function of the form is
. Then, applying the properties of derivatives, we have
So, we want to solve the following equation
which implies . Using a calculator we get the value
REcall that to check if the point is a minimum, we use the second derivative criteri. It states that the point x is a mininimum if f'(x) =0 and f''(x)>0
Note that , note that A''(46) = 6>0, so x=46 is the minimum value of the area.
