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anygoal [31]
2 years ago
6

What is the area of this figure? 4 cm 5 cm 5 cm​

Mathematics
1 answer:
Zarrin [17]2 years ago
5 0

Answer:

100

Step-by-step explanation:

5*5*4=100

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A box with a square base and open top must have a volume of 48668 cm3. We wish to find the dimensions of the box that minimize t
marin [14]

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

x^2\cdot y = 48668. From here, we know that y=\frac{48668}{x^2}

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 x^2. Recall that each side is a rectangle of base x and height y. So, the are of one side is x\cdot y. Then, the area of the box is given by the function

A(x,y) = x^2+4xy. Since we can relate y to x through the volume, this can be a function of one variable. Namely

A(x) = x^2+4x\cdot \frac{48668}{x^2}= x^2+4\cdot \frac{48668}{x} = x^2+48668x^{-1}

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 x^n is nx^{n-1}. Then, applying the properties of derivatives, we have

A'(x) = 2x-4\cdot \frac{48668}{x^2} = \frac{2x^3-4\cdot 48668}{x^2}

So, we want to solve the following equation

2x^3-4\cdot 48668 =0

which implies x^3 = 97336. Using a calculator we get the value x= 46

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 A''(x) = 2+8\cdot\cdot 48668}{x^3}, note that A''(46) = 6>0, so x=46 is the minimum value of the area.

3 0
3 years ago
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