a box has a square base of side x and height h find the dimensions x, h for which the volume is 14 and the surface area is as small as possible

The dimension for which the volume is 14 and the surface area is as small as possible are 2.41 by 2.41 by 2.41

The formula for calculating the volume of the square box is expressed as:

Given that the volume of the box is 14, on substituting the value we will have:

The surface area of the box is expressed as:

To get the surface area is as small as possible, then dS/dx = 0

Get the height of the box. Recall that;

Hence the dimension for which the volume is 14 and the surface area is as small as possible are 2.41 by 2.41 by 2.41

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Volume of cuboid :

Surface area of this cuboid :

Putting in above equation, we get :

For minimum area, :

Therefore, value of x is 2.14 .

Hence, this is the required solution.

the largest number of the four options is 1.8 x 10-2

you have it in scientific notation.

1.80 x 10-6   = 0.00000180

1.80 x 10-4   = 0.000180 (bigger, or "less small", than the former one)

so

1.8 x 10-2       = 0.0180. as you see. the "less small", or the biggest one of them all.

as you have negative exponent, it means you are dividing 1.8 by 100 (i.e. moving the dot towards left, two times).

if you had positive exponent for the scientific notation, power of tenths, you would be multiplying times 100 (moving the dot towards right) as below:

1.8 x (10)2 = 180

so. if you have negative exponent in scientific notation, move the dot toward left. if it is positive, toward right as many times the exponent tells you.