Answer:
Dimensions: 
Perimiter: 
Minimum perimeter: [16,16]
Step-by-step explanation:
This is a problem of optimization with constraints.
We can define the rectangle with two sides of size "a" and two sides of size "b".
The area of the rectangle can be defined then as:

This is the constraint.
To simplify and as we have only one constraint and two variables, we can express a in function of b as:

The function we want to optimize is the diameter.
We can express the diameter as:

To optimize we can derive the function and equal to zero.

The minimum perimiter happens when both sides are of size 16 (a square).
Answer:
V= 42.41
Step-by-step explanation:

where r is the radius = 3 and h is the height = 4.5
×
× 
V = 42.41
Answer:
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Step-by-step explanation:
Answer:
; minimum
Step-by-step explanation:
Given:
The function is, 
The given function represent a parabola and can be expressed in vertex form as:

The vertex form of a parabola is
, where,
is the vertex.
So, the vertex is
.
In order to graph the given parabola, we find some points on it.
Let 




So, the points are
.
Mark these points on the graph and join them using a smooth curve.
The graph is shown below.
From the graph, we conclude that at the vertex
, it is minimum.
Answer:
Angle 9: 60°
Angle 10: 30°
Side = radius = 40sqrt(x/3)
Area = [800sqrt(3)]x
Step-by-step explanation:
Total angle in a hexagon:
(6 - 2) × 180
720
Each interior angle:
720/6 = 120
angle 9 = 120/2 = 60
Angle 10 = 60/2 = 30
sin(60) = 20sqrt(x)/r
r = 20sqrt(x) ÷ sqrt(3)/2
r = 40sqrt(x/3)
Side:
sin(30) = ½s/(40sqrt(x/3))
½s = 20sqrt(x/3)
s = 40sqrt(x/3)
Area = (3sqrt(3))/2 × s²
Area = 3sqrt(3)/2 × 1600x/3
Area = [800sqrt(3)]x