A storage shed is to be built in the shape of a box with a square base. It is to have a volume of 343 cubic feet. The concrete for the base costs $2 per square foot, the material for the roof costs $3 per square foot, and the material for the sides costs $2.50 per square foot. Find the dimensions of the most economical shed.

x = 7 ft

h = 7 ft

Step-by-step explanation:

Let call  "x " side of the square base and " h " the height of the shed, then

V(s) =  x² * h     ⇒   343  =  x² * h    ⇒  h  =  343 / x²

Total cost of the shed is

C(t)  =  Cost of the base (C₁)  +  Cost of the roof  (C₂ ) + Cost of 4 sides (C₃)

Then

C₁  = 2*x²             C₂  =  3*x²          and     C₃  =  4*2,50*x*h

In C₃  as  h  =  343 / x²       Then    C₃  =  10 *343/x     C₃ = 3430 /x

Total cost of the shed  as a function of x is:

C(x) = 2*x²   +  3*x²   +  3430 / x      (1)

Taking derivatives on both sides of the equation we get:

C´(x)  =  10*x  -  3430 / x²

C´(x)  =  0   ⇒   10*x -  3430 / x²  = 0

x  - 343 /x²   =  0     ⇒  x³  = 343

x  =∛ 343

x = 7 ft

If we plug this value in equation 1 we see C(x) > 0 then for x = 7 , we get a minimun in the function

h = 343 /x²

h = 7 ft

pentagon

step-by-step explanation:

the coordinates   a(5, -2), b(3,-1),c(-4,-4) d(-3,8) and e(-1,4) forms the vertices of the polygon when they are connected in order.

in order to classify that which polygon is it with the given coordinates, first count the number of vertices and then count the number of sides it can make with these vertices.

since, the given coordinates have five vertices, therefore, it will make five sides. moreover, pentagon is regular in nature therefore it will be convex.

now, a polygon with five sides is known as pentagon, therefore, the given polygon is the pentagon.

2,160 π

step-by-step explanation: