To create an open-top box out of a sheet of cardboard that is 25 inches long and 13 inches wide, you make a square flap of side length x inches in each corner by
13 in.
cutting along one of the flap's sides and folding along the other. (In the diagram,
a solid line segment in the interior of the rectangle indicates a cut, while a
dashed line segment indicates a fold.) Once you fold up the four sides of the
box, you glue each flap to the side it overlaps. To the nearest tenth, find the value of x that maximizes the
volume of the box.

Step-by-step explanation:

length of the box is a = 25 -2x

width of the box is b =13 -2x

height of the box is x

and x < 6,5

so the volume of the box V=f(x) = x.(25-2x)(13-2x)= 4x^3 -76x^2 +325x

f'(x) = 12x^2 - 152x + 325

to maximize the volume of the box ( f(x) max ) then f'(x) = 0 --> x ≈ 2,72 and x ≈ 9,94

so x = 2,72 inches

length of the box is a = 25 -2x

width of the box is b =13 -2x

height of the box is x

and x < 6,5

so the volume of the box V=f(x) = x.(25-2x)(13-2x)= 4x^3 -76x^2 +325x

f'(x) = 12x^2 - 152x + 325

to maximize the volume of the box ( f(x) max ) then f'(x) = 0 --> x ≈ 2,72 and x ≈ 9,94

so x = 2,72 inches

Step-by-step explanation:

clock

step-by-step explanation:

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.