What is the maximum volume in cubic inches of an open box to be made from a 12-inch by 16-inch piece of cardboard by cutting out squares of equal sides from the four corners and bending up the sides? your work must include a statement of the function and its derivative. give one decimal place in your final answer.

step-by-step explanation:

the volume of a box is: v = l * w * h. we are starting with a piece of cardboard 10 inches by 20 inches. if we let x represent the height of the box, x will also be the side of the square taken out of each corner.

the length of the box becomes 20 - 2x. the width of the box becomes 10 - 2x. the height of the box is x.

substituting in the equation for the volume gives:

v = x * (20 - 2x) * (10 - 2x)

multiply the right side to get:

v(x) = 4x3 - 60x2 + 200x

take the first derivative of v:

v'(x) = 12x2 - 120x + 200

to find x, set the first derivative equal to 0:

12x2 - 120x + 200 = 0

use the quadratic formula to solve for x:

a = 12

b = -120

c = 200

x = (120 ± √)2 - 4 * 12 * 200) / (2 * 12)

use the positive value first:

x = (120 + √(14400 - 9600) / 24

x = (120 + 69.3) / 24

x = 189.3 / 24

x = 7.9 inches

this solution can be discarded because it would make the width a negative number.

use the negative value next:

x = (120 - √(14400 - 9600) / 24

x = (120 - 69.3) / 24

x = 50.7 / 24

x = 2.1 inches

take the second derivative of v to check for a local maximum.

v''(x) = 24x - 120

substitute for x:

v''(2.1) = 24 * 2.1 - 120

the second derivative is less than zero, x is a maximum.

sides of the box:

length = 20 - 2 * 2.1 = 17.8 inches

width = 10 - 2 * 2.1 = 7.8 inches

height = 2.1 inches

let's solve your equation step-by-step.

24−3x=−27

step 1: simplify both sides of the equation.

24−3x=−27

24+−3x=−27

−3x+24=−27

step 2: subtract 24 from both sides.

−3x+24−24=−27−24

−3x=−51

step 3: divide both sides by -3.

−3x

−3

=

−51

−3

x=17

answer:

x=17

step-by-step explanation:

mark brainliest and have a great day!

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.