lifeoflashay1659
14.04.2021 •
Mathematics
What is the surface area?
1 mm
7 mm
3 mm
square millimeters
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Ответ:
0.62 cm
Step-by-step explanation:
Ответ:
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