Did you make a sketch?
Then clearly you can see that
Volume = x(100-2x)^2
SA = (100 - 2x)^2 + 4(x)(100-2x)
"for each x-value", you gave none, but if you want only whole numbers
and obviously 100-2x > 0, or 0 < x < 50
here is one example:
let x = 10
volume = 10(100-20)^2 = 64,000 cm^3
SA = (100-20)^2 + 4(10)(80) = 9,600 cm^2
A square sheet of cardboard 100 cm by 100 cm is to have corners of side length x cut out. These cutouts will create flaps that can be folded up to form a box with no lid. The length and width of the box are given by the expression (100 – 2x), and the height of the box is x, where x is in centimeters.) Find the volume and surface area of the box for each x-value.
2 answers
V = L W H
The length and width are equal so:
V = L ∙ W ∙ H = ( 100 - 2 x ) ∙ ( 100 - 2 x ) ∙ x = x ( 100 - 2 x )²
Sufrace area:
SA = base + 4 sides = ( 100 - 2 x )² + 4 ∙ ( 100 - 2 x ) ∙ x =
( 100 - 2 x ) ∙ [ ( 100 - 2 x ) + 4 x ] = ( 100 - 2 x ) ∙ ( 100 + 2 x )
Since:
(a + b ) ( a - b ) = a² - b²
SA = ( 100 - 2 x ) ∙ ( 100 + 2 x ) = 100² - ( 2 x )²
The length and width are equal so:
V = L ∙ W ∙ H = ( 100 - 2 x ) ∙ ( 100 - 2 x ) ∙ x = x ( 100 - 2 x )²
Sufrace area:
SA = base + 4 sides = ( 100 - 2 x )² + 4 ∙ ( 100 - 2 x ) ∙ x =
( 100 - 2 x ) ∙ [ ( 100 - 2 x ) + 4 x ] = ( 100 - 2 x ) ∙ ( 100 + 2 x )
Since:
(a + b ) ( a - b ) = a² - b²
SA = ( 100 - 2 x ) ∙ ( 100 + 2 x ) = 100² - ( 2 x )²
2 answers