If 1,200 cm2 of material is available to make a box with a square base and an open top, find the maximum volume of the box in cubic cm. For example, if the answer is 2,000 cm3 write 2000.

Step-by-step explanation:

Volume of the rectangular box = x²h

Surface area  S = x²+4xh

x is the side length of the square base

h is the height

Given

Surface area = 1200cm²

Substitute into the formula S = x²+4xh

1200 = x²+4xh

From V = x²h

h= V/x²

1200 = x² + 4x (V/x² )

1200 = x² + 4V/x

1200x = x³ + 4V

1200x - x³ = 4V

V = 1200x/4- x³/4

V = 300x - x³/4

To maximize the volume, dV/dx = 0

dV/dx = 300 -3x²/4

0 =  300 -3x²/4

0 = 1200 - 3x²

3x² = 1200

x² = 400

x = 20cm

Since 1200 = x²+4xh

1200 = 400+80h

80h = 1200-400

80h = 800

h = 10cm

Find the volume;

Volume = 400 * 10 Volume -= 4000

Hence the maximum volume of the box in cubic cm is 4000 cm³

=====================================

explanation

the graph has a slope of 1/3 because each time you move up 1, you go to the right 3. so, slope = rise/run = 1/3. we can cross choice a off the list since we want a slope of 3, rather than 1/3.

the first table shown has a slope of -3. this is because the rise is -6 and the run is 2, so slope = rise/run = -6/2 = -3. note how x increases by 2 each time (run) and y drops by 6 each time (rise). choice b can be eliminated.

the second table has a slope of 3. each time x goes up by 1, the y values increase by 3. slope = rise/run = 3/1 = 3. alternatively you can use the slope formula to prove that the slope is 3 here. use any two rows of values to generate the two points. choice c is one of the answers

the third table also has a slope of 3. let's use the slope formula to prove as such. i'm going to use the first two rows of the table. so i'll use the two points (2,1) and (5,10)

m = slope

m = (y2-y1)/(x2-x1)

m = (10-1)/(5-2)

m = 9/3

m = 3 so we get a slope of 3

choice d is the other answer

they are simplified

step-by-step explanation: