A designer is creating various open-top boxes by cutting four equally-sized squares from the corners of a standard sheet of 8.5 inch by 11 inch paper, and then folding up and securing the resulting 'flaps' to be the sides of the box. Let x represent the varying side length of the square cutouts in inches. Let l , w , and h represent the varying length, width and height of the box (in inches), respectively. Note that the width and length dimensions are such that w < l . Let V represent the varying volume of the box in cubic inches. Write formulas for length, width, and volume of the box, each in terms of x .

L(x) = 11 - 2x

W(x) = 8.5 - 2x

V(x) = 4x³ - 39x² + 93.5x

Step-by-step explanation:

If a square of x sides length is cut from each corner of the sheet, then it meas the length and width of the box a shortened by x inces from each side, giving a total subtraction of 2 times x inches.

FORMULA FOR LENGTH:

Since, Length is the bigger side of the box. Therefore, it will be taken on 11 inches side of the paper:

L(x) = 11 - 2x

FORMULA FOR WIDTH:

Since, width is the smaller side of the box. Therefore, it will be taken on 8.5 inches side of the paper:

W(x) = 8.5 - 2x

FORMULA FOR VOLUME:

Volume is the product of length, width and time:

V(x) = L(x)*W(x)*H(x)

Height must be equal to the side folds which are equal to length of the side of square = x:

H(x) = x

Therefore,

V(x) = (11 - 2x)(8.5 - 2x)(x)

V(x) = (93.5 -22x -17x + 4x²)(x)

V(x) = 4x³ - 39x² + 93.5x

step-by-step explanation:

12: 00 p.m. to 7: 00 p.m. is a total of 7 hours.

4: 00 p.m. to 7: 00 p.m. is a total of 3 hours.

this means that from 4: 00 p.m to 7: 00 p.m. this represents 3/7 of what was lost from 12: 00 p.m. to 7: 00 p.m.

after knowing this, after doing 11,600 - 6,800 = 4,800

4,800 represents 3/7 of what was lost for 7 hours.

we then divide 4,800 by 3 which gets 1,600.

because we know that 1,600 represents 1/7 of how much water was lost, we just have to multiply by 7, which gets 11,200.   after adding 11,200 to how much water was left at 7: 00 p.m. which was 6,800. 11,200 + 6,800 = 18,000

this means that the pool originally had 18,000 gallons in the pool.

you can also check your work by doing 18,000 - 4(1600), which becomes           1800 - 6400 which gets you 11,600.