A box with a square base and an open top is to be constructed using 7500 sq. in. of cardboard. Find the dimensions of the box that will maximize its volume. What is the maximum volume

Step-by-step explanation:

Let the length of one side of the square base be x

Let the height of the box by y

Volume of the box V = x²y

Since the box is opened at the top, the total surface area S = x² + 2xy + 2xy

S = x² + 4xy

Given

S = 7500sq in.

Substitute into the formula for calculating the total surface area

7500 = x² + 4xy

Make y the subject of the formula;

7500 - x² = 4xy

y = (7500-x²)/4x

Since V = x²y

V = x² (7500-x²)/4x

V = x(7500-x²)/4

V = 1/4(7500x-x³)

For us to maximize the volume, then dV/dx = 0

dV/dx = 1/4(7500-3x²)

1/4(7500-3x²) = 0

(7500-3x²) = 0

7500 = 3x²

x² = 7500/3

x² = 2500

x = √2500

x = 50in

Since y = (7500-x²)/4x

y = 7500-2500/4(50)

y = 5000/200

y = 25in

Hence the dimensions of the box that will maximize its volume is 50in by 50in by 25in.

The Volume of the box V = 50²*25

V = 2500*25

V= 62,500in³

Hence the maximum volume is 62,500in³

easy math just plug in a and b

step-by-step explanation:

$3720

step-by-step explanation:

taxable income = $31000

as per the 2018-2019 us tax bracket, the tax rate is 12% for an individual whose taxable income is between $9,526 to $38,700.

so, the tax to be paid by him = 12% of 31000

= 0.12 * 31000     [12% converted to decimal as 0.12]

= $3720