Dimensions of a Box A box is made from a rectangular piece of metal with length 50 inches and width 40 inches by cutting out square corners of length x and folding up the sides.

(a) Write an expression that gives the surface area of the inside of the box.
(b) If the surface area of the box is 1900 square inches, find x.

No top ?

So I see a base which is (50-2x) by (40-2x)
and I see a height of x

total surface area
= (50-2x)(40-2x) + 2x(50-2x) + 2x(40-2x)
=2000 - 180x + 4x^2 + 100x - 4x^2 + 80x - 4x^2
= -4x^2 + 2000

-4x^2 + 2000 = 1900
-4x^2 = -100
x^2 = 25
x = √25 = 5

check my arithmetic

(a) The box is made by cutting out square corners of length x from a rectangular piece of metal with dimensions 50 inches and 40 inches. After folding up the sides, the dimensions of the box will be (50 - 2x) inches for the length, (40 - 2x) inches for the width, and x inches for the height.

The surface area of the inside of the box can be calculated by finding the sum of the areas of each face. The box has 6 faces: 2 rectangular faces of dimensions (50 - 2x) inches by (40 - 2x) inches, 2 rectangular faces of dimensions (50 - 2x) inches by x inches, and 2 rectangular faces of dimensions (40 - 2x) inches by x inches.

Expression for the surface area of the inside of the box:
Surface Area = 2(50 - 2x)(40 - 2x) + 2(50 - 2x)x + 2(40 - 2x)x

(b) Given that the surface area of the box is 1900 square inches, we can set up the equation and solve for x.

2(50 - 2x)(40 - 2x) + 2(50 - 2x)x + 2(40 - 2x)x = 1900

Simplifying the equation:

(50 - 2x)(40 - 2x) + (50 - 2x)x + (40 - 2x)x = 950

Expanding and combining like terms:

(2000 - 180x + 4x^2) + (50x - 2x^2) + (40x - 2x^2) = 950

Combining like terms:

2000 + 50x + 40x - 180x + 4x^2 - 2x^2 - 2x^2 = 950

Simplifying:

100x - 360x + 2x^2 = -1050

Rearranging:

2x^2 - 260x + 1050 = 0

Now we can solve this quadratic equation.

To find the dimensions of the box and the value of x, we can use the given information and set up equations.

(a) The inside surface area of the box consists of the bottom, top, and four sides.

The bottom of the box will have the dimensions (50 - 2x) inches by (40 - 2x) inches.
The top of the box will have the same dimensions as the bottom.
The two sides along the length of the box will have the dimensions x inches by (40 - 2x) inches.
The two sides along the width of the box will have the dimensions x inches by (50 - 2x) inches.

So, the expression for the surface area of the inside of the box is:

Surface Area = 2[(50 - 2x)(40 - 2x)] + 2[x(40 - 2x)] + 2[x(50 - 2x)]
Surface Area = 2[(50 - 2x)(40 - 2x)] + 2x(40 - 2x) + 2x(50 - 2x)
Surface Area = 2[(50 - 2x)(40 - 2x)] + 80x - 4x^2 + 100x - 4x^2

Simplifying further:

Surface Area = 2(2000 - 180x + 4x^2) + 180x - 8x^2
Surface Area = 4000 - 360x + 8x^2 + 180x -8x^2
Surface Area = 4000 - 180x

(b) We know that the surface area of the box is 1900 square inches.

So, we can set up the equation:

1900 = 4000 - 180x

Solving for x:

1900 - 4000 = -180x
-2100 = -180x
x = (-2100)/(-180)
x = 11.67 inches (approximately)

Therefore, the value of x is approximately 11.67 inches.