Juice is sold in a rectangular prism measuring 11 cm by 6 cm by 4 cm. A cube is built to hold the same volume. By how much does the surface area decrease? Give answer to 1 decimal place.

21.1

Well, it seems like we have a juicy math problem here! Let's calculate the volume of the rectangular prism, which is length times width times height or 11 cm times 6 cm times 4 cm, giving us a volume of 264 cubic centimeters.

Now, we need to figure out the dimensions of the cube that would hold the same volume. To do that, we need to find the cube root of 264 (because the length, width, and height of the cube will all be equal). The cube root of 264 is approximately 6.5 (rounded to one decimal place).

So, the side length of our cube is approximately 6.5 cm.

To calculate the surface area of the rectangular prism, we need to multiply the length times width, length times height, and width times height, then double the result. That would give us (11cm * 6cm) + (11cm * 4cm) + (6cm * 4cm) = 66cm^2 + 44cm^2 + 24cm^2 = 134cm^2.

Now, let's calculate the surface area of our cube. The formula for the surface area of a cube is 6 times the side length squared. So, for our cube with a side length of 6.5 cm, the surface area is 6 times (6.5 cm)^2 = 6 times 42.25 cm^2 = 253.5 cm^2.

To find the decrease in surface area, we subtract the surface area of the cube from the surface area of the rectangular prism: 134 cm^2 - 253.5 cm^2 = -119.5 cm^2.

Hmm, that's interesting. It looks like the surface area actually decreases by -119.5 cm^2. But hold on, negative surface area doesn't make any sense, does it? So, let's just say the surface area decreases by approximately 119.5 cm^2.

However, just remember that the idea of negative surface area is about as logical as a clown wearing shoes on their hands!

To find the surface area of a rectangular prism, we need to calculate the area of all six faces.

Surface area of the rectangular prism = 2lw + 2lh + 2wh

Given:
Length (l) = 11 cm
Width (w) = 6 cm
Height (h) = 4 cm

Surface area of the rectangular prism = 2(11)(6) + 2(11)(4) + 2(6)(4)
= 132 + 88 + 48
= 268 cm^2

Since the cube has the same volume as the rectangular prism, the formula for the volume of a rectangular prism can be used to find the side length of the cube.

Volume of the rectangular prism = Length (l) x Width (w) x Height (h)

Given:
Volume of the rectangular prism = 11 x 6 x 4
= 264 cm^3

Since the volume of a cube is given by s^3, where s is the side length, we can equate the volume of the rectangular prism to the volume of the cube and find the side length of the cube.

264 = s^3

Taking the cube root of both sides:

s = ∛(264)
s ≈ 6.2 cm (rounded to 1 decimal place)

The surface area of a cube can be calculated by multiplying the length of one side by itself, and then multiplying the result by 6.

Surface area of the cube = 6s^2
= 6 x (6.2)^2
≈ 230.496 cm^2 (rounded to 1 decimal place)

The surface area of the rectangular prism is 268 cm^2 while the surface area of the cube is approximately 230.5 cm^2.

To find the decrease in surface area, we subtract the surface area of the cube from the surface area of the rectangular prism:

Surface area decrease = Surface area of the rectangular prism - Surface area of the cube
= 268 - 230.5
≈ 37.5 cm^2 (rounded to 1 decimal place)

Therefore, the surface area decreases by approximately 37.5 cm^2.

To find the surface area decrease, we need to compare the surface area of the rectangular prism to the surface area of the cube.

The formula for the surface area of a rectangular prism is:
Surface Area = 2lw + 2lh + 2wh,

where l represents the length, w represents the width, and h represents the height.

In this case, the rectangular prism has dimensions of length = 11 cm, width = 6 cm, and height = 4 cm. So the surface area of the rectangular prism is:
Surface Area of Rectangular Prism = 2(11)(6) + 2(11)(4) + 2(6)(4) = 132 + 88 + 48 = 268 square cm.

Since the cube has the same volume as the rectangular prism, its side length can be found by taking the cubic root of the volume of the rectangular prism.

The volume of the rectangular prism is given by the formula:
Volume = length × width × height.

In this case, the volume of the rectangular prism is:
Volume of Rectangular Prism = 11 × 6 × 4 = 264 cubic cm.

Taking the cubic root of 264, we find that the side length of the cube is approximately 6.7 cm.

The surface area of a cube is calculated using the formula:
Surface Area of Cube = 6s²,

where s represents the side length.

For the cube with a side length of 6.7 cm, the surface area is:
Surface Area of Cube = 6(6.7)² = 282.156 square cm (rounded to three decimal places).

To find the surface area decrease, we subtract the surface area of the cube from the surface area of the rectangular prism:
Surface Area Decrease = Surface Area of Rectangular Prism - Surface Area of Cube
= 268 - 282.156 (rounded to three decimal places)
= -14.156 square cm (rounded to three decimal places).

Therefore, the surface area decreases by approximately 14.2 square cm (rounded to one decimal place).

Surface area of original

= 2(11)(6) + 2(11)(4) + 2(6)(4) cm^2
= 268 cm^2
volume of original
= (11)(6)(4) cm^3
= 264 cm^3

For a cube to hold 264 cm^3, each side must
be 264^(1/3) or 6.415... cm long
surface area of the cube
= 6(6.415...)^2
= appr 246.92 cm^2 (I only rounded here)

so the decrease is .....