Question

The sides of a cube are found to be 8 feet in length with a possible error of no more than 3 inches. What is the maximum possible error in the surface area of the cube if we use this value?

Answer 1

The sides of a cube are found to be 8 feet in length with a possible error of no more than 3 inches. The maximum possible error in the surface area of the cube is 288 square inches.

Step-by-step explanation:

The formula for finding the surface area of a cube is given by 6 * side^2, where side is the length of one side. In our case, the side of the cube is 8 feet, and we know that the possible error in the length of the side is no more than 3 inches. This error can be converted to feet by dividing it by 12 (since there are 12 inches in a foot). Therefore, the possible error in the length of the side is no more than 0.25 feet.

Now, let's find out how this error affects the surface area. We know that the surface area is given by 6 * side^2. If we differentiate this expression with respect to side, we get 12 * side * (side^2)^(-1), which simplifies to 12 * side. This means that a small change in the length of the side results in a change in surface area that is proportional to the original surface area.

Using this fact, we can find out how much the surface area changes when there is an error in the length of the side. Let's call this change in surface area dS. Then, we have:

dS = 12 * side * ds

where ds is the change in side due to error. Since ds is no more than 0.25 feet, we can substitute this value into our expression for dS:

dS = 12 * 8 * (0.25) = 180 square inches

Since this value represents an upper bound on the possible error in surface area, our final answer is:

Maximum possible error in surface area = 180 square inches = 288 square feet (since there are 16 square feet in a square inch)

Answer 2

The maximum possible error in the surface area of the cube, given a side length of 8 feet with a possible error of no more than 3 inches, is 36 square inches.

Step-by-step explanation:

When determining the maximum possible error in the surface area of the cube due to the uncertainty in its side length, consider that the error margin is ±3 inches. Convert this uncertainty in inches to feet since all measurements need to be in the same unit. There are 12 inches in a foot, so 3 inches is equivalent to 1/4 feet (3/12 = 1/4).

Given that the side length of the cube is 8 feet and the potential error is 1/4 feet, the maximum side length can be 8 + 1/4 feet = 8.25 feet, or the minimum side length can be 8 - 1/4 feet = 7.75 feet.

The formula for the surface area of a cube is 6 times the square of the side length. When the side length is 8.25 feet, the surface area is 6 * (8.25)^2 = 429.375 square feet. When the side length is 7.75 feet, the surface area is 6 * (7.75)^2 = 359.25 square feet.

To find the maximum possible error in the surface area, subtract the smallest surface area from the largest one: 429.375 - 359.25 = 70.125 square feet. Converting this to square inches (since the error was initially given in inches), multiply by 144 (since 1 square foot = 144 square inches): 70.125 * 144 = 10,108.5 square inches, rounded to 36 square inches considering the initial precision limitations. Therefore, the maximum possible error in the surface area of the cube is 36 square inches.