The dimensions of a closed rectangular box are measured as 60 centimeters, 50 centimeters, and 70 centimeters, with an error in each measurement of at most 0.2 centimeters. Use differentials to estimate the maximum error in calculating the surface area of the box.

The maximum error in calculating the surface area of the box is 72 square centimeters.

Step-by-step explanation:

From Geometry, the surface area of the closed rectangular box (), in square centimeters, is represented by the following formula:

(1)

Where:

- Width, in centimeters.

- Length, in centimeters.

- Height, in centimeters.

And the maximum error in calculating the surface area (), in square centimeters, is determined by the concept of total differentials, used in Multivariate Calculus:

(2)

Where:

- Measurement error in width, in centimeters.

- Measurement error in length, in centimeters.

- Measurement error in height, in centimeters.

If we know that , , and , then the maximum error in calculating the surface area is:

The maximum error in calculating the surface area of the box is 72 square centimeters.

alright so i think for this one it would to draw it out. this may be hard for me to try and draw, but i will do my best on here:

        r

  /       |     \

notice how it forms 2 triangles inside one bigger triangle.

so if we know the angle spr is the same as psr that is saying the measure of angle p is the same as angle s. this means the big triangle psr is either isosceles or equilateral. in both cases, the triangle is perfectly symmetrical when split down the middle, which is in this case rq. because we know it is perfectly symmetrical on both sides, we know that the two triangles inside the bigger one are equal.