Differential Approximation: A cubic box is to be constructed with a volume of 100cm^3

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I finished a differential approximation problem but I'm not sure I plugged in the right values.

"A cubic box is to be constructed with a volume of 100cm^3. Find the differential approximation to the maximum percentage error in the length of an edge of the cube if the maximum allowable percentage error in the volume of the cube is 0.75%."

So I take this to mean that the change in volume would be 0.75, (100 * 0.0075), which means I can approximate the change in a side-length. I'll display my work below and the answer, please tell me if this is correct.

Let S be a function of Volume representing a side-length, then S(V) = V^(1/3) since volume of a cube is the side-length cubed. Then:

dS/dV = 1/3V^(-2/3)

dS = 1/3V^(-2/3) * dV

dS = 1/3(100)^(-2/3) * 0.75 approximately equals 0.0116039721

Should the actual change, dS, be 100^(1/3) - 99.25^(1/3) which equals, rounded to 10 decimals, 0.0116331035 or 100.75^(1/3) - 100^(1/3) which equals, rounded to 10 decimals, 0.0115750824?
 
I finished a differential approximation problem but I'm not sure I plugged in the right values.

"A cubic box is to be constructed with a volume of 100cm^3. Find the differential approximation to the maximum percentage error in the length of an edge of the cube if the maximum allowable percentage error in the volume of the cube is 0.75%."

So I take this to mean that the change in volume would be 0.75, (100 * 0.0075), which means I can approximate the change in a side-length. I'll display my work below and the answer, please tell me if this is correct.

Let S be a function of Volume representing a side-length, then S(V) = V^(1/3) since volume of a cube is the side-length cubed. Then:

dS/dV = 1/3V^(-2/3)

dS = 1/3V^(-2/3) * dV

dS = 1/3(100)^(-2/3) * 0.75 approximately equals 0.0116039721

Should the actual change, dS, be 100^(1/3) - 99.25^(1/3) which equals, rounded to 10 decimals, 0.0116331035 or 100.75^(1/3) - 100^(1/3) which equals, rounded to 10 decimals, 0.0115750824?

You weren't asked for the actual change (error), but for the differential approximation to it (unless you left out part of the problem). And it is in fact close to the actual changes in both directions, as you have found (in fact, between them, which is what I would expect).

But you haven't answered the question: What is the approximate percentage error?
 
You weren't asked for the actual change (error), but for the differential approximation to it (unless you left out part of the problem). And it is in fact close to the actual changes in both directions, as you have found (in fact, between them, which is what I would expect).

But you haven't answered the question: What is the approximate percentage error?

But the question is asking for the differential approximation TO the percentage error, of which I calculated about 0.25% error between my value and the 100^(1/3) - 99.25(1/3) values. Doesn't that mean the question just wants the approximate no bigger in error than that of the maximum allowed for the volume? My main question is, are my steps and subsequent approximation correct? I just listed the actual change because I'm curious about how this topic works.
 
But the question is asking for the differential approximation TO the percentage error, of which I calculated about 0.25% error between my value and the 100^(1/3) - 99.25(1/3) values. Doesn't that mean the question just wants the approximate no bigger in error than that of the maximum allowed for the volume? My main question is, are my steps and subsequent approximation correct? I just listed the actual change because I'm curious about how this topic works.

The differential approximation to the percentage error in S is just dS/S*100%.

Your calculation of dS is correct. It is not necessary to calculate anything exactly.
 
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