Question

You have a set of calipers that can measure thicknesses of a few inches with an uncertainty of 0:005 inches. I mesure the thickness of a deck of 52 cards and get 0.590 in: (a) If you now calculate the thickness of 1 card, what is my answer, including its uncertaintyb) I can improve this result by measuring several decks together. If I want to know the thickness of 1 card with an uncertainty of only 0.00002 inch, how many decks do i need to measure?

Answer 1

To solve this problem we will apply the concepts related to the calculation of significant figures under tolerance levels. We will also take into account that the number of significant digits at the end of an answer must not be greater than the number of significant digits of a number:

PART A) The thickness of the 52 cards is

$T = 0.590 \pm 0.005 in$

The thickness, t, of 1 card can be:

$t = \frac{0.590}{52} \pm \frac{0.005}{52} in$

$t = 0.0113461\pm 0.000096 in$

The thickness of 52 cards has 3 significant figures and the uncertainty has 1 significant digit. So

the significant figures of the thickness of one card and the uncertainty should also be 3 and 1 respectively

$t = 0.01134 \pm 0.005 in$

$t = 0.0113 \pm 0.0001 in$

Therefore the thickness of one card is $\mathbf {t = 0.0113 \pm 0.0001 in }$

PART B) One card has uncertainty of 0.0001 in if measured using 1 deck.

The number of decks, n, required to create the uncertainty of 0.00002 in is

$n = \frac{0.0001}{0.00002}$

$n = 5$

Therefore, 5 decks are required to measure the thickness of one card with an uncertainty of 0.00002 in