You are 3D printing six-sided dice for a Yahtzee tournament. After some research, you find out thatmost dice have a volume of 4 cm3each. An example of a single Yahtzee dice is illustrated below.Note that each side length, s, has the same measure.a(a) In the blank below, write the function that relates the volume of a single dice, V , and thedice’s side length, s.V (s) =(b) What is the perfect side length? i.e. What side length will result in a volume of 4 cm3? Youranswer should be written to 6 decimal places.s0=(c) Due to the nature of your 3D printer, the volume of each dice will be within 0.01 cm3 of 4cm3. Algebraically determine the interval around s0 that will ensure that correspondingoutput values are within 0.01 cm3 of 4 cm3. If you use decimals, your answer should bewritten to 6 decimal places.(d) What is the maximum amount of error that can occur on either side of s0 so that the volumevalues still lie within 0.01 of 4 cm3? (i.e. what is the δ value?) If you use decimals, youranswer should be written to 6 decimal places. (e) Below is the portion of the graph of V (s) relevant for this problem. Using the graph, label thefollowing:(i) The axes(ii) The point corresponding to s0 and V = 4 cm3(iii) The interval of ±0.01 cm3 around 4 cm3.(iv) The interval you found in part (c).   (f) In Section 2.2, we said that limx→af(x) = L if f(x) is arbitrarily close to L for all x sufficientlyclose to a. How does the Yahtzee problem relate to this definition?In your discussion, be sure to identify f(x), x, L, and a in the context of the Yahtzee problem,but don’t just restate the definition using the contextual f, x, L, and a. It may be helpful toconnect the definition to your results above.

Question

You are 3D printing six-sided dice for a Yahtzee tournament. After some research, you find out that
most dice have a volume of 4 cm3
each. An example of a single Yahtzee dice is illustrated below.
Note that each side length, s, has the same measure.
a
(a) In the blank below, write the function that relates the volume of a single dice, V , and the
dice’s side length, s.
V (s) =
(b) What is the perfect side length? i.e. What side length will result in a volume of 4 cm3
? Your
answer should be written to 6 decimal places.
s0=
(c) Due to the nature of your 3D printer, the volume of each dice will be within 0.01 cm3 of 4
cm3
. Algebraically determine the interval around s0 that will ensure that corresponding
output values are within 0.01 cm3 of 4 cm3
. If you use decimals, your answer should be
written to 6 decimal places.

(d) What is the maximum amount of error that can occur on either side of s0 so that the volume
values still lie within 0.01 of 4 cm3
? (i.e. what is the δ value?) If you use decimals, your
answer should be written to 6 decimal places.

(e) Below is the portion of the graph of V (s) relevant for this problem. Using the graph, label the
following:
(i) The axes
(ii) The point corresponding to s0 and V = 4 cm3
(iii) The interval of ±0.01 cm3 around 4 cm3
.
(iv) The interval you found in part (c).

 

(f) In Section 2.2, we said that limx→a
f(x) = L if f(x) is arbitrarily close to L for all x sufficiently
close to a. How does the Yahtzee problem relate to this definition?
In your discussion, be sure to identify f(x), x, L, and a in the context of the Yahtzee problem,
but don’t just restate the definition using the contextual f, x, L, and a. It may be helpful to
connect the definition to your results above.

 

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