Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Transcribed Image Text:_) Jane bakes sourdough loaves, which are full of gluten. Unfortunately, John is
allergic to gluten and wants to bake gluten-free loaves. John does not have their own “starter”
(a mix of yeast, flour, and water that makes a loaf rise in the oven). Jane's starter is 50g of
water and 50g of flour with gluten, and a microscopic amount of yeast that we will ignore.
Jane gives John 50g of her starter. For John's first loaf, they replace the missing half of their
starter with 25 g of gluten-free flour and 25g of water. This means their starter is now 25g of
gluten-free flour, 25g of flour with gluten, and 50g of water. Once the yeast in this starter is
fully activated, John is ready to make their first loaf. To bake the loaf they mix 50
g (half)
of the starter with 225g of gluten-free flour and 225g of water, meaning the loaf is always
Transcribed Image Text:10 percent starter. Because John removed half of the starter from their container to bake
the loaf, they have to "feed" their starter, to get it back to 100g. John does this by mixing
in 25g of gluten-free flour and 25g of water. Now John's starter is back to normal size, and
once the yeast is activated, it is ready to use in the next loaf. Notice that every time John
bakes a new loaf the starter has less and less gluten. Therefore, each sequential loaf has less
gluten than the previous loaf.
(a)
Let {n} be a sequence where xn is defined as the amount (in g) of glutenous
flour in John's starter, just before it is used in the n loaf. Write a formula for
is recursive. Be sure to write down what x₁ is.
Xn
that
(b)
Solve the recursive equation in part (a), so that your answer is a function of
n. (Hint: It may be useful to go back to the example on half-life for inspiration)
(c)
Note that questions (a) and (b) were about the starter. How much glutenous
flour is in the nth loaf (in g)?
(d)
Write a formula for the inverse of the function in part (c). Explain its mean-
ing, and use it to find the number of loaves your friend must bake so that there is less
than 1 g of glutenous flour. It is acceptable to use a calculator for this last part (note
on the exam we would not give you a problem that would need a calculator).
(e)
If it were possible for John to repeat this process infinitely many times,
would John's loaves, in the limit, approach "gluten-free?" In other words, for any small
tolerable threshold amount of gluten e is there a number of loaves John can bake such
that they can be confident their next loaf has less than epsilon amount of gluten? (no
need to give a formal proof. An explanation or calculation suffices)
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Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question
With repspect to the equation for c, the question mentions 10% of the loaf will always be starter. Should this be encorporated into the equation as it doesnt not exclusively mention that the loaves will always be the same weight?
Solution
by Bartleby Expert
Follow-up Question
In your explanation of the other students follow up question regarding where (1/2) came from in part b (attached) I do not understand why the subscript on the RHS and LHS are different. should it not be n-1 on LHS and RHS and same for n-2 and so on?
Thank you!
![Step 2
Similarly, we will proceed for xn-1 , Xn-2, Xn-3 and so on.
Now, using equation (2), we will get x₁₁-1 as:
Xn-1 = Xn₂-2
.3
[Here, the subscript of x in R.H.S. i.e., n - 2 is 1 less than the subscript of x in L.H.S. i.e.,
n - 1.]
Using equation (3), we will get x₁-2 as:
Xn-2 = Xn-3
.4
[Here, the subscript of x in R.H.S. i.e., n - 3 is 1 less than the subscript of x in L.H.S. i.e.,
n-2.]
Using equation (4), we will get x-3 as:
Xn-3 = Xn-4
.5
[Here, the subscript of x in R.H.S. i.e., n - 4 is 1 less than the subscript of x in L.H.S. i.e.,
n - 3.]](https://content.bartleby.com/qna-images/question/f838eeb0-09f3-4680-97d9-c9dc2f60da27/2a6b0684-5ce1-4d12-931b-fd5dc0c70d9c/1ob7j8f_processed.png)
Transcribed Image Text:Step 2
Similarly, we will proceed for xn-1 , Xn-2, Xn-3 and so on.
Now, using equation (2), we will get x₁₁-1 as:
Xn-1 = Xn₂-2
.3
[Here, the subscript of x in R.H.S. i.e., n - 2 is 1 less than the subscript of x in L.H.S. i.e.,
n - 1.]
Using equation (3), we will get x₁-2 as:
Xn-2 = Xn-3
.4
[Here, the subscript of x in R.H.S. i.e., n - 3 is 1 less than the subscript of x in L.H.S. i.e.,
n-2.]
Using equation (4), we will get x-3 as:
Xn-3 = Xn-4
.5
[Here, the subscript of x in R.H.S. i.e., n - 4 is 1 less than the subscript of x in L.H.S. i.e.,
n - 3.]
Solution
by Bartleby Expert
Follow-up Question
Why is it X_(n+1) in question a?
Doesnt the question ask for the nth loaf? Sorry, just a little confused.
Solution
by Bartleby Expert
Follow-up Question
Could you please explain what happened in b and where the 1/2 came from thank you :)
Solution
by Bartleby Expert
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question
With repspect to the equation for c, the question mentions 10% of the loaf will always be starter. Should this be encorporated into the equation as it doesnt not exclusively mention that the loaves will always be the same weight?
Solution
by Bartleby Expert
Follow-up Question
In your explanation of the other students follow up question regarding where (1/2) came from in part b (attached) I do not understand why the subscript on the RHS and LHS are different. should it not be n-1 on LHS and RHS and same for n-2 and so on?
Thank you!
Transcribed Image Text:Step 2
Similarly, we will proceed for xn-1 , Xn-2, Xn-3 and so on.
Now, using equation (2), we will get x₁₁-1 as:
Xn-1 = Xn₂-2
.3
[Here, the subscript of x in R.H.S. i.e., n - 2 is 1 less than the subscript of x in L.H.S. i.e.,
n - 1.]
Using equation (3), we will get x₁-2 as:
Xn-2 = Xn-3
.4
[Here, the subscript of x in R.H.S. i.e., n - 3 is 1 less than the subscript of x in L.H.S. i.e.,
n-2.]
Using equation (4), we will get x-3 as:
Xn-3 = Xn-4
.5
[Here, the subscript of x in R.H.S. i.e., n - 4 is 1 less than the subscript of x in L.H.S. i.e.,
n - 3.]
Solution
by Bartleby Expert
Follow-up Question
Why is it X_(n+1) in question a?
Doesnt the question ask for the nth loaf? Sorry, just a little confused.
Solution
by Bartleby Expert
Follow-up Question
Could you please explain what happened in b and where the 1/2 came from thank you :)
Solution
by Bartleby Expert
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