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 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 {xn} 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 th loaf. Write a formula for xn that is recursive. Be sure to write down what x1 is. 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 n th loaf (in g)? d)Write a formula for the inverse of the function in part (c). Explain its meaning, 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 ϵ 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)
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 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 {xn} 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 th loaf. Write a formula for xn that is recursive. Be sure to write down what x1 is.
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 n th loaf (in g)?
d)Write a formula for the inverse of the function in part (c). Explain its meaning, 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 ϵ 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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Can you please show me how you made the inverse function?
by Bartleby ExpertAre you able to show me what steps you took to make the inverse function? How did you get there? Why do we add + 1?
by Bartleby ExpertHi, thank you so much for this. Why do you +25 in question a?
The amount of glutenous flour will keep decreasing as the loaves are baked. This question asks for just before the nth loaf is baked. So even still, 25 should be the absolute maximum it can be right? I just don't understand why we are adding this :)
by Bartleby Expert