Suppose you have a choice of two coffee cups of the type shown, one that bends o

Question

Suppose you have a choice of two coffee cups of the type shown, one that bends outward and one inward, and you notice that they have the same height and their shapes fit together snugly. You wonder which cup holds more coffee. Of course you could fill onecup with water and pour it into the other one but, being a calculus student, you decide on a more mathematical approach. Ignoring thehandles, you observe that both cups are surfaces of revolution, so you can think of the coffee as a volume of revolution.

a) Suppose the cups have height h, cup A is formed by rotating the curve x = f(y) about the y-axis, and cup B is formed by rotating the same curve about the line x = k.  Find the value of k such that the two cups hold the same amount of coffee.

Suppose you have a choice of two coffee cups of the type shown, one that bends outward and one inward, and you notice that they have the same height and their shapes fit together snugly. You wonder which cup holds more coffee. Of course you could fill onecup with water and pour it into the other one but, being a calculus student, you decide on a more mathematical approach. Ignoring thehandles, you observe that both cups are surfaces of revolution, so you can think of the coffee as a volume of revolution. Suppose the cups have height h, cup A is formed by rotating the curve x = f(y) about the y-axis, and cup B is formed by rotating the same curve about the line x = k. Find the value of k such that the two cups hold the same amount of coffee. What does your result from Problem I say about the areas A1 and A2 shown in the figure? Based on your own measurements and observations, suggest a value for h and an equation for x = f (y)and calculate the amount of coffee that each cup holds.

Explanation / Answer

Once you set the volumes equal to each other you have this:

? ?[f(y)]^2 dy = ? ?[k - f(y)]^2 dy
?[f(y)]^2 dy = ?[k - f(y)]^2 dy

[f(y)]^2 = [k - f(y)]^2 (differentiate both sides to get rid of the ?}

f(y)^2 = k^2 - 2kf*y) + f(y)^2
0 = k^2 - 2kf(y)
0 = k[k - 2f(y)]
either k = 0 (which means the second cup is also rotated around the y-axis, making it the first cup)
or 0 = k - 2f(y) --> k = 2f(y)

When k = 2f(y) both cups will have the same volume. I don't see any images, so I can't say anything about the surface areas of the two cups.

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.