AP Calculus AB @ ATDP: maximum and minimum problems-Optimization problems

maximum and minimum problems-Optimization problems

2005/11/06 08:50:51 PST by Baps [0/20]
Awards: 10 from gandalf

Question #1
A printed page will have margins of 2 cm at the top and sides and 4 cm at the bottom. If the printed area is 150cm squared, find the dimentions of the whole page so that its area will be a minimum.

Question #2
Painters are painting the second floor exterior wall of the building that adjoins a busy sidewalk. A corridor 2m wide and 3 m high is built to protect pedestrians. What is the length of the shortest ladder that will reach from the ground over the corridor of the wall of the building?

2005/12/12 12:08:41 PST by showbunny [0/20]
Awards: 20 from gandalf

An apple orchard has an average yield of 36 bushels/tree when there are 22 trees per acre. For each additional x trees per acre, yield decreases by 2x bushels per tree.

a) If you plant 30 trees per acre, how many bushels of apples would you harvest from an acre of trees?

b) Write a formula for total apples harvested from an acre, and use it to find the maximum possible harvest.

2005/12/21 14:51:14 PST by gandalf [manager]

Hey, it's great that you guys are still using this site. I hope the resources on this site have been a help to you in your study of calculus.

Quote from showbunny:

An apple orchard has an average yield of 36 bushels/tree when there are 22 trees per acre. For each additional x trees per acre, yield decreases by 2x bushels per tree.

a) If you plant 30 trees per acre, how many bushels of apples would you harvest from an acre of trees?

b) Write a formula for total apples harvested from an acre, and use it to find the maximum possible harvest.

In response to your question,

(a) x = 8; avg yield = 36 - 2x = 20; total trees = 30; bushels harvested/acre = avg yield * total trees = (20)(30) = 600

(b) Let A(x) be the apples harvested when there are 22 + x trees per acre. A(x) = (22+x)(36 - 2x) = 22(36) -8x - 2x^2.
Calculate the derivative to get dA/dx = -8 - 4x = -4(x + 2) = 0 when x = -2. We know then that the max occurs at x = -2 (though to be rigorous you might want to apply the first or second derivative test here), max harvest = A(-2) = (20)(40) = 800 apples/acre.

I'll try to check up on the site a bit more frequently, so feel free to keep posting your questions. Also, you can look up some of your old classmates e-mails on the roster and ask them to use the site. That way, you guys could also help each other with your questions.

- Arkajit
Site Manager

2007/07/25 07:08:03 PDT by dks2114 [0/0]

For a fish swimming at a speed v relative to the water, the energy expenditure per until time is proportional to v^3. It is believed that migrating fish try to minimize the total energy required a fixed distance. If the fish are swimming against a current u (u<v) then the time required to swim a distance L is L / (u-v) and the total energy required to swim the distance is given by:

E(v)= (av^3) ( L/ v-u)

where a is a proportionality constant.

a) Determine the value of v that minimizes E.

"What do I do and why to solve this problem?"