D3 – The Tangent to a Line at a Point Activity

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					F2 – Optimization Problems
The great boat race.
Have the calculus class challenge another math class in a boat competition. The class is to
build a boat made of cardboard that will support one class member. Each group will need
a 2 m by 2 m sheet of cardboard, a roll of duct tape. (different dimensions will work but
the bigger the better.) The students must build a boat that will carry the student across the
width of the pool in the fastest time. The oars must be made from the cardboard given.
Encourage the calculus students to construct their boat using their Optimization problem
solving skills.

1.) Use

a) Find the price that will result in the maximum revenue.
b) Find the price that will result in the maximum profit.
c) Why are these prices not the same?

2.) Find the rectangle with the maximum area inscribed inside a semicircle with a radius of 6cm.
Go to the following site for a visual and support calculations of this problem.
Area of a rectangle inscribed in a semicircle

Try these optimization problems:

Problem 1. Find the dimensions of the largest isosceles triangle that can be inscribed in a circle of
radius 4 centimeters.

Problem 2. Find the length of the shortest ladder that will reach over an 8-foot high fence to a
large wall that is 3 feet behind the fence.

Problem 3. A rectangular box with a square base and an open top is to be constructed. Find the
dimensions of the largest box that can be made from 675 cm of material.

Problem 4. Two posts, one 12m high and the other 28m high stand 30m apart. They are to be
held by two wires, attached to a single stake, running from ground level to the top of each post.
Where should the stake be placed to use the least wire?

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