Maxima and Minima

04 Largest Right Triangle of Given Hypotenuse

Problem
Find the area of the largest right triangle whose hypotenuse is fixed at c.
 

03-largest-right-triangle-given-hypotenuse.gif           03-largest-right-triangle-given-hypotenuse-theta.gif

 

03 Maximum Revenue for Tour Bus of 80 Seats

Problem
A tour bus has 80 seats. Experience shows that when a tour costs P28,000, all seats on the bus will be sold. For each additional P1,000 charged, however, 2 fewer seats will be sold. Find the largest possible revenue.
A. P29,000
B. P28,500
C. P28,900
D. P28,700
 

Trapezoidal gutter made from 24 m wide iron sheet

A gutter whose cross-section is trapezoidal is to be made of galvanized iron sheet of 24 m wide. If its carrying capacity is maximum, find the dimension of the base.
A.   4 m
B.   6 m
C.   8 m
D.   10 m
 

diffcalc-maxmin-gutter.gif

02 Location of the third point on the parabola for largest triangle

Problem
The line y = 2x + 8 intersects the parabola y = x2 at points A and B. Point C is on the parabolic arc AOB where O is the origin. Locate C to maximize the area of the triangle ABC.
 

02-largest-triangle-line-parabola.jpg

 

01 Minimum length of cables linking to one point

Problem 01
A 5-m line AD intersect at 90° to line BC at D so that BD is 2 m and DC = 3 m. Point P is located somewhere on AD. The total length of the cables linking P to points A, B, and C is minimized. How far is P from A?
 

01-lines-minimum-length.gif

 

01 Maximum area of triangle of given perimeter

Problem 1
Show that the largest triangle of given perimeter is equilateral.
 

004-triangle-geven-perimeter.gif

 

Length of one side for maximum area of trapezoid (solution by Calculus)

Problem
BC of trapezoid ABCD is tangent at any point on circular arc DE whose center is O. Find the length of BC so that the area of ABCD is maximum.
 

55 - Greatest angle subtended by a picture

Problem 55
The lower edge of the picture is a ft, the upper edge is b ft, above the eye of an observer. At what horizontal distance should he stand, if the vertical angle subtended by the picture is to be greatest?
 

26-27 Horizontal rod entering into a room from a perpendicular corridor

Problem 26
A corridor 4 ft wide opens into a room 100 ft long and 32 ft wide, at the middle of one side. Find the length of the longest thin rod that can be carried horizontally into the room.

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