Maxima and Minima

Smallest Triangular Portion From A Square Lot

Problem
A farmer owned a square field measuring exactly 2261 m on each side. 1898 m from one corner and 1009 m from an adjacent corner stands Narra tree. A neighbor offered to purchase a triangular portion of the field stipulating that a fence should be erected in a straight line from one side of the field to an adjacent side so that the Narra tree was part of the fence. The farmer accepted the offer but made sure that the triangular portion was a minimum area. What was the area of the field the neighbor received and how long was the fence? Hint: Use the Cosine Law.

A.   A = 972,325 m2 and L = 2,236 m
B.   A = 950,160 m2 and L = 2,122 m
C.   A = 946,350 m2 and L = 2,495 m
D.   A = 939,120 m2 and L = 2,018 m

 

Dimensions of the Lot for a Given Cost of Fencing

Problem
A rectangular waterfront lot has a perimeter of 1000 feet. To create a sense of privacy, the lot owner decides to fence along three sides excluding the sides that fronts the water. An expensive fencing along the lot’s front length costs Php25 per foot, and an inexpensive fencing along two side widths costs only Php5 per foot. The total cost of the fencing along all three sides comes to Php9500. What is the lot’s dimensions?

A.   300 feet by 100 feet C.   400 feet by 200 feet
B.   400 feet by 100 feet D.   300 feet by 200 feet

 

Longest Day of the Year: Summer Solstice

Problem
The number of hours daylight, D(t) at a particular time of the year can be approximated by
 

$D(t) = \dfrac{K}{2}\sin \left[ \dfrac{2\pi}{365}(t - 79) \right] + 12$

 

for t days and t = 0 corresponding to January 1. The constant K determines the total variation in day length and depends on the latitude of the locale. When is the day length the longest, assuming that it is NOT a leap year?

A.   December 20 C.   June 20
B.   June 19 D.   December 19

 

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.
 

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