Differential Calculus - PART 3 - Maxima-minima-timeratesFull description...

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Mar Lou Galinato

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PART [3]

MAXIMA-MINIMA In solving a problem under maxima/minima, the following steps are to be considered: 1. Draw a figure when necessary 2. Identify what/which to maximize or minimize. 3. Formulate the equation. 4. Differentiate. 5. Equate to zero.

PROBLEMS: [1] Find the minimum distance from the point (4,2) to the parabola = 8.

[A] 4√ 3 [B] 2√ 2 [C] √ 3 [D] 2√ 3 [2]

NOTE: When the first derivative (slope) is equated to zero, it results to either maximum point or minimum points.

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At maximum point:

= 0 " ( ) [5]

At minimum point:

= 0 " ( ) At point of inflection:

"=0 " are the first and second derivatives respectively.

Where:

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The sum of two positive numbers is 50. What are the numbers if their product is to be the largest possible. [A] 24 26 [B] 28 22 [C] 25 25 [D] 20 30 A triangle has variable sides ,, subject to the constraint such that the perimeter is fixed to 18 cm. What is the maximum possible area for the triangle? [A] 15.59 [B] 18.71 [C] 17.15 [D] 14.03 A farmer has enough money to build only 100 meter of fence. What are the dimensions of the field he can enclose the maximum area? [A] 25 25 [B] 15 35 [C] 20 30 [D] 22.5 27.5 Find the minimum amount of tin sheet that can be made into a closed cylinder having a volume of 108 cubic inches in square inches. [A] 125.50 [B] 127.50 [C] 129.50 [D] 123.50

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A box is to be constructed from a piece of zinc which is 2020 by cutting equal squares from each corner and turning up the zinc to form the side. What is the volume of the largest box that can be constructed? [A] 599.95 [B] 592.59 [C] 579.50 [D] 622.49 A printed page must contain 60 .. of printed material. There are to be margins of 5 on either side and margins of 3 on top and bottom. How long should the printed lines be in order to minimize the amount of paper used? [A] 10 [B] 18 [C] 12 [D] 15 Find the absolute maximum and minimum value of () = − 12 + 5 on the interval [−1,4] . [A] −25 25 [B] −11 16 [C] −11 21 [D] −16 21 Jodi wishes to use 100 of fencing to enclose a rectangular garden. Determine the maximum possible area of her garden. [A] 525 [B] 625 [C] 725 [D] 825

[10] What is the minimum possible perimeter for a rectangle whose area is 100 ? [A] 10 ℎ [B] 20 ℎ [C] 30 ℎ [D] 40 ℎ

SUPPLEMENTARY PROBLEMS: [1]

An open field is bounded by a lake with a straight shoreline. A rectangular enclosure is to be constructed using 500 of fencing along three sides and the lake as a natural boundary on the fourth side. What dimensions will maximize the enclosed area? What is the maximum area?

[2]

Ryan has 800 of fencing. He wishes to form a rectangular enclosure and then divide it into three sections by running two lengths of fence parallel to one side. What should the dimensions of the enclosure be in order to maximize the enclosed area?

[3]

20 meters of fencing are to be laid out in the shape of a right triangle. What should its dimensions be in order to maximize the enclosed area? A piece of wire 100 ℎ long is to be used to form a square and/or a circle. Determine their (a) maximum and (b) minimum combined area.

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Find the maximum area of a rectangle inscribed in a semicircle of radius 5 ℎ if its base lies along the diameter of the semicircle. An open box is to be constructed from a

12×12−ℎ piece of cardboard by cutting away squares of equal size from the four corners and folding up the sides. Determine the size of the cutout that maximizes the volume of the box. [7]

A window is to be constructed in the shape of an equilateral triangle on top of a rectangle. If its perimeter is to be 600 , what is the maximum possible area of the window?

[8]

Postal regulations require that the sum of the length and girth of a rectangular package may not exceed 108 ℎ (the girth is the perimeter of an end of the box). What is the maximum volume of a package with square ends that meets this criteria?

stamped from square pieces of metal and the rest of the square discarded. What dimensions will minimize the amount of metal needed in the construction of the can?

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A weight W is attached to a rope 12 m long and the rope is passed over a pulley 6 m above the ground. The end E of the rope is pulled along the ground at the rate of 1.2 m/s. How fast is the weight rising when the end of the rope has moved away 3 m? [A] 0.735 / [B] 0.753 / [C] 0.573 / [D] 0.737 /

[7]

The dimensions of a rectangle are continuously changing. The width increases at the rate of 3 in/sec while the length decreases at the rate of 2 in/sec. At one instant the rectangle is a 20-inch square. How fast is its area changing 3 seconds later? Is the area increasing or decreasing? [A] increasing at 14 / [B] decreasing at 14 / [C] increasing at 16 / [D] decreasing at 16 /

[4]

A ladder 20 feet long is placed against a wall. The foot of the ladder begins to slide away from the wall at the rate of 1 ft/sec. How fast is the top of the ladder sliding down the wall when the foot of the ladder is 12 feet from the wall? [A] 0.72 / [B] 0.73 / [C] 0.74 / [D] 0.75 /

[8]

A trough filled with water is 2mlong and has a cross section in the shape of an isosceles trapezoid 30 cm wide at the bottom, 60 cm wide at the top, and a height of 50 cm. If the trough leaks water at the rate of 2000 cm3/min, how fast is the water level falling when the water is 20 cm deep? [A] 0.22 cm/min [B] 0.23 cm/min [C] 0.24 cm/min [D] 0.25 cm/min

[16] A rectangle is to be inscribed in the ellipse [9]

A rectangle is inscribed in a right triangle whose sides are 5, 12, and 13 inches. Two adjacent sides of the rectangle lie along the legs of the triangle. What are the dimensions of the rectangle of maximum area? What is the maximum area?

[10] What is the minimum amount of fencing needed to construct a rectangular enclosure containing 1800 using a river as a natural boundary on one side? [11] An open rectangular box is to have a base twice as long as it is wide. If its volume must be 972 , what dimensions will minimize the amount of material used in its construction? [12] Find the points on the parabola closest to the point (0,1).

=

[13] A publisher wants to print a book whose pages are each to have an area of 96 . The margins are to be 1 in on each of three sides and 2 in on the fourth side to allow room for binding. What dimensions will allow the maximum area for the printed region? [14] A closed cylindrical can must have a volume of 1000 . What dimensions will minimize its surface area? [15] A closed cylindrical can must have a volume of 1000 . Its lateral surface is to be constructed from a rectangular piece of metal and its top and bottom are to be

+

= 1. Determine its maximum

possible area. PART [4]

TIME-RATES The predominant tool used in the solution of related rates problems is the chain rule. Since most related rates problems deal with time as the independent variable, we state the chain rule in terms of t:

= ∙

() = ′() In solving a problem under time rates, the following steps are to be considered: 1. Draw a figure when necessary 2. Formulate the equation. 3. Differentiate with respect to time. 4. Substitute the boundary condition(s) to the equation. IMPORTANT: Substitute the given values only after differentiating. PROBLEMS: [1] A baseball field is a square of side 27.44 m. A player on second base runs toward third base at the rate of 6 m/s. How fast is his distance from home plate changing when he is half-way to third base? [A] −6.28 / [B] −8.26 / [C] −6.82 / [D] −2.68 /

A point is moving along the circle + = 25 in the first quadrant in such a way that its x coordinate changes at the rate of 2 cm/sec. How fast is its y coordinate changing as the point passes through

At a certain instant the three dimensions of a rectangular parallel piped are 150 mm, 200 mm, 250 mm and these are increasing at the rates of 5 mm/s, 4 mm/s, and 3 mm/s respectively. How fast is the volume changing? [A] 940,000 / [B] 490,000 / [C] 860,000 / [D] 340,000 /

[5]

A plane, P, flies horizontally at an altitude of 2 miles with a speed of 480 mi/h. At a certain moment it passes directly over a radar station, R. How fast is the distance between the plane and the radar station increasing 1 minute later? [A] 464.66 /ℎ [B] 465.67 /ℎ [C] 466.68 /ℎ [D] 467.69 /ℎ

(3,4)? [A] −1.5 / [B] −1.6 / [C] −1.7 / [D] −1.8 /

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