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Steps in Solving Worded Problems in Maxima and Minima

This document provides guidance on solving optimization problems involving maxima and minima as well as time-rate problems. For maxima and minima problems, it outlines steps to (1) draw a diagram, (2) write an equation representing the quantity to be optimized, (3) express the equation in terms of a single variable, and (4) differentiate and set equal to zero. For time-rate problems, it outlines steps to (1) draw a diagram labeling constants, (2) determine known and unknown rates, (3) relate variables in an equation, (4) differentiate the equation, and (5) substitute knowns and solve for the unknown rate. Several examples of each type of problem are provided

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Joel Rodello Dugenia
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138 views2 pages

Steps in Solving Worded Problems in Maxima and Minima

This document provides guidance on solving optimization problems involving maxima and minima as well as time-rate problems. For maxima and minima problems, it outlines steps to (1) draw a diagram, (2) write an equation representing the quantity to be optimized, (3) express the equation in terms of a single variable, and (4) differentiate and set equal to zero. For time-rate problems, it outlines steps to (1) draw a diagram labeling constants, (2) determine known and unknown rates, (3) relate variables in an equation, (4) differentiate the equation, and (5) substitute knowns and solve for the unknown rate. Several examples of each type of problem are provided

Uploaded by

Joel Rodello Dugenia
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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CALCULUS circle that will maximize the area of the

window.
A. How to Solve Worded Problems in Maxima
and Minima 7. The range of a projectile is R =
where initial velocity V and the acceleration
Steps in Solving Worded Problems in Maxima
due to gravity are constant. Find the angle
and Minima
that maximizes the range of the projectile.
1. Draw a diagram if necessary
2. Write an equation representing the
quantity to be maximized or minimized.
This quantity will typically be
represented in terms of two or more
variables.
3. Use any relationships between the
variables to express the equation
obtained in step 2 into a function of
single variable.
4. Differentiate and equate the function to
zero.

Example:
1. Find the area of the largest rectangle that
can be inscribed in an equilateral triangle of
side 20.

2. In how many equal parts into which a given


number N must be divided so that the
successive product of its parts will be a
maximum?

3. What is the radius of a cylindrical can with a


volume of 512 cu.inches that will use the
minimum material.

4. An open rectangular box is to be made


from a 9 x 12 inch piece of tin by cutting
square x inches from the corner and folding
up the sides. What should be the value of x
to maximize the volume of the box?

5. Find the point on the curve x2 + y2 = 1


closest to point (2,1)

6. A window consists of an open rectangle


topped by a semi-circle and ahs a perimeter
of 288 inches. Find the radius of the semi-
6. Sand is pouring from a spout at the rate of
25 cc/sec. It forms a cone whose height is
B. Steps in Solving Problems Involving Time- always 1/3 the radius of its base. At what
rates. rate in cm/sec is the height increasing,
when the cone is 50 cm high?
1. Draw a diagram if necessary. Label
constants with their numerical value. 7. A man is riding a car at the rate of 30 kph
2. Determine which rates are given and towards the foot of a monument 6 m high.
which rate you need to find. At what rate is he approaching the top
3. Find an equation relating the variables when he is 36 m from the foot of the
defined in step 1 monument?
4. Differentiate the equation in step 3
5. Substitute all the given information into 8. A girl on a wharf is pulling a rope tied to a
the result of step 4 and find the raft at the rate of 0.9 m/s. If the hands of
unknown rate. the girl pulling the rope at 3.8 m. above the
level of water, how fast is the raft
Example: approaching the wharf when there is 6 m.
1. Sand escapes at the rate of 8ft3/min from of rope out?
an inverted conical container whose depth
is 15 ft. and whose base is a circle of radius
5 ft. Find how fast is the level of the sand 9. A car drives east from point A at 30 kph.
sinking when there are 10 ft of sand in the Another car starting from B at the same
tank. time, drives S.30 W toward A at 60 kph. B is
30 km away from A. How fast in kph is the
2. The top of a 25 ft ladder, leaning against a distance between two cars changing after
vertical wall is slipping down at the rate of 30 minutes?
2ft/s. Find how fast is the bottom end of the
ladder slipping along the ground when it is 8 10. A man is driving a car at a speed of 45 kph
ft away from the base of the wall. along a straight line passing 20 m in front of
a 15 m high monument. At what rate is his
distance from the top of the monument
3. A cylindrical tank radius 10 ft. is being filled changing 4 seconds after he passed in front
with water at the rate of 314 cu.ft. per of the monument?
minute. How fast is the depth of the water
increasing in ft. per minute?

4. A point on the rim of a 0.60 inch diameter


wheel is travelling at 75 ft/sec. What is the
angular velocity of the wheel in radians per
second?

5. A 3 m. long steel pipe has its upper end


leaning against a vertical wall and lower end
on a level ground. The lower end moves
away at a constant rate of 2 cm/sec. How
fast is the pipe rotating, when the lower
end is 2 m. from the wall in rad/sec?

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