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Calculus and Analysis

Working with growth and decay rates and decay rate expressions.

A crude-oil refinery has an underground storage tank which has a fixed volume of 'V' liters. Due to pollutants, it gets contaminated with 'P(t)' kilograms of chemical waste at time 't' which is evenly distributed throughout the tank. Oil containing a variety of pollutants with concentration of 'k' kilograms per liter enters

Differential Equations and Newton's Law of Cooling

At 4:30 PM on Monday, a Virginia criminalist was called to the scene of a homicide. She noted that the body temperature of the deceased was 85.5 deg. while the air temperature was 78 deg. Thirty minutes later, the deceased's body temperature was 82 deg. Assuming the air temperature stayed constant, what is the estimated time of

Find the maximum area of a window.

A special window has the shape of a rectangle surrmounted by an equilateral triangle. If the perimeter is 16 feet, what dimensions will admit the most light? (hint: Area of equilateral triangle = the square root of 3/4 times x squared.)

Newton's Law of Cooling relating to differential equations.

At 10:00 AM, an object is removed from a furnace and placed in an environment with a constant temperature of 68 degrees. Its core temperature is 1600 degrees. At 11:00 AM, its core temperature is 1090 degrees. Find its core temperature at 5:00 PM on the same day.

Maximization to Construct a Fence

A Farmer wants to construct a fence. The area that he is going to enclose is rectangular, but one of the sides is a river (assumed to be straight). If he has 120 m of fence, what is the maximum surface that he can enclose?

Sample Question: Word problems

1. The manager of a large apartment complex knows from experience that 80 units will be occupied if the rent is 320 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 8 dollar increase in rent. Similarly, one additional unit will be occupied for each 8 dollar decreas


1. A weather balloon is rising vertically at a constant rate of 4 ft/s directly above a straight and level road. When the balloon is 75 ft above the road, a car moving at 55 ft/s passes directly under the balloon. Based on this information find: a. the rate the distance between the balloon and the car is changing 3 sec after t

A minimization fencing problem.

A rectangular field is going to be enclosed and divided into two separate rectangular areas. (Areas do not have to be equal). Find the minimum fencing that is required if the total area of the field is 1200m2.

Infinite Sequences and Series Example Problem

In the figure (see attachment) there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length 1, find the total area occupied by the circles.

Rate of production of eggs

Eggs are produced at a rate of R(t)eggs per hour,where t=0 represents 12:00 midnight and R(t)(in thousands of eggs) is :- R(t)= -10cospi/12t+10 a)how many eggs are produced in one day. b)When are the eggs produced at the fastest rate c)A machine can produce eggs at a constant rate. At the end of 1 week the same

Tan line and velocity problems

The parabola y = (x^2) + 3 has two tangents which pass through the point (0, -2). One is tangent to the to the parabola at (A, A^2 + 3) and the other at (-A, A^2 + 3). Find (the positive number) ? If a ball is thrown vertically upward from the roof of 64ft foot building with a velocity of 96 ft/sec, its height after t seconds

Related Rates Word Problem

A highway patrol helicopter hovers 3/10 mile above a level, straight interstate highway which has a posted speed limit of 65 miles per hour. The helicopter pilot sees a car on the highway and determines with radar that at that particular instant, the distance between the helicopter and the car is 1/2 mile and is increasing at a

Limit points

Prove that the point p is a limit point of the point set X if and only if each open point set containing p contains a point in X which is different from p. Prove without using sequences. Only use the def. of open set, open interval, and that the point p is a limit point of the point set X means that each open interval containing

Area of a rectangle

A rectangular lot, 54 square yds. in area with a perimeter fence, is divided into 2 rectangular sections by a single connecting fence costing $2.00/yd. the perimeter fence costs $5.00/yd. Find the dimensions of the lot which minimizes the cost of fencing.

Determining the best place to sit at the movies based on a set of criteria.

A movie theater has a screen that is positioned 10 feet off the floor and is 25 feet high. The first row of seats is placed 9 feet from the screen and the rows are 3 feet apart. The floor of the seating area is inclined at an angle above the horizontal and the distance up the incline that you sit is x. The theater has 21 ro

Working with position functions using acceleration and velocity.

A model rocket is fired vertically upward from rest. It's acceleration for the first three seconds is a(t)=60t at which time the fuel is exhausted and it becomes a free falling body. After 17 seconds, the rocket's parachute opens and the velocity slows linearly to -18 ft/sec in 5 seconds. The rocket then floats to the ground

Compound Interest Problems

If the tuition at a certain college is determined to cost $ 32000 in 10 years, how large must a trust fund that pays 7.5% compounded continuously be, in order for a child on her 8th birthday to ensure sufficient funds at age 18?


I have two problems (well, one problem with three parts and another one): 1. (a) Let f(x)=ax^2+bx+c, a does not equal zero, be a quadratic polynomial. How many points of inflection does the graph of f have? (b)Let f(x)=ax^3+bx^2+cx+d, a does not equal zero, be a cubic polynomial. How many points of inflection does the grap

Mu''(t) + ku(t) = 0

Suppose m and k are positive numbers. Find u so that mu''(t) + ku(t)= 0 for all numbers t and u(0)=1 and u'(0)=2. (note: u'' = second derivative of u) Please clarify any shorthand that you are using. Thanks!