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Basic Calculus

Mathematics - Calculus - Minimization

An open-topped cylindrical pot is to have volume 125in^3. What dimensions will minimize the total amount of material used in making this pot? Neglect the thickness of the material and possible wastage.

Area Between Productivity Curves

After t hours on the job, one factory worker is producing Q'1(t) = 60 - 2(t - 1)^2 units per hour, while a second worker is producing Q'2(t) = 50 - 5t units per hour. a) If both arrive on the job at 8:00 AM how many more units will the first worker have produced by noon than the second worker? b) Interpret the answer in

Mathematics - Calculus - Minimizing Cost

A surfboard manufacturing company has determined that the cost function c(x), in hundreds of dollars, for producing x hundreds of boards is c(x) = .2x^2 -.84x + 3.625. How many surfboards should the company make to minimize production costs?

Mathematics - Application of Calculus in Commerece

A manufacturer estimates marginal revenue to be R'(q) = 100q^ -1/2 dollars per unit when the level of production is q units. The corresponding marginal cost has been found to be 0.4q dollars per unit. Suppose the manufacturer's profit is $520 when the level of production is 16 units. What is the manufacturer's profit when the le

Calculus

A long rectangular sheet of metal is to be made into a rain gutter by turning up two sides at right angles to the remaining center strip. The rectangular cross section of the gutter is to have area 18in^2. Find the minimum possible width of the sheet.

Maximizing Area of a Fence

A developer wants to enclose a rectangular grassy lot that borders a city street for parking. If the developer has 640 feet of fencing and does not fence the side along the street, what is the largest area that can be enclosed?

Mathematics - Calculus Sales Problem

Please help with the following problem. Provide a step by step calculation. Sales and Saturation Point : Integration using Separation of Variables - The rate of increase in sales ... (Please see the attached file) Sales: The rate of increase in sales S (in thousands of units) of a product is proportional to the current le

Application of Derivatives

The prong of a tuning fork moves back and forth when it is set into vibration. The distance the prong moves between its extreme positions is 2.22 mm. If the frequency of the tuning fork is 441.0 Hz, what are the maximum velocity and the maximum acceleration of the prong? Assume SHM. ______m/s ______m/s^2^

Optimization ...

Suppose that you are to make a rectangular box with a square base from two different materials. The material for the top and four sides of the box costs $1/ft^2; the material for the base costs $2/ft^2. FInd the dimensions of the box of greatest possible volume if you are allowed to spend $144 for the material to make it.

Curve Sketching ...

Sketch the graph of y = x^4/4 - 2x^2 + 1. i) determine and show on your graph all relative minima, maxima, and points of inflection. ii) find intervals where the graph is increasing, decreasing, concave up, and concave down. iii) find the absolute maximum and absolute minimum values of y on the interval [-1,3].

Mathematics - Calculus - Maximum Value..

8. An entrepreneur buys an apartment building with 40 units. The previous owner charged $240 per month for a single apartment and on average rented 32 apartments at that price. The entrepreneur discovers that for every $20 he raises the price another apartment stands vacant. A. Let x represent the number of $20 price increases

Mathematics - Calculus - Functions

C(x)=2x+56 R(x) = 20x-x2 a)Find the marginal cost function ?what does this function predict? b)Find the profit function. c)Find the marginal profit function ?what does this function predict? d)What are the breakeven points? e)Find the value of x, where the graph R(x) has a horizontal tangent.

Mathematics - Calculus - Maximum Value

A zookeeper needs to add a rectangular outdoor pen to an animal house with a corner notch, as shown in the figure. If 85 m of new fence is available, what dimensions of the pen will maximize its area? No fence will be used along the walls of the animal house.

Mathematics - Calculus - Rate measures

A plane flying at an altitude of 25,000 ft. has a defective airspeed indicator. To determine her speed, the pilot sights a fixed point on the ground. At the moment when the angle of depression (From the horziontal) of her line of sight is 65 degrees, she notes that this angle is increasing at 1.5 degrees per second. What is the

Calculus

A large storage tank, open to the atmosphere at the top and filled with water, develops a small hole in its side at a point 16.0 m below the water level. If the rate of flow from the leak is 2.50 x 10^ -3 m^3/ min, determine (a) the speed at which the water leaves the hole and (b) the diameter of the hole.

Mathematics - Calculus - Exponential Functions

A cancer patient undergoing chemotherapy is injected with a particular drug. The function D(h)=4e^-0.35h gives the number of milligrams D of this drug that is in the patient's blood stream h hours after the drug has been administered. How many milligrams of the drug were injected? To the nearest milligram, how much of the drug w

Mathematics - Calculus - Rate Measure..

A baseball diamond is a square whose sides are 60 ft long. a player running from first to second base has a speed of 25 ft per second. at what rate is the player's distance from home base changing when he is 10 ft from second base?

Solving Polynomials by factorizing

Factor 1. x^2 + 12x +36 - y^6 2.9y^2 + 12y + 4 - x^2 solve each equation n^2 + n = 72 (4x + 9)(x - 4)(x + 1) = 0 m^3 = m^2 + 12m (x + 4)(5x - 1 ) = 0 x^ + 6x - 7 = 0 A stereo system installer needs to run speaker wire along the two diagonals of a rectangular room whose dimensions are 40 feet by 75

Mathematics - Calculus..

A) Give a possible formula for a cubic function. Assume the graph crosses the x-axis at exactly -3, 1, and 4. Assume it crosses the y-axis at exactly -12. B) The desert temperature, H, oscillates daily between 42ºF at 4am and 82ºF at 4pm. Write a possible formula for H in terms of t, measured in hours from 4am. Give your

Polynomial functions

Your manager is very pleased with your presentations and being able to show your calculations in a detailed fashion. He has asked that you help him prepare for the upcoming company summit. Before you get together, he has asked that you give him a little review on polynomial functions and how you would apply them to everyday use.

Riemann sum and volume

See attached file Write a Riemann sum and then a definite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly.

Calculus problem

Please help me in attached problems. A customer ordered a quadratic metal sheet with side length x0=10cm. What is the maximal acceptable error of the side length if the area of the metal sheet should be?... For each of the following functions: state if the function is monotone Decide if it is injective, surjective or bij

Mathematics - Calculus Rate Problem

14. A spherical balloon is inflated so that its volume is increasing at the rate of 3 ft^3/min. How fast is the diameter of the balloon increasing when the radius is 1 ft? 16. A 17-ft ladder is leaning against a wall. If the bottom of the ladder is pulled along the ground away from the wall at a constant rate of 5 ft/s, how

Pilot Off Course

A pilot, attempting to fly from Middletown to Westburg, a distance of 200 miles, notices after flying 180 miles that West burg is still 35 miles away because the heading was off by a few degrees. How many degrees was the heading off the desired course (to one decimal place)?

Spherical coordinates and electrical charge

An electric charge is distributed throughout 3-space with density inversely proportional to the distance from the origin. Show that the total charge inside a sphere of radius R is proportional to R^2.

Solving Extraneous Equations

Please solve below exercises using the Equation Editor. Answers are already provided for you from the actual book. You must show however how you have arrived at the answer. Section 6.5 Exercise 55 Sophomore math. A survey of college sophomores showed that of the males were taking mathematics class and of the females

Geometry's Role in the Math

The importance of geometry's role in the math curriculum is debated in many high schools and colleges. Some schools offer the course while others have done away with it. Based on what you have learned, do you think geometry is a valuable tool for students to learn? Choose one side of this debate, state your view, support your vi

Converging Subsequence

Theorem: Suppose that a sequence S of real numbers has a subsequence that converges to a real number a. Then S converges to a. I know this is true as an if and only if statement, but I need a counter example to show that just one converging subsequence isn't enough. Here are two sequences I'm considering: {1,-1,1,-1,1,-1..