As dry air moves upwards, it expands and then cools at a rate of 1°C for each 100 m rise, up to about 12 km.

a. If the ground temperature is 20°C, find an expression for the temperature T as a function of the height h. Assume a linear function.

b. Use this function to predict the temperature at 5 km up from ground level.

c. Find the derivative of the function with appropriate units. What does the derivative of the function represent?

d. Is the rate of air cooling increasing as air moves upwards (to 12 km)? Explain.

An oil company wants to build a pipeline to take oil from an oil well to a refinery. Unfortunately, the well and the refinery are on either side of a straight river which is 10 miles wide, and they are 50 miles apart along the coastline (that is, if you want to go from the well to the refinery you must first cross 10 miles of river and then go 50 miles along the side of the river). The company has hired you to figure out the cheapest way to build the pipeline, and you need to clearly explain your solution so that the less mathematically-sophisticated oil people will understand.

It costs $200 per mile to lay pipe across the river but only $160 per mile to lay the pipe over land. There is also one other potential cost: It costs extra money each time that you have a bend in the pipeline. If you go straight across the river and use an L-shaped bend it costs an extra $150. If you lay the pipe diagonally across the river but hit land before you get to the refinery, you have to use a slanted L-shaped bend (like an obtuse angle). These have to be custom made and they cost $975. If you go directly diagonally across the river to the refinery without touching any land, then you do not have to pay extra for a bend (since there will not be one).

Exercise 1. Find the cheapest path to lay the pipeline by doing the following:

(a) Draw a diagram of the situation. Include any variables that you are going to use in the rest of your answer.

(b) Calculate the cost of building the pipeline for a few different situations: (i) How much would it cost to build the pipeline 10 miles straight across the river, then make a 90 degree bend and go 50 miles along the side of the river? (ii) How much would it cost to go diagonally across the river, without going on land at all? (iii) Suppose P is the point directly across the river from the oil well. How much would it cost to go diagonally across the river to a point 10 miles along the bank from P, then bend the pipeline and go the remaining 40 miles along the side of the river? (iv) How much would it cost to go diagonally across the river to a point 40 miles along the bank from P, then bend the pipeline and go the remaining distance along the side of the river?

(c) Calculate the cheapest way to build the pipeline for the situation given above by finding the minimum cost among ALL possible locations for P. (i) You will need to write a general expression for the cost of building the pipeline1 , and then use calculus to minimize the cost. (ii) Use either the first or second derivative test to prove that your result is a minimum. (d) What is the effect of the extra cost for a bend in the pipeline? If there was no extra cost for the bend, would you have a different answer for what the cheapest path would be?

Two objects are traveling in elliptical paths given by the following parametric equations.

x1 = 4cost and y1 = 2sint

x2 = 2sin2t and y2 = 3cos2t

At what rate is the distance between the two objects changing when t = π.

Use Euler's method with step size 0.2 to estimate y(0.4), where  y(x) is the solution of the initial-value problem y' = 3x − 4xy, y(0) = 0. (Round your answer to four decimal places.)

y(0.4) =

(b) Repeat part (a) with step size 0.1. (Round your answer to four decimal places.)

y(0.4) =

The project will be graded both for mathematical quality and for expository quality. Look at the six examples in section 4.7 of our text for good ideas about expository style. Notice how the author mixes the necessary mathematical equations with sentences of logic and explanation. Notice that the exposition is almost always in present tense. And, the author frequently uses we referring to the reader and author working together on the steps of solution. These are all standard conventions in mathematical writing.

The Problem. You are a landscape designer. A client has asked you to design a plan to enclose a rectangular garden having 10,000 m2 of area. The north and south sides are to be bounded by wooden fencing, which costs $20/m; the east and west sides are to be bounded by rhododendrons, which cost $50/m. Find the dimensions of the garden that minimize the total cost of the fencing and shrubbery. The client is getting estimates from several designers, so you need to prove that your plan is guaranteed to result in the lowest cost (given those per-meter costs for installing the fencing and the plants). Accordingly, when you write up your plan include a careful description of how you find these dimensions and how you verify that these dimensions actually minimize the cost.

Having finished your plan for the garden to be enclosed by rhododendrons and wooden fencing, you realize that rectangular gardens with different ”boundary material” on the north and south than on the east and west have become popular. To simplify the design process, you want to create a template for these problems. The general situation is as follows. A 10,000 m2 garden is to be enclosed on the north and south by material A, which costs a dollars per meter, and on the east and west by material B, which costs b dollars per meter. Find the dimensions that minimize the total cost of enclosing the garden. (The dimensions will be in terms of the constants a and b.) As before, in order to prove to your clients that your plan is the best possible, include a careful description of how you find these dimensions and how you verify that these dimensions actually minimize the cost.

What does your template tell you about the dimensions of the garden when a = b? When a = 4b? When a = (1/4)b? Does it agree with the dimensions you found for the rhododendron/wooden fence-enclosed garden? What happens if the area of the garden is to be something other than 10,000 m2?

The audience for your paper is the client, but assume this client is someone who is familiar with calculus. Begin with a brief introduction as a reminder of the problem. Incorporate into your report a computer-drawn graph of the cost function used in the first part of the analysis (with costs $20/m and $50/m). You will need to experiment with the window dimensions in order to include the important details of the function. The graph should exhibit the domain of the function and the minimum point.

(a) Estimate the area under the graph of f(x) = 4 cos(x) from x = 0 to x = π/2 using four approximating rectangles and right endpoints. (Round your answers to four decimal places.) R4 = Sketch the graph and the rectangles. WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot Is your estimate an underestimate or an overestimate? underestimate overestimate (b) Repeat part (a) using left endpoints. L4 = Sketch the graph and the rectangles. WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot Is your estimate an underestimate or an overestimate? underestimate overestimate

Consider the function f(x,y) = e^xy and closed triangular region D with vertices (2,0), (0,2) an (0,-2). Find the absolute maximum and minimum values of f on this region.

Need an explanation pls

Find the volume of the solid enclosed by the two paraboloids y=x2+z2y=x2+z2 and y=2−x2−z2y=2−x2−z2.

You have been asked to design a can with a volume of 500cm3 that is shaped like a right circular cylinder. The can will have a closed top. What radius r and height h, in centimeters, would minimize the amount of material needed to construct this can? Enter an exact answer.

A) Find the equation of the plane that passes through (2, -1,3) and is perpendicular to the line x = 2-3t, y = 3 + t, z = 5t
B) Find the equation where the planes 2x-3y + z = 5 and x + y-z = 2 intersect.
C) Find the distance from the point (2,3,1) to the x + y-z = 2 plane.
D) Find the angle between the planes x + y + z = 1 and x-2y + 3z = 1

Use the Alternating Series Estimation Theorem to estimate the range of values of x for which the given approximation is accurate to within the stated error. Check your answer graphically

X^2

Let z=e^(x) tan y.

a. Compute the first-order partial derivatives of z.

b. Compute the second-order partial derivatives of z.

c.∗ Convert z = f(x,y) into polar coordinates and then compute the first- order partial derivatives fr and fθ by directly differentiating the com- posite function, and then using the Chain Rule.