Use Lagrange multipliers to find the point on the given plane that is closest to the following point.

x − y + z = 6;    (2, 7, 3)

RATES: 2.1 Q:1

A) A stone dropped into a still pond sends out a circular ripple whose radius increases at a constant rate of 2.3 ft/s.

How rapidly is the area enclosed by the ripple increasing when the radius is 2 feet?

The area is increasing at

B) Oil spilled from a ruptured tanker spreads in a circle whose area increases at a constant rate of 8 mi^2/hr. How rapidly is radius of the spill increasing when the area is 5 mi^2?

The radius is increasing at

C) A 15 ft ladder leans against a wall. The bottom of the ladder slides away from the wall at a rate of 2ft/sec

How fast is ladder sliding down the wall when the base of the ladder is 66 ft away from the wall?

If cosα=0.082 and sinβ=0.802 with both angles’ terminal rays in Quadrant-I, find the values of cos(α+β) and sin(β−α)

Use the Left and Right Riemann Sums with 3 rectangles to estimate the area under the curve of y=lnx on the interval of [2,10] Round your answers to the second decimal place.

\sum _{n=1}^{\infty }\:\frac{4\left(-1\right)^n+2^n}{3^n} Determine whether the series is convergent or divergent. If it is convergent, find its sum.

4. This problem is about some function. All we know about the function is that it exists everywhere and we also know the information given below about the derivative of the function. Answer each of the following questions about this function. Be sure to justify your answers. f ′(−5) = 0 f ′(−2) = 0 f ′(4) = 0 f ′(8) = 0 f ′(x) < 0 on (−5,−2), (−2,4), (8,∞) f ′(x) > 0 on (−∞,−5), (4,8)

a. Identify the critical points of the function.

b. Determine the intervals on which the function increases and decreases.

c. Classify the critical points as relative maximums, relative minimums or neither.

a) Determine whether the given differential equation is exact. If it is exact, solve it. (If it is not exact, enter NOT.)

(2xy² − 5) dx + (2x²y + 4) dy = 0

b) Solve the given differential equation by finding, as in Example 4 of Section 2.4, an appropriate integrating factor.

(6 − 20y + e^−5x) dx − 4 dy = 0

Show that the cylinder x 2 + y 2 = 4 and the sphere x 2 + y 2 + z 2 − 8y − 6z + 21 = 0 are tangent at the point (0, 2, 3). That is, show that the cylinder and sphere intersect at that point, and that they share the same tangent plane at that point.

1. Given the function M(t) = 2t3 - 3t2 - 36t, find the critical values and determine, using both the second derivative test and a sign chart, the nature of these values.
2. A projectile is launched with a velocity of 22 m/s at 50° to the ground. Determine its horizontal and vertical velocities.
3. Two trains start from the same point at the same time, one going east at a rate of 40 km/h and the other going south at 60 km/h, as shown in the diagram at right. Find the rate at which they are separating after 1 h of travel.
4. A professional basketball team plays in a stadium that holds 23,000 spectators. With ticket prices at $60, the average attendance had been 18,000. When ticket prices were lowered to $55, the average attendance rose to 20,000. Based on this pattern, how should ticket prices be set to maximize ticket revenue?
5. Corey is asked to find the maximum value of a function. Not having a complete understanding of the process, Corey decides to find the derivative of the function, set it equal to zero, and solve. The resulting value, Corey reasons, will yield the maximum point. Explain fully why Corey's method is flawed.
6. A 5,000 m_ rectangular area of a field is to be enclosed by a fence, with a moveable inner fence built across the narrow part of the field, as shown.The perimeter fence costs $10/m and the inner fence costs $4/m. Determine the dimensions of the field to minimize the cost.
7. The following table displays the number of HIV diagnoses per year in a particular country.
Year 1997 1998 1999 2000 2001 2002 2003 2004 2005
Diagnoses 2512 2343 2230 2113 2178 2495 2496 2538 2518
a. Using Curve Expert or another curve modelling program, determine an equation that can be used to model this data.
b. Using this model, estimate the number of diagnoses in 1996 and in 2006.
c. At what rate would the number of diagnoses be changing in 2006?
d. Halfway through 2006, the number of new HIV diagnoses was found to be 1232. Assuming this rate stays fairly constant for the remainder of the year, does this new information change the modelling equation? If so, how would this change your answer to part (c)? If you were an advocate for furthering HIV and AIDS research and treatment programs, would you be encouraged or discouraged by these results?

Find the local maximum and minimum values of f using the First Derivative Test.

f(x) =

Find the center of mass of the solid bounded by the surfaces z = x ^ 2 + y ^ 2 and z = 8-x ^ 2-y ^ 2. Consider that the density of the solid is constant equal to 1.

Mass= ?

x=?

y=?

z=?

Step by step please

a virus is spreading on a small deserted island. The total number of people infected by the virus t days after the outbreak is given by the model p(t)=5200/(1+20^(-t+15)).

the different parts can be solved independently.

a. How many people will be infected in the long run? (hint: compute the limit of p as t goes too infinity.)

b. The island only has a small hospital with a capacity of 250 patients. When will the total number of infections reach 250?

c. After how many days does the rate of new infections reach a maximum? Justify your answer carefully using calculus. Approximately how many people will be newly infected on that day?

A pentagon is formed by adding two sides of the same length to attach an isoceles triangle to one side of a rectangle. If the area is fixed at A, what are the dimensions (in terms of A) that will minimize the pentagon's perimeter?