Find the equation of the tangent plane and the parametric equations for the normal line to the surface x² + y² - z = 0 at the point P(4,-1, 6). Show all steps

The count in a bacteria culture was 800 after 20 minutes and 1100 after 30 minutes. Assuming the count grows exponentially,

What was the initial size of the culture?

Find the doubling period.

Find the population after 70 minutes.

When will the population reach 10000. You may enter the exact value or round to 2 decimal places.

solve using the multiplication principle first. Then add

1/3x+3/2y=5/6

1/4x+1/2y=3/8

Use differentials to approximate the value of the expression. Compare your answer with that of a calculator. (Round your answers to four decimal places.)

1. The Arizona Fish and Game department has a lottery for elk tags in an effort to control the population of elk in Northern Arizona. There are two proposed methods of deciding how many tags to auction:
A) Auction 6 % of the current population each year.
B) Auction 7500 tags per year regardless of the population.
Clearly one method is a better policy since it monitors the current population.
However; ignoring the birth rate of elk, decide if policy A) and B) lead to exponential or linear decay in the population of elk.
Write EXP for exponential and LIN for linear growth.. Answer for A and B

2.  Let P=f(t)=800(1.027)tP=f(t)=800(1.027)t be the population of a community in year tt.(a) Evaluate f(0)=f(0)= (b) Evaluate f(10)=f(10)= (c) Which of these statements correctly explains the practical meaning of the value you found for f(10)f(10) in part (b)? (select all that apply if more than one is correct)

3. The rat population in a major metropolitan city is given by the formula n(t)=43e0.015tn(t)=43e0.015twhere tt is measured in years since 1993 and nn is measured in millions. (a) What was the rat population in 1993? (b) What is the rat population going to be in the year 2002?. answer for A and B

4. If 6900 dollars is invested at an interest rate of 10 percent per year, compounded semiannually, find the value of the investment after the given number of years.
(a) 5 years:
Your answer is  
(b) 10 years:
Your answer is  
(c) 15 years:
Your answer is

5.  A city had a population of 5,295 at the begining of 1948 and has been growing at 7.4% per year since then.

(a) Find the size of the city at the beginning of 1999.
Answer:

(b) During what year will the population of the city reach 13,195,713 ? (Plug answer into a calculator and round.)
Answer:

6. If 4000 dollars is invested in a bank account at an interest rate of 7 percent per year, compounded continuously. How many years will it take for your balance to reach 20000 dollars?

7. Find the doubling time for a city whose population is growing by 15% per year.

The doubling time is  years.

Use Laplace transforms, solve the differential equation y'' + 16y = 4 sin 4? , where y(0)=2.    y′(0)=0.

The price of a product in a competitive market is $200. If the cost per unit of producing the product is 80 + 0.1x dollars, where x is the number of units produced per month, how many units should the firm produce and sell to maximize its profit?

use the method of Lagrange multipliers to find the absolute maximum and minimum values of the function subject to the given constraints f(x,y)=x^2+y^2-2x-2y on the region x^2+y^2≤9 and y≥0

A heavy rope, 40 feet long, weighs 0.3 lb/ft and hangs over the edge of a building 110 feet high. Let x be the distance in feet below the top of the building.
Find the work required to pull the entire rope up to the top of the building.

1. Draw a sketch of the situation.
We can look at this problem two different ways. In either case, we will start by thinking of approximating the amount of work done by using Riemann sums. First, let’s imagine “constant force changing distance.”

2. Imagine chopping the rope up into n pieces of length ∆x. How much does each little
piece weigh? (This is the force on that piece of rope. It should be the SAME for each
piece of rope.)
3. How far does the piece of the rope located at xi have to travel to get to the top of the
building? (Notice that this is DIFFERENT for each piece of rope; it depends on the
location of the piece.)
4. How much work is done (approximately) to move one piece of rope to the top of the
building?
5. Find the amount of work required to pull the entire rope to the top of the building,
using an integral.

Now, let’s do the same problem, but this time, imagining “constant distance, changing force.” Imagine pulling up the rope a little bit at a time, say, we pull up ∆x feet of rope with each pull.
6. After you have pulled up xi feet of rope, how much of the rope remains to be pulled?
7. What is the force on the remaining amount of rope?
8. Remembering that each pull moves the rope ∆x feet, how much work is done for each
pull?
9. Find the amount of work required to pull the entire rope to the top of the building,
using an integral.

Help 6 to 9 and write it in order with number.

Solve this system of equations. Write the solution as an ordered triple.
6x + 3y + z = 36
x - 3y + 2z = -12
17x - 2y + 3z = 80

A business that produces color copies is trying to minimize its average cost per copy​ (total cost divided by the number of​ copies). This average cost in cents is given by

​f(x)equals=0.00000093x^2−0.0146+60​,

where x represents the total number of copies produced.

​(a) Describe the graph of f.

​(b) Find the minimum average cost per copy and the corresponding number of copies made.

1)

Let y be the solution of the equation

y ′ = 4(x^4)*sin(x^4) satisfying the condition y ( 0 ) = − 1.

Find y ( pi^1/3 ).

2)

Find the largest value of the parameter r

for which the function y = e^(rx) is a solution of the

equation y ″ − 14 y ′ + 28 y = 0.

3)

Let y ′ = − 4x^2*e^(-x^4) and let y ( 0 ) = 1.

Find ln ⁡ ( y ( 2 ) ).

Could you clarify what you mean by "typing error"?

Find the general solution of this differential equation: y'''+y''+y'+y=4e^(-t)+4sin(t)

The following exercise is designed to be solved using technology such as calculators or computer spreadsheets.

Interest paid on a home mortgage is normally tax deductible. That is, you can subtract the total mortgage interest paid over the year in determining your taxable income. This is one advantage of buying a home. Suppose you take out a 30-year home mortgage for $250,000 at an APR of 8% compounded monthly. The mortgage payments details for the first year are given below.

Suppose that your marginal tax rate is 30%. What is your actual tax savings due to mortgage payments? (Round your answer to the nearest cent.)

Use Green’s theorem in the plane to evaluate∮C(4x+ 1)x2ydx+x(x3−y3)dy where C is the circle whose equation is x2+y2= 2x transversed once in an anti cloickwise direction.