1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
julsineya [31]
2 years ago
9

calculus optimization problem I have a lot of trouble with these so an explanation would be really helpful thanks!

Mathematics
1 answer:
Alexandra [31]2 years ago
7 0
Ok so for optimization problems, there are 3 steps:

1) Find an objective function in terms of a single variable
2) Find the derivative of this function and set it equal to zero.
3) Solve for the variable.

What this does is allow us to find the point on the function where the slope(derivative) is 0, indicating a local minimum or maximum.

For this problem, it will give us the minimum since we want to minimize Cost.

Now for step 1)
First determine the variable. Here we want the distance to point P on north bank. Lets call it 'x'. The total east-west distance from refinery to storage is 9 km. So we know that 'x' must be less than 9.
The distance from P to the south bank is diagonal distance 'd', which can be found using pythagorean theorem. 
d^2 = 1^2 + (9-x)^2
The function we will use is the cost function.
Simplify it in terms of 100,000 units.
Cost = 2x + 4d
C = 2x + 4\sqrt{x^2-18x+82}
This function will tell us the cost of laying pipe given how far downriver point P is.

Step 2)
To minimize the cost, we need to find when slope is 0.
Find derivative of cost function.
Note: \frac{d}{dx} \sqrt{u} = \frac{du}{2 \sqrt{u}}
Therefore
\frac{dC}{dx} = 2 + \frac{4(2x-18)}{2 \sqrt{x^2 -18x+82}} = 0

Step 3)
Solve derivative equation for x.
Usually this involves a lot of algebra and quadratic formula.
With a square root term, you want to isolate it on one side first. Then square both sides to get rid of the radical.

(\sqrt{x^2-18x+82})^2 = (18-2x)^2 \\  \\ x^2 -18x +82 = 324 -72x+4x^2 \\  \\ 3x^2 -54x +242 = 0 \\  \\ x = 9 - \frac{\sqrt{3}}{3} = 8.42

Remember x < 9, so the other solution from quadratic formula does not apply.

Ok we're done. The Cost is minimized when point P is 8.42 km downriver.
You might be interested in
1. The sum of Teresa's and Cathy's weight is
sergeinik [125]

Answer:

d

Step-by-step explanation:

E is ththe answer

the first equation should be T + C = 118 and the second equation should be C + 26 = T

none of these equations are correct

4 0
2 years ago
Check the picture of the necessary information
denis-greek [22]
The probability would be out of 100,
in this case if it refers to the data from 2009 it would be 39:61
4 0
2 years ago
Help!!!!!! red car travels 17 2/5 miles useing 0.75 of gas. please answer today
Nataliya [291]

I think you just divide

7 0
2 years ago
I’m confuse plz help
Ad libitum [116K]

Answer:

As in an equilateral triangle all sides are equal so 9+3x = 5x-5

9+3x = 5x-5

9+5 = 5x-3x

14 = 2x

14/2 = x

7= x

So,As we know that x is 15 so we can substitute the values 9+3x or 5x-5

9+3x

=9+3*7

=30

So, one side of the triangle = 30

Perimeter of an equilateral triangle = 3*side

=3*30 = 90

So, perimeter of the equilateral triangle = 90

6 0
3 years ago
Please help thank you
worty [1.4K]

Answer:

Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named as Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°.

7 0
2 years ago
Other questions:
  • A jar contains 2 red marbles, 5 blue marbles, and 1 green marble. What is the probability of selecting a red
    12·1 answer
  • The manager of a grocery store wants to determine the average amount of money spent at his store as it compares to the average d
    14·1 answer
  • Pls help with this question. I forgot
    11·1 answer
  • Can Anyone help me with this question?!
    11·1 answer
  • 4 1/2 cups of flour make 2 batches of cookies. How much flour per batch
    13·2 answers
  • It takes Naomi 4 minutes to run 5 laps. How long does it take her to run 30 laps? Use a table, double number line, tape diagram
    9·1 answer
  • Pls help!! i will give brainliest answer to the first right answer
    12·1 answer
  • Which is more, 6 inches or 1 foot? 6 inches 1 foot Neither, they are equal ​
    7·2 answers
  • Which equivalent expression to 2w + 1/2 ( -40 + 12 ) ?
    10·1 answer
  • julie sprros age 21 and unmarried drives her vechicle to and from work. her driver rating factor is 1.85 . her vechicle is class
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!