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
koban [17]
2 years ago
8

A flat circular plate has the shape of the region x squared plus y squared less than or equals 1.The​ plate, including the bound

ary where x squared plus y squared equals 1​,is heated so that the temperature at the point (x comma y )is Upper T (x comma y )equalsx squared plus 2 y squared minus three halves x.Find the temperatures at the hottest and coldest points on the plate.
Mathematics
1 answer:
rjkz [21]2 years ago
4 0

Answer:

We have the coldest value of temperature T(\frac{3}{4},0) = -9/16. and the hottest value is T(-(3/4),\frac{\sqrt{7}}{4})=\frac{5}{16}.

Step-by-step explanation:

We need to take the derivative with respect of x and y, and equal to zero to find the local minimums.

The temperature equation is:

T(x,y)=x^{2}+2y^{2}-\frac{3}{2}x

Let's take the partials derivatives.

T_{x}(x,y)=2x-\frac{3}{2}=0

T_{y}(x,y)=4y=0

So, we can find the critical point (x,y) of T(x,y).

2x-\frac{3}{2}=0

x=\frac{3}{4}

4y=0

y=0

The critical point is (3/4,0) so the temperature at this point is: T(\frac{3}{4},0)=(\frac{3}{4})^{2}+2(0)^{2}-(\frac{3}{2})(\frac{3}{4})

T(\frac{3}{4},0)=-\frac{9}{16}    

Now, we need to evaluate the boundary condition.

x^{2}+y^{2}=1

We can solve this equation for y and evaluate this value in the temperature.

y=\pm \sqrt{1-x^{2}}

T(x,\sqrt{1-x^{2}})=x^{2}+2(1-x^{2})-\frac{3}{2}x  

T(x,\sqrt{1-x^{2}})=-x^{2}-\frac{3}{2}x+2

Now, let's find the critical point again, as we did above.

T_{x}(x,\sqrt{1-x^{2}})=-2x-\frac{3}{2}=0            

x=-\frac{3}{4}    

Evaluating T(x,y) at this point, we have:

T(-(3/4),\sqrt{1-(-3/4)^{2}})=-(-\frac{3}{4})^{2}-\frac{3}{2}(-\frac{3}{4})+2  

T(-(3/4),\frac{\sqrt{7}}{4})=\frac{5}{16}

Now, we can see that at point (3/4,0) we have the coldest value of temperature T(\frac{3}{4},0) = -9/16. On the other hand, at the point -(3/4),\frac{\sqrt{7}}{4}) we have the hottest value of temperature, it is T(-(3/4),\frac{\sqrt{7}}{4})=\frac{5}{16}.

I hope it helps you!

You might be interested in
Jen buys 4 tires for 272$
Nostrana [21]
What are you asking if all together they are 272 then each tire would cost 68 dollars
8 0
2 years ago
If a sample mean is 32, which of the following is most likely the range of
Nataly [62]

Answer:

(28.36)

Step-by-step explanation:

3 0
1 year ago
What is x squared minus 18x equal to negative 32
lisabon 2012 [21]

Answer:

x=16

x=2

Step-by-step explanation:

x^2-18x=-32\\x^2-18x+32=0\\(x-16)(x-2)=0\\\\x-16=0\\x=16\\\\x-2=0\\x=2

8 0
2 years ago
Solve this please and explain how you do it thank you
andrew11 [14]

Answer:

Step-by-step explanation:

7x=14

X=2

5 0
2 years ago
Statistics:
Dima020 [189]

Answer:

Step-by-step explanation:

a) sub in 600 for 6

600-25/40 =14.375

b) switch x and y

x=y-25/40

Solve for y

40x=y-25

y=40x+25

c)Plug in 2.5 for z(x)

2.5= x-25/40

100=x-25

x=125

8 0
3 years ago
Other questions:
  • How many different ways can the line above be named? What are those names?
    7·1 answer
  • Please someone help ASAP
    5·1 answer
  • Maria is attempting to factor the expression In her first step, she rewrites the expression as. Which of the following statement
    5·1 answer
  • Harry is at the depot in Dover and has deliveries to make in Edinburgh, Glasgow and Aberdeen. From Dover which of the deliveries
    11·1 answer
  • A school paid $31.25 for each calculator. If the school spent $500 on calculators, how many did the school buy?
    7·2 answers
  • Divide.<br> 1/3 divide 1/2
    15·1 answer
  • Lois is four times as old as her son Don. The sum of their ages is 40. How old is Don?
    5·2 answers
  • I need help with solving an equation and solution for this someone please
    10·1 answer
  • PLEASE HELP IS THIS CORRECT GUYS YES OR NO!
    9·1 answer
  • Over the weekend, Braden drank 5/6 of a bottle of soda and Zachary drank 1/3 of a bottle.
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!