Answer:
the minimum is located in x = -5/3 , y= -5/3
Step-by-step explanation:
for the function
f(x,y)=2x + 2y
we define the function g(x)=9x² - 9xy + 9y² - 25 ( for g(x)=0 we get the constrain)
then using Lagrange multipliers f(x) is maximum when
fx-λgx(x)=0 → 2 - λ (9*2x - 9*y)=0 →
fy-λgy(x)=0 → 2 - λ (9*2y - 9*x)=0
g(x) =0 → 9x² - 9xy + 9y² - 25 = 0
subtracting the second equation to the first we get:
2 - λ (9*2y - 9*x) - (2 - λ (9*2x - 9*y))=0
- 18*y + 9*x + 18*x - 9*y = 0
27*y = 27 x → x=y
thus
9x² - 9xy + 9y² - 25 = 0
9x² - 9x² + 9x² - 25 = 0
9x² = 25
x = ±5/3
thus
y = ±5/3
for x=5/3 and y=5/3 → f(x)= 20/3 (maximum) , while for x = -5/3 , y= -5/3 → f(x)= -20/3 (minimum)
finally evaluating the function in the boundary , we know because of the symmetry of f and g with respect to x and y that the maximum and minimum are located in x=y
thus the minimum is located in x = -5/3 , y= -5/3
Answer:
x = 3
y = 1
Domain: x < 3 U x > 3
Range: y < 1 U y > 1
Step-by-step explanation:
Vertical asymptote: x = 3
Horizontal asymptote: y = 1
Domain: x < 3 U x > 3
Range: y < 1 U y > 1
Answer:
Step-by-step explanation:
Surface area is finding the area of each face and adding them up. So a cube will have 6 faces, each of them a square. So find the area of one face and you can just add it 6 times. If the faces are not all the same you have to find them seperately. I won't do all of your questions, but let me know if you need help with a specific one
1) there are two trianglular faces and three rectangular ones. The triangles are the same so let's do those. They have the base and height of 4 and 3, so area of both will be .5*4*3 = 6, so together that makes 12. now the three rectangular sides. Right away you can see there is a 3 by 10 one and a 4 by 10 one. These ahve areas of 30 and 40 respectfully. The final face has one length of 10 but the last one is actually the last side of the right triangle faces. Use the Pythagorean theorem to find it, it word out to be 5. So now we know the last face is 10 by 5 which means an area of 50. So let's add them all together. 5 faces total with areas of 30, 40, 50, 6 and 6. Adding them together makes 136.
If you don't get how I did any of that let me know. In general it's a good idea to mark every face and then find what their sides are. I'd be happy to help with specific problems.
Now the second part, again I will do the second one because cones are little less intuitive. a cone has two faces, the circular base and the actual cone shape. the surface area of the cone shape is pi*r*L where r is the radius of the base and L is the length of one of the slanted sides. Now, let's find the area of the two faces. Area of a circle is pi*r^2 so that's pi*4^2 = 16*pi in number 2. Then the area of the cone part is pi*r*L = pi*4*14 = 56*pi. now we sum all the areas of the faces up and get 16*pi + 56*pi = 72*pi.
Again, I'm happy to answer specific questions, so let me know if you have any. just keep in mind what I told you so far, find each face, then find each face's area then sum them up. For cylinders the tubes turn into rectangles where one side is the length of the tube and the other is the circumference, while cones fave two faces, the circular base and the cone shape itself with an area of pi*r*L. if you want me to check your answers I will totally do that too.
500 is what you already have if it grows 2.75% a month you multiply it by the month and add it to 500
500+2.75%m
