At what point on the paraboloid y = x2 + z2 is the tangent plane parallel to the plane 3x + 2y + 7z = 2? (if an answer does not exist, enter dne.) (x, y, z) =

Final answer:

The point on the paraboloid y = x^2 + z^2 where the tangent plane is parallel to the plane 3x + 2y + 7z = 2 is found by setting the normal vectors proportional. Solving the equations, the point is (3/4, 29/8, 7/4).

Explanation:

To find the point on the paraboloid y = x2 + z2 where the tangent plane is parallel to the plane 3x + 2y + 7z = 2, we first need to determine the normal vector of the given plane. The normal vector of the plane is defined by its coefficients, which are (3, 2, 7). For the paraboloid, we can find the normal vector at any point by taking the gradient of the function y.

The gradient of y with respect to x and z is (2x, 1, 2z). A tangent plane to the paraboloid at point (x, y, z) will have this gradient as its normal vector. To find the point where this tangent plane is parallel to the given plane, we set the gradients equal to each other up to a constant factor because parallel planes have proportional normal vectors.

Therefore, we solve the equations:

2x = 3k

1 = 2k

2z = 7k

From the second equation, k = 1/2. Substituting k into the other equations, we find x = 3/4 and z = 7/4. Now we can substitute x and z into the equation of the paraboloid to find y:

y = (3/4)2 + (7/4)2 = 9/16 + 49/16 = 58/16 = 29/8.

The point on the paraboloid where the tangent plane is parallel to the given plane is (3/4, 29/8, 7/4).

The product of two numbers is 44. find the sum of the two numbers s(x) as a function of one of the numbers, x. s(x)

Hank's pickup can travel 42 miles on 3 gallons of gas. How many gallons will​ Hank's pickup need to travel 28 ​miles?

Frederick needs to borrow $3000 for a down payment on a new car. His uncle decided to loan him the money as a 4.5% loan. Frederick doesn't want to pay more than $810 in interest. How many years does Frederick have to pay back the loan?

WHAT IS 7.06 DIVIDED BY 0353

Simplify (x3+ 8x + 5) − (2x2− 3x − 12). −x2+ 11x + 17 x3− 2x2+ 11x + 17 −x2+ 5x − 7 x3− 2x2+ 5x − 7

Write a linear model for the cost of tickets based on the number of tickets a fan buys.

What is the order of the set A= (-1,20,3,17,5)
A. 3
B. 5
C. -1, 20, 3, 17, 5
D. -1,3, 5, 17, 20

If two lines don’t intersect, then is it parrell (( true or false ))

Mayumi is constructing a line through point P that is perpendicular to RS←→. A horizontal line segment R S and a point P are drawn. The point P is above the horizontal line segment R S. She places her compass on point P to construct an arc. What must be true about the width of the compass opening when Mayumi draws the arc? It must be greater than 12RS. It must be equal to PS . It must be wide enough so that the arc intersects RS←→ in two points. It must be equal to PR .

What is the degree of 12x⁴-8x+4x²-3
A)12
B)4
C)3
D)8

if kyra cuts a 12" string into 3/4 strips for her art project , how many strips will she have
A) 9
B) 12
C) 15
D) 16

3(x−7)=6(x−10)
Solve For X
3x-21=6x-60

What 8 letter word can have a letter taken away and it still makes a word. take another letter away and it still makes a word. keep on doing that until you have one letter left - that is still a word. what is the 8 letter word...and what are the other smaller words?

Evaluate the expression for x = –3

What is the value for x?
Enter your answer in the box.

The current diameter of a tree trunk is 10 inches. the trunk diameter increases at a rate of 2 inches per year. an insect burrows into the trunk and lays its eggs in the secondary xylem, 1 inch out from the center of the tree. how far from the center will the dormant eggs be located in two years?

Suppose A and B are points. How many lines contain both A and B?

Can someone please help me with an algebra 1 problem. I'm really stuck.
Write each interval as an inequality. Then graph the solutions
24. [-4, 5]
please help, thank you so much.

at the start of his diet kerrin weighed 15 stone 10 pounds three months later he weighed 14 stone 8 pounds there are 14 pounds in a stone

What is the value of x?

x−25=7

Enter your answer in the box in simplest form.
x =

Solve each system by substitution {4p+2q=8 and q=2p+1} {x+6y=2 and 5x+4y=36}

is the choice vertical angle theorem correct?

what is f(-5) if f(x) equals |2x-1|+10?

Find the 40th term of the following arithmetic sequence:16,6,-4,-14

If x y z are positive integers and 3x = 4y = 7z, then the least possible value of x + y + z is

Solve for y, give the slope and y intercept 5y=20, -y + 3=15x, y +1/3x=17

what is m Help Please

PLEASE HELP!

Translate each phrase into a numerical expression.

Twice the opposite of eleven.

Which set of integers is a pythagorean triple?
a. 2, 3, 6
b. 3, 4, 5
c. 4, 5, 9
d. 5, 7, 12?

3tens 12ones= —— tens 2 ones