The function f(t)=-16t^2+10

Find the max and min values of f(x,y,z)=x+y-z on the sphere x^2+y^2+z^2=81

Anton [14]

Using Lagrange multipliers, we have the Lagrangian

$L(x,y,z,\lambda)=x+y-z+\lambda(x^2+y^2+z^2-81)$

with partial derivatives (set equal to 0)

$L_x=1+2\lambda x=0\implies x=-\dfrac1{2\lambda}$
$L_y=1+2\lambda y=0\implies y=-\dfrac1{2\lambda}$
$L_z=-1+2\lambda z=0\implies z=\dfrac1{2\lambda}$
$L_\lambda=x^2+y^2+z^2-81=0\implies x^2+y^2+z^2=81$

Substituting the first three equations into the fourth allows us to solve for $\lambda$ :

$x^2+y^2+z^2=\dfrac1{4\lambda^2}+\dfrac1{4\lambda^2}+\dfrac1{4\lambda^2}=81\implies\lambda=\pm\dfrac1{6\sqrt3}$

For each possible value of $\lambda$ , we get two corresponding critical points at $(\mp3\sqrt3,\mp3\sqrt3,\pm3\sqrt3)$ .

At these points, respectively, we get a maximum value of $f(3\sqrt3,3\sqrt3,-3\sqrt3)=9\sqrt3$ and a minimum value of $f(-3\sqrt3,-3\sqrt3,3\sqrt3)=-9\sqrt3$ .

5 0

3 years ago

LM is a perpendicular bisector of NP The length of LN is 12w +7, and the length of LP is 15w 5. What is the length of LN?

Mazyrski [523]

Answer:

$LN = 55$

Step-by-step explanation:

Given

$LN = 12w + 7$

$LP = 15w - 5$

Required

Determine LN

Since LM is a bisector, then we have:

$LP = LN$ (See attachment for illustration)

$15w - 5 = 12w + 7$

Collect Like Terms

$15w - 12w = 5 +7$

$3w = 12$

Solve for w

$w = 12/3$

$w = 4$

LN is calculated as thus:

$LN = 12w + 7$

Substitute 4 for w

$LN = 12 * 4 + 7$

$LN = 48 + 7$

$LN = 55$

8 0

2 years ago

Solve the system of linear equations.

sweet-ann [11.9K]

Answer:

dependent system
x = 2 -a
y = 1 +a
z = a

Step-by-step explanation:

Let's solve this by eliminating z, then we'll go from there.

Add 6 times the second equation to the first.

(3x -3y +6z) +6(x +2y -z) = (3) +6(4)

9x +9y = 27 . . . simplify

x + y = 3 . . . . . . divide by 9 [eq4]

Add 13 times the second equation to the third.

(5x -8y +13z) +13(x +2y -z) = (2) +13(4)

18x +18y = 54

x + y = 3 . . . . . . divide by 18 [eq5]

Equations [eq4] and [eq5] are identical. This tells us the system is dependent, and has an infinite number of solutions. We can find them in terms of z:

y = 3 -x . . . . solve eq5 for y

x +2(3 -x) -z = 4 . . . . substitute into the second equation

-x +6 -z = 4

x = 2 - z . . . . . . add x-4

y = 3 -(2 -z)

y = z +1

So far, we have written the solutions in terms of z. If we use the parameter "a", we can write the solutions as ...

x = 2 -a

y = 1 +a

z = a

_____

<em>Check</em>

First equation:

3(2-a) -3(a+1) +6a = 3

6 -3a -3a -3 +6a = 3 . . . true

Second equation:

(2-a) +2(a+1) -a = 4

2 -a +2a +2 -a = 4 . . . true

Third equation:

5(2-a) -8(a+1) +13a = 2

10 -5a -8a -8 +13a = 2 . . . true

Our solution checks algebraically.

6 0

2 years ago

Harold wrote this equation to model the level of water in a pool over time. The variable x represents time in hours. f(x) = 3,50

lina2011 [118]

Answer:

The water level is falling.

The initial level of water in the pool was 3,500 units

The water was 2,600 units high after 4 hours.

Step-by-step explanation:

The given function that models the water level is

$f(x)=3,500-225x$

where $x$ represents time in hours.

The function represents a straight line that has slope $m=-225$

Since the slope is negative, it means the water level is falling.

The initial level of water in the pool can found when we put $x=0$ into the function.

$f(0)=3,500-225(0)$

$f(0)=3,500$ , hence the initial level is 3,500.

To determine the level of water in the pool after 14 hours, we put $x=14$ into the equation to get;

$f(14)=3,500-225(14)$

$f(14)=3,500-3150$

$f(14)=350$

To determine the water level after 4 hours we put $x=4$

$f(4)=3,500-225(4)$

$f(4)=3,500-900$

$f(14)=2,600$

4 0

3 years ago

Read 2 more answers

Yuet solved the equation 6 a minus 2 b = 12 for a. Her steps are shown below. 1. Subtract 2b: 6 a = 12 minus 12 b 2. Divide by 6

Ber [7]

Question:

Yuet solved the equation $6a - 2b = 12$ for a.

Her steps are shown below.

1. Subtract 2b: $6a = 12 - 2b$

2. Divide by 6: $a = 2 - \frac{b}{3}$

Answer:

In step 1 she needed to add 2b to both sides of the equation.

Step-by-step explanation:

Given:

The above steps shows how Yuet solved an equation

Required:

True statement about Yuet's work

The implication of the statement is to state what Yuet should have done instead of what she did.

In step 1, she subtracted 2b from both sides of the equation; this is wrong.

Instead of subtracting, she ought to add 2b to both sides of the equation.

The correct steps and result is as follows:

1. Add 2b: $6a = 12 + 2b$

2. Divide by 6: $a = 2 + \frac{b}{3}$

Hence, we can conclude that in step 1 she needed to add 2b to both sides of the equation.

4 0

3 years ago

Read 2 more answers