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3 years ago

What is the constant of proportionality for the relationship shown in the table? x 2 4 6 8 y 1 2 3 4

1 answer:

6 0

What is the constant of proportionality for the relationship shown in the table? x 2 4 6 8 y 1 2 3 4

☆☆☆☆☆☆1/2
2
4
8

Y=kx where k is the constant proportionality
y=kx
(1)=k(2)
1/2=k2/2
1/2=k

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Which equation is represented by the table

lyudmila [28]

Answer:

B. b = 3a + 2

Step-by-step explanation:

We can write the equation in slope-intercept form as b = ma + c, where,

m = slope/rate of change

c = y-intercept/initial value

✔️Find m using any two given pair of values, say (2, 8) and (4, 14):

Rate of change (m) = change in b/change in a

m = (14 - 8)/(4 - 2)

m = 6/2

m = 3

✔️Find c by substituting (a, b) = (2, 8) and m = 3 into b = ma + c. Thus:

8 = 3(2) + c

8 = 6 + c

8 - 6 = c

2 = c

c = 2

✔️Write the equation by substituting m = 3 and c = 2 into b = ma + c. Thus:

b = 3a + 2

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3 years ago

Find the maximum and minimum values attained by f(x, y, z) = 5xyz on the unit ball x2 + y2 + z2 ≤ 1.

Allushta [10]

Check for critical points within the unit ball by solving for when the first-order partial derivatives vanish:
$f_x=5yz=0\implies y=0\text{ or }z=0$
$f_y=5xz=0\implies x=0\text{ or }z=0$
$f_z=5xy=0\implies x=0\text{ or }y=0$

Taken together, we find that (0, 0, 0) appears to be the only critical point on $f$ within the ball. At this point, we have $f(0,0,0)=0$ .

Now let's use the method of Lagrange multipliers to look for critical points on the boundary. We have the Lagrangian

$L(x,y,z,\lambda)=5xyz+\lambda(x^2+y^2+z^2-1)$

with partial derivatives (set to 0)

$L_x=5yz+2\lambda x=0$
$L_y=5xz+2\lambda y=0$
$L_z=5xy+2\lambda z=0$
$L_\lambda=x^2+y^2+z^2-1=0$

We then observe that

$xL_x+yL_y+zL_z=0\implies15xyz+2\lambda=0\implies\lambda=-\dfrac{15xyz}2$

So, ignoring the critical point we've already found at (0, 0, 0),

$5yz+2\left(-\dfrac{15xyz}2\right)x=0\implies5yz(1-3x^2)=0\implies x=\pm\dfrac1{\sqrt3}$
$5xz+2\left(-\dfrac{15xyz}2\right)y=0\implies5xz(1-3y^2)=0\implies y=\pm\dfrac1{\sqrt3}$
$5xy+2\left(-\dfrac{15xyz}2\right)z=0\implies5xy(1-3z^2)=0\implies z=\pm\dfrac1{\sqrt3}$

So ultimately, we have 9 critical points - 1 at the origin (0, 0, 0), and 8 at the various combinations of $\left(\pm\dfrac1{\sqrt3},\pm\dfrac1{\sqrt3},\pm\dfrac1{\sqrt3}\right)$ , at which points we get a value of either of $\pm\dfrac5{\sqrt3}$ , with the maximum being the positive value and the minimum being the negative one.

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What is 9 and 19/18 simplified

Anastasy [175]

10 1/18...........................................

...................................................

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3 years ago

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A first number plus twice a second number is 13. Twice the first number plus the second totals 14. find the numbers

sashaice [31]

Answer: