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4 years ago

What is 2/3 ÷ 4/5 = ?

Mathematics

1 answer:

11Alexandr11 [23.1K]4 years ago

3 0

Answer:

5/6

Step-by-step explanation:

2/3 ÷ 4/5 = ⇒ changing division to multiplication by reciprocal of the second fraction
2/3 × 5/4 = ⇒ simplify and multiply numerators and denominators
5/6 ⇒ here is the answer

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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.

5 0

3 years ago

(-5)^2-2x(-9)+6 Please show work and step by step.

Varvara68 [4.7K]

Answer:

Step-by-step explanation:

Use PEMDAS:

P Parentheses first

E Exponents (ie Powers and Square Roots, etc.)

MD Multiplication and Division (left-to-right)

AS Addition and Subtraction (left-to-right)

==============================================

(-5)² - 2 × (-9) + 6 first Exponents

25 - 2 × (-9) + 6 next Multiplication

25 + 18 + 6 next Addition

43 + 6 = 49

5 0

3 years ago

Two bicyclists are 7/8 of the way through a mile long tunnel when a train approaches the closer end at 40 mph. The riders take o

Anastaziya [24]

Answer:

the cyclists rode at 35 mph

Step-by-step explanation:

Assuming that the cyclists stopped, and accelerated instantaneously at the same speed than before but in opposite direction , then

distance= speed*time

since the cyclists and the train reaches the end of the tunnel at the same time and denoting L as the length of the tunnel :

time = distance covered by cyclists / speed of cyclists = distance covered by train / speed of the train

thus denoting v as the speed of the cyclists :

7/8*L / v = L / 40 mph

v = 7/8 * 40 mph = 35 mph

v= 35 mph

thus the cyclists rode at 35 mph

6 0

4 years ago

What are the coordinates of the centroid of a triangle with vertices J(−4, 2) , K(2, 4) , and L(0, −2) ?

Kryger [21]

Given the coordinates of the three vertices of a triangle ABC,
the centroid coordinates are (x1+x2+x3)/3, (y1+y2+y3)/3
so (-4+2+0)/3=-2/3, ]2+4+(-2)]/3=4/3
so the coordinates are (-2/3, 4/3)

3 0

3 years ago

Read 2 more answers

For four weeks in June Cameron baked 3 1/4 miles each week and swim 2 1/2 miles each week for three weeks in July he baked 4 3/4

Tems11 [23]

For June:
Bike:
(4) * (3 1/4) = 13 miles
Swim:
(4) * (2 1/2) = 10 miles
Total:
13 + 10 = 23 miles
For July:
Bike:
(3) * (4 3/4) = 14.25 miles
Swim:
(3) * (3 1/2) = 10.5 miles
Total:
14.25 + 10.5 = 24.75 miles
The difference between both months is:
24.75 - 23 = 1.75 miles
Answer:
the total distance Cameron bike and swim in July compared to the total distance he bike in swim in June was 1.75 miles greater

6 0

3 years ago