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3 years ago

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Mathematics

1 answer:

Fynjy0 [20]3 years ago

4 0

The total is 219 miles.

He rides at 38 mph and walks at 4 mph for 8 hours. If he rides for h hours, he walks for 8-h hours.

Set them equal to each other:

219=38h+4(8-h)

Solve and you should get 5.5 hours on the bicycle and 2.5 hours walking.

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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

Someone pls help! <br> write the expression in radical form. 2 4/5

kipiarov [429]

Answer:

$\sqrt[5]{2^4}$

Step-by-step explanation:

Maybe you want 2^(4/5) in radical form.

The denominator of the fractional power is the index of the root. Either the inside or the outside can be raised to the power of the numerator.

$2^{\frac{4}{5}}=\boxed{\sqrt[5]{2^4}=(\sqrt[5]{2})^4}$

In many cases, it is preferred to keep the power inside the radical symbol.

7 0

3 years ago

You are constructing deck chairs that require 12 2/3 feet of lumber each. How many feet of lumber will you use to make 15 chairs

Mrrafil [7]

As each chair needs 12.6667 feet lumber.So 15 chairs will need 15*12.6667=190.0005 lumber

5 0

3 years ago

A visual-effects model maker for a movie draws a spaceship using a ratio of 1 : 24. The drawing of the spaceship is 22 inches lo

Kay [80]

Answer:

Length of spaceship in movie = 528

Step-by-step explanation:

Given that:

Drawing ratio of spaceship = 1 : 24

It means that 1 inch on drawing equals 24 inches in movie

Drawing of spaceship = 22 inches

Length of spaceship in movie = x

As the relationship is proportional,

1 : 24 :: 22 : x

Product of mean = Product of extreme

24*22 = x

x = 528

Hence,

Length of spaceship in movie = 528

8 0

3 years ago

The following data points represent the number of toppings each customer at I Scream Sundae Shop put on their ice cream.

dolphi86 [110]

Answer:

5.5

Step-by-step explanation:

6 0

3 years ago

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