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Sergeeva-Olga [200]

3 years ago

14

Three postal workers can sort a stack of mail in 30 minutes 45 minutes, and 90 minutes, respectively Find how long it takes them

to sort the mail if all three work together

1 answer:

gizmo_the_mogwai [7]3 years ago

5 0

Person 1: They can sort the mail in 30 minutes, so they do 1/30th of the job in one minute.

Person 2: They can sort the mail in 45 minutes, so they do 1/45th of the job in one minute.

Person 3: They can sort the mail in 90 minutes, so they do 1/90th of the job in one minute.

Together, they do (1/30 + 1/45 + 1/90)th of the job in one minute.

1/30 + 1/45 + 1/90 = 3/90 + 2/90 + 1/90 = 6/90

So they do 6/90 of the job together in 1 minute.

If "t" is the time it takes them working together to do the job, then we need to solve

6/90 · t = 1 (that's 1 for 1 whole job)

Solving that, you get t = 90/6 = 15

So, it'd take them 15 minutes working together to get that job done.

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Use the method of Lagrange multipliers to find the dimensions of the rectangle of greatest area that can be inscribed in the ell

Tanzania [10]

Answer:

Length (parallel to the x-axis): $2 \sqrt{2}$ ;

Height (parallel to the y-axis): $4\sqrt{2}$ .

Step-by-step explanation:

Let the top-right vertice of this rectangle $(x,y)$ . $x, y >0$ . The opposite vertice will be at $(-x, -y)$ . The length the rectangle will be $2x$ while its height will be $2y$ .

Function that needs to be maximized: $f(x, y) = (2x)(2y) = 4xy$ .

The rectangle is inscribed in the ellipse. As a result, all its vertices shall be on the ellipse. In other words, they should satisfy the equation for the ellipse. Hence that equation will be the equation for the constraint on x and y.

For Lagrange's Multipliers to work, the constraint shall be in the form: $g(x, y) =k$ . In this case

$\displaystyle g(x, y) = \frac{x^{2}}{4} + \frac{y^{2}}{16}$ .

Start by finding the first derivatives of $f(x, y)$ and $g(x, y)$ with respect to x and y, respectively:

$f_x = y$ ,
$f_y = x$ .

$\displaystyle g_x = \frac{x}{2}$ ,
$\displaystyle g_y = \frac{y}{8}$ .

This method asks for a non-zero constant, $\lambda$ , to satisfy the equations:

$f_x = \lambda g_x$ , and

$f_y = \lambda g_y$ .

(Note that this method still applies even if there are more than two variables.)

That's two equations for three variables. Don't panic. The constraint itself acts as the third equation of this system:

$g(x, y) = k$ .

$\displaystyle \left\{ \begin{aligned} &y = \frac{\lambda x}{2} && (a)\\ &x = \frac{\lambda y}{8} && (b)\\ & \frac{x^{2}}{4} + \frac{y^{2}}{16} = 1 && (c)\end{aligned}\right.$ .

Replace the $y$ in equation $(b)$ with the right-hand side of equation $(b)$ .

$\displaystyle x = \lambda \frac{\lambda \cdot \dfrac{x}{2}}{8} = \frac{\lambda^{2} x}{16}$ .

Before dividing both sides by $x$ , make sure whether $x = 0$ .

If $x = 0$ , the area of the rectangle will equal to zero. That's likely not a solution.

If $x \neq 0$ , divide both sides by $x$ , $\lambda = \pm 4$ . Hence by equation $(b)$ , $y = 2x$ . Replace the $y$ in equation $(c)$ with this expression to obtain (given that $x, y >0$ ) $x = \sqrt{2}$ . Hence $y = 2x = 2\sqrt{2}$ . The length of the rectangle will be $2x = 2\sqrt{2}$ while the height will be $2y = 4\sqrt{2}$ . If there's more than one possible solutions, evaluate the function that needs to be maximized at each point. Choose the point that gives the maximum value.

7 0

3 years ago

I have 5 minuets please answer (image included)

Alex_Xolod [135]

Answer:

The second one !!

Step-by-step explanation:

You always start from the origin... (0, 0) !! From the bottom of that function, from the origin, you would move to the right 1, down 4 in order to get where the graph is in the picture !!

i hope this made sense... have a good day !

8 0

3 years ago

the perimeter of a rectangle is 210cm. the ratio of the width to the length is 3:4. what is the length of the rectangle?

sveta [45]

Answer:

length = 60 cm

Step-by-step explanation:

ratio of width : length = 3 : 4 = 3x : 4x

perimeter = (2 × width ) + (2 × length) = 6x +8x = 14x

now perimeter = 210, hence

14x = 210 ( divide both sides by 14 )

x = 15

hence length = 4 × 15 = 60 cm

4 0

4 years ago

What mathematicians helped to discover alternatives to euclidean geometry in the nineteenth century

Likurg_2 [28]

Answer:

Nikolai Lobachevsky and Bernhard Riemann

Step-by-step explanation:

Nikolai Lobachevsky (A russian mathematician born in 1792) and Bernhard Riemann (A german mathematician born in 1826) are the mathematicians that helped to discover alternatives to euclidean geometry in the nineteenth century.

7 0

4 years ago

Which is not a flavor that your taste buds sense?<br> A.sour<br> B. salty<br> C. bitter<br> D. warm

levacccp [35]

Answer:

warm

Step-by-step explanation:

even though our mouth can tell that the food is warm, the tastebuds don't. they can only tell the other 3.

8 0

3 years ago