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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

a coffee retailer has two grades of decaffeinated coffee beans, one wholesales at $4.00 a pound and the other for 6.50 a pound.

RoseWind [281]

Answer:

Step-by-step explanation:

To solve mixture problems like this the best way is to make a table. Here's the basic format of the table we will use, calling the first coffee C1 and the second coffee C2:

#lbs x $/lb = Total

C1

C2

Mix

This is the table we will fill in. First, we were given the cost per pound of each type of coffee, so we put that in first:

#lbs x $/lb = Total

C1 4

C2 6.50

Mix

Next, we are told that she wants to make a mix of these coffees so she has a total number of pounds as 150 and that she wants it to cost $4.75 per pound. That goes in next:

#lbs x $/lb = Total

C1 4

C2 6.50

Mix 150 * 4.75

That asterisk is the multiplication sign.

Here's where we are now responsible for filling out the rest of the chart. If she needs to mix these 2 coffees to get a mix that is `150 pounds, she can have x pounds of C1 which means that she has to have 150 - x of C2:

#lbs x $/lb = Total

C1 x * 4

C2 150 - x * 6.50

Mix 150 * 4.75

Now all we have left to do is what the table tells us to do, which is to multiply the first 2 columns and put that product under the Total column:

#lbs x $/lb = Total

C1 x * 4 = 4x

C2 150 - x * 6.50 = 975 - 6.50x

Mix 150 * 4.75 = 712.50

Since we have to add the 2 coffees together to get the pounds of mix in the first column, we also have to add the cost of those coffees together to get the total cost of the mix. The equation looks like this:

4x + 975 - 6.50x = 712.50 and combining like terms:

-2.5x = -262.5 so

x = 105

That means that there is 105 pounds of coffee 1 and 150 - 105 pounds of coffee 2.

C1 = 105 pounds, C2 = 45 pounds

3 0

3 years ago

Here is a 5 by 8 grid

Ksivusya [100]

On solving the linear equation, There are in total 24 unshaded squares and 16 shaded squares.

<h3>What is a linear equation?</h3>

Ax+By=C is the usual form for two-variable linear equations. A typical form linear equation is, for instance, 2x+3y=5. When an equation is provided in this format, finding both intercepts is rather simple (x and y).

squares, where x= number of squares

2x = shaded squares

Number of unshaded squares = 3x.

This is due to the fact that there are 3 times as many unshaded squares as there are shaded squares (2 times more unshaded = 3*). (shaded)

Overall, there are 40 squares (5 * 8), therefore 40 total squares—shaded and unshaded—are present.

2x+3x = 40

5x = 40

x = 40/5

x = 8

There are 3×8 = 24 unshaded squares

8×2 = 16 shaded squares in all.

Therefore, you may shade in 8 squares whatever you choose.

To learn more about linear equation from given link

brainly.com/question/2030026

#SPJ9

5 0

1 year ago

What is nearest whole number to 63.62

kvv77 [185]

Answer:

64

Step-by-step explanation:

Looking at the number 63.62, you see that 6 comes immediately after the decimal point.

Now if the number after the decimal point is more than 5 we round up by adding one to the number just before the decimal point which is 3 in this case.

Assuming the number after the decimal point was less than 5 we would round down adding nothing to the number just before the decimal point.

8 0

4 years ago

Find the largest interval which includes x = 0 for which the given initial-value problem has a unique solution. (Enter your answ

Ivan

Answer:

$(-\infty,3)$