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4 years ago

Please help! I only need the equations not the graphing

Mathematics

1 answer:

Anvisha [2.4K]4 years ago

6 0

Answer:

x + y ≥ 4

8x + 10y ≤ 70

Step-by-step explanation:

1. State the variables

According to the graph, "Number of green towels" is on the x-axis, and "Number of blue towels" is on the y-axis, so:

let x be the number of green towels.

let y be the number of blue towels

2. Write an equation for the total number of towels.

The sum of number of green towels and the number of blue towels is the total number of towels.

x + y ≥ 4

"She wants to buy at least 4 towels" means she needs 4 or more. Use the ≥ symbol instead of > (which would mean more than 4 towels, not only 4).

3. Write an equation for the money she will spend.

Multiply each variable by the cost of the towel colour. Add them together. Equate the total as 70 because she is only willing to spend 70.

8x + 10y ≤ 70

"She does not want to spend more than $70" means she is willing to spend 70, but no more. This is why we use ≤ and not < (which would mean she would not spend 70, only less than 70).

The equations are x + y ≥ 4 and 8x + 10y ≤ 70.

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$\bf g(x,y)=(x+1/2)^2+(y+1/2)^2-25/2=0$

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$\bf \nabla f=(\displaystyle\frac{\partial f}{\partial x},\displaystyle\frac{\partial f}{\partial y})=(2x,2y)\\\\\nabla g=(\displaystyle\frac{\partial g}{\partial x},\displaystyle\frac{\partial g}{\partial y})=(2x+1,2y+1)$

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$\bf \nabla f=\lambda\nabla g$

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$\bf (2x,2y)=\lambda(2x+1,2y+1)\Rightarrow 2x=\lambda (2x+1)\;,2y=\lambda (2y+1)$

we can assume (x,y)≠ (-1/2, -1/2) since that point is not in the restriction, so

$\bf \lambda=\displaystyle\frac{2x}{(2x+1)} \;,\lambda=\displaystyle\frac{2y}{(2y+1)}\Rightarrow \displaystyle\frac{2x}{(2x+1)}=\displaystyle\frac{2y}{(2y+1)}\Rightarrow\\\\\Rightarrow 2x(2y+1)=2y(2x+1)\Rightarrow 4xy+2x=4xy+2y\Rightarrow\\\\\Rightarrow x=y$

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Replacing these values in f, we see that

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We have then,

(-3,-3) is the farthest from the origin

(2,2) is the closest to the origin.

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