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ExtremeBDS [4]
3 years ago
11

The cost of floor material is $2840 and the installation cost $1.90 per square foot. If the total is $5120, find the area of the

room in square feet.
Mathematics
1 answer:
djyliett [7]3 years ago
6 0
1,200.
$5120-$2840=2280
2280/1.90= 1,200
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Consider the function f(x, y) = x2 + xy + y2 defined on the unit disc, namely, d = {(x, y)| x2 + y2 ≤ 1}. use the method of lagr
Pani-rosa [81]
First we note that the partial derivatives vanish simultaneously at one point:

\begin{cases}f_x=2x+y=0\\f_y=x+2y=0\end{cases}\implies(x,y)=(0,0)

so there is one critical point within the region D.

The Lagrangian is

L(x,y,\lambda)=x^2+xy+y^2+\lambda(x^2+y^2-1)

and has partial derivatives

L_x=2x+y+2\lambda x
L_y=x+2y+2\lambda y
L_\lambda=x^2+y^2-1

Set each partial derivative to 0 to find the possible critical points within the disk D. Then we notice that

yL_x=2xy+y^2+2\lambda xy=0
xL_y=x^2+2xy+2\lambda xy=0
L_\lambda=0\implies x^2+y^2=1

\implies xL_y-yL_x=x^2-y^2=0

Since x^2+y^2=1, we have

x^2-y^2=x^2+y^2-2y^2=0\implies1=2y^2\implies y^2=\dfrac12\implies y=\pm\dfrac1{\sqrt2}

And since x^2-y^2=0, or x^2=y^2, we also have

x=\pm\dfrac1{\sqrt2}

So we have four possible additional critical points to consider:

f(0,0)=0
f\left(-\dfrac1{\sqrt2},-\dfrac1{\sqrt2}\right)=\dfrac32
f\left(-\dfrac1{\sqrt2},\dfrac1{\sqrt2}\right)=\dfrac12
f\left(\dfrac1{\sqrt2},-\dfrac1{\sqrt2}\right)=\dfrac12
f\left(\dfrac1{\sqrt2},\dfrac1{\sqrt2}\right)=\dfrac32

It should be clear enough which of these correspond to the absolute extrema of f over D.
8 0
3 years ago
The two shortest sides of a right triangle are 10 in. And 24 in. long. What is the length of the shortest side of a similar righ
ivanzaharov [21]
If the two shortest sides of the triangle are 10in and 24in, then using Pythagoras' theorem, the longest side =
\sqrt{10^2 + 24^2} 
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= 26

Now we know the two longest sides of the first triangle (24in and 26in) we can compare them with the two longest sides of the second triangle.
If x = the scale factor the first triangle is enlarged by then
26x = 39 and 24x = 36 
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Finally, we need to multiply the smallest side of the first triangle by the scale factor to find the shortest side of the second triangle.
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You could, instead, calculate the length of the shortest side of the second triangle by using Pythagoras' theorem and ignoring the first triangle completely.
5 0
4 years ago
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Please help me on this
Lina20 [59]
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Answer:

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Answer:

I can try to answer.

Step-by-step explanation:

1. 23

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29-23=6

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Im not rlly sure about 5, but ill answer it if I figure it out.

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