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Natasha_Volkova [10]
3 years ago
13

Let the (x; y) coordinates represent locations on the ground. The height h of

Mathematics
1 answer:
grigory [225]3 years ago
7 0

The critical points of <em>h(x,y)</em> occur wherever its partial derivatives h_x and h_y vanish simultaneously. We have

h_x = 8-4y-8x = 0 \implies y=2-2x \\\\ h_y = 10-4x-12y^2 = 0 \implies 2x+6y^2=5

Substitute <em>y</em> in the second equation and solve for <em>x</em>, then for <em>y</em> :

2x+6(2-2x)^2=5 \\\\ 24x^2-46x+19=0 \\\\ \implies x=\dfrac{23\pm\sqrt{73}}{24}\text{ and }y=\dfrac{1\mp\sqrt{73}}{12}

This is to say there are two critical points,

(x,y)=\left(\dfrac{23+\sqrt{73}}{24},\dfrac{1-\sqrt{73}}{12}\right)\text{ and }(x,y)=\left(\dfrac{23-\sqrt{73}}{24},\dfrac{1+\sqrt{73}}{12}\right)

To classify these critical points, we carry out the second partial derivative test. <em>h(x,y)</em> has Hessian

H(x,y) = \begin{bmatrix}h_{xx}&h_{xy}\\h_{yx}&h_{yy}\end{bmatrix} = \begin{bmatrix}-8&-4\\-4&-24y\end{bmatrix}

whose determinant is 192y-16. Now,

• if the Hessian determinant is negative at a given critical point, then you have a saddle point

• if both the determinant and h_{xx} are positive at the point, then it's a local minimum

• if the determinant is positive and h_{xx} is negative, then it's a local maximum

• otherwise the test fails

We have

\det\left(H\left(\dfrac{23+\sqrt{73}}{24},\dfrac{1-\sqrt{73}}{12}\right)\right) = -16\sqrt{73} < 0

while

\det\left(H\left(\dfrac{23-\sqrt{73}}{24},\dfrac{1+\sqrt{73}}{12}\right)\right) = 16\sqrt{73}>0 \\\\ \text{ and } \\\\ h_{xx}\left(\dfrac{23+\sqrt{73}}{24},\dfrac{1-\sqrt{73}}{12}\right)=-8 < 0

So, we end up with

h\left(\dfrac{23+\sqrt{73}}{24},\dfrac{1-\sqrt{73}}{12}\right)=-\dfrac{4247+37\sqrt{73}}{72} \text{ (saddle point)}\\\\\text{ and }\\\\h\left(\dfrac{23-\sqrt{73}}{24},\dfrac{1+\sqrt{73}}{12}\right)=-\dfrac{4247-37\sqrt{73}}{72} \text{ (local max)}

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A sphere and a cylinder have the same radius and height. The volume of the cylinder is 27 pi feet cubed.
LenaWriter [7]

Answer:

C. V = two-thirds (27)

Step-by-step explanation:

Given

Solid Shapes: Cylinder and Sphere

Volume of Cylinder = 27π ft³

Required

Volume of the sphere.

From the question,

<u>We have that</u>

1.  The volume of the sphere is the same as the volume of the cylinder

2. The height of the sphere is the same as the height of the cylinder.

From (2) above;

This means that the height of the cylinder equals the diameter of the sphere.

Let h represent the height of the sphere and d represent the diameter of sphere.

Mathematical, d = h

Recall that radius, r = r = \frac{d}{2}

Substitute h for d in the above expression

r = \frac{h}{2}. ----- (take note of this)

Calculating the volume of a cylinder.

V = πr²h

Recall that  V = 27; This gives us

27 = πr²h

Divide both sides by h

\pi r^2 = \frac{27}{h}

-------------------

Calculating the volume of a sphere

V = \frac{4}{3}\pi r^3

Expand the above expression

V = \frac{4}{3}\pi r^2 * r

Substitute \pi r^2 = \frac{27}{h}

V = \frac{4}{3} * \frac{27}{h} * r

Recall that r = \frac{h}{2}

So,

V = \frac{4}{3} * \frac{27}{h} * \frac{h}{2}

V = \frac{4}{3} * \frac{27}{2}

V = \frac{2}{3} * 27

V = \frac{2}{3} (27)

V = two-third (27)

4 0
3 years ago
4 glue sticks cost $7.76<br><br> Which equation would help determine the cost of 13 glue sticks?
Elza [17]

Answer:

y=1.94x

13 glue sticks cost $25.22.

Step-by-step explanation:

4 blue sticks cost $7.76, this means that one glue stick costs

$7.76/4 = $1.94.

Let y be the cost of the glue sticks, and x be the number of glue sticks; then

\boxed{y=1.94x}

We can use this equation to find the cost of 13 glue sticks; we just put x=13\ into our equation and it gives:

y=1.94(13)=25.22

So 13 glue sticks cost $25.22.

4 0
3 years ago
Read 2 more answers
A designer enlarged both the length and the width of a rectangular carpet by 60 percent. The new carpet was too large so the des
Snowcat [4.5K]

x= length, y=width

Both the length and width were increased by 60% so now they are:

1.6x and 1.6y

Then they were both reduced by 25% so now they are 75%(100-25) of their previous size so they are:

0.75(1.6x) and 0.75(1.6y)

Their area would be length * width so it would be:

0.75(1.6x) * 0.75(1.6y)

Simplified this would be:

1.2x * 1.2y

Simplified more it would be :

1.44xy which is also 144% of xy which is a 44% increase of the original area (xy)


So the answer is:

44%



5 0
3 years ago
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Ratling [72]
X = 28% * 80

x = 28/100 * 80

x = 28/10 * 8

x = 2.8 * 8

x = 22.4


28% of 80 is 22.4.

Your final name is a. 22.4
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Help me plz Will be marked brainliest
Dmitriy789 [7]

Answer:

The answer is obviously letter A!

8 0
3 years ago
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