Help pleaseeeee

Find the maximum and minimum values of the function f(x,y)=2x2+3y2−4x−5 on the domain x2+y2≤100. The maximum value of f(x,y) is:

ryzh [129]

First find the critical points of f :

$f(x,y)=2x^2+3y^2-4x-5=2(x-1)^2+3y^2-7$

$\dfrac{\partial f}{\partial x}=2(x-1)=0\implies x=1$

$\dfrac{\partial f}{\partial y}=6y=0\implies y=0$

so the point (1, 0) is the only critical point, at which we have

$f(1,0)=-7$

Next check for critical points along the boundary, which can be found by converting to polar coordinates:

$f(x,y)=f(10\cos t,10\sin t)=g(t)=295-40\cos t-100\cos^2t$

Find the critical points of g :

$\dfrac{\mathrm dg}{\mathrm dt}=40\sin t+200\sin t\cos t=40\sin t(1+5\cos t)=0$

$\implies\sin t=0\text{ OR }1+5\cos t=0$

$\implies t=n\pi\text{ OR }t=\cos^{-1}\left(-\dfrac15\right)+2n\pi\text{ OR }t=-\cos^{-1}\left(-\dfrac15\right)+2n\pi$

where n is any integer. We get 4 critical points in the interval [0, 2π) at

$t=0\implies f(10,0)=155$

$t=\cos^{-1}\left(-\dfrac15\right)\implies f(-2,4\sqrt6)=299$

$t=\pi\implies f(-10,0)=235$

$t=2\pi-\cos^{-1}\left(-\dfrac15\right)\implies f(-2,-4\sqrt6)=299$

So f has a minimum of -7 and a maximum of 299.

4 0

3 years ago

Un árbol proyecta una sombra de 24 pies al mismo tiempo que una vara de medir proyecta una sombra de 2 pies encuentre la altura

Zigmanuir [339]

Hola soy Dora amigo hola mark me as brainlest

4 0

2 years ago

What is the volume of the right triangular prism shown?

ollegr [7]

Given:

The sides of the base of the triangle are 8, 15 and 17.

The height of the prism is 15 units.

We need to determine the volume of the right triangular prism.

Area of the base of the triangle:

The area of the base of the triangle can be determined using the Heron's formula.

$S=\frac{a+b+c}{2}$

Substituting a = 8, b = 15 and c = 17. Thus, we have;

$S=\frac{8+15+17}{2}$

$S=\frac{40}{2}=20$

Using Heron's formula, we have;

$Area = \sqrt{S(S-a)(S-b)(S-c)}$

$Area = \sqrt{20(20-8)(20-15)(20-17)}$

$Area = \sqrt{20(12)(5)(3)}$

$Area = \sqrt{3600}$

$Area = 36$

Thus, the area of the base of the right triangular prism is 36 square units.

Volume of the right triangular prism:

The volume of the right triangular prism can be determined using the formula,

$V=\frac{1}{2}A_b h$

where $A_b$ is the area of the base of the prism and h is the height of the prism.

Substituting the values, we have;

$V=\frac{1}{2}(60\times 15)$

$V=450$

Thus, the volume of the right triangular prism is 450 cubic units.

8 0

3 years ago

Joel Bradley purchased a new cooler for his florist shop for $4,500. He wants to finance it with an installment loan from his ba

Bas_tet [7]

Answer:

62.22% I think.

Step-by-step explanation:

Have a good day.

5 0

2 years ago

I need help with exercises 3,4, and 6 please show all work and don’t answer number 5 just tell me if it is correct

joja [24]

*If you need me to do #5, DM me!

3. The area of a triangle can be given by (just plug and chug as always):
$A_t = \frac{1}{2} bh = \frac{1}{2}(3)(4) = 6ft^2$
The area of the triangle is 6ft².

4. I will divide into a triangle and a rectangle (because the actual equation for the area of a pentagon requires it to be a perfect pentagon). Let's do the triangle first (height is 3 because you subtract 12 from 15):
$A_t = \frac{1}{2} bh = \frac{1}{2}(8)(3) = 12m^2$
$A_r = bh = 12(8) = 96m^2$
Now we just add them:
$96+12 = 108m^2$

So, the area of that pentagon is 108m².

5. You are actually wrong on this one because the area of a triangle is:
$A_t = \frac{1}{2}bh$
So, just halve your answer and it will be correct.

6. We can just split it into 4 triangles of equal area and then multiply the area of 1 triangle by 4 to get the total area. Let's do just that:
$A_t = \frac{1}{2}bh = \frac{1}{2}(2.5)(5) = 6.25cm^2$
Multiply by 4 to get total area:
$6.25*4 = 25cm^2$

So, the area of the given rhombus is 25cm².

8 0

3 years ago