Is a basket ball a polyhedron

Find the absolute maximum and minimum values of f(x, y) = x+y+ p 1 − x 2 − y 2 on the quarter disc {(x, y) | x ≥ 0, y ≥ 0, x2 +

Andreas93 [3]

Answer:

absolute max: f(x,y)=1/2+p1 ; at P(1/2,1/2)

absolute min: f(x,y)=p1 ; at U(0,0), V(1,0) and W(0,1)

Step-by-step explanation:

In order to find the absolute max and min, we need to analyse the region inside the quarter disc and the region at the limit of the disc:

Region inside the quarter disc:

There could be Minimums and Maximums, if:

∇f(x,y)=(0,0) (gradient)

we develop:

(1-2x, 1-2y)=(0,0)

x=1/2

y=1/2

Critic point P(1/2,1/2) is inside the quarter disc.

f(P)=1/2+1/2+p1-1/4-1/4=1/2+p1

f(0,0)=p1

We see that:

f(P)>f(0,0), then P(1/2,1/2) is a maximum relative

Region at the limit of the disc:

We use the Method of Lagrange Multipliers, when we need to find a max o min from a f(x,y) subject to a constraint g(x,y); g(x,y)=K (constant). In our case the constraint are the curves of the quarter disc:

g1(x, y)=x^2+y^2=1

g2(x, y)=x=0

g3(x, y)=y=0

We can obtain the critical points (maximums and minimums) subject to the constraint by solving the system of equations:

∇f(x,y)=λ∇g(x,y) ; (gradient)

g(x,y)=K

Analyse in g2:

x=0;

1-2y=0;

y=1/2

Q(0,1/2) critical point

f(Q)=1/4+p1

We do the same reflexion as for P. Q is a maximum relative

Analyse in g3:

y=0;

1-2x=0;

x=1/2

R(1/2,0) critical point

f(R)=1/4+p1

We do the same reflexion as for P. R is a maximum relative

Analyse in g1:

(1-2x, 1-2y)=λ(2x,2y)

x^2+y^2=1

Developing:

x=1/(2λ+2)

y=1/(2λ+2)

x^2+y^2=1

So:

(1/(2λ+2))^2+(1/(2λ+2))^2=1

$\lambda_{1}=\sqrt{1/2}*-1 =-0.29$

$\lambda_{2}=-\sqrt{1/2}*-1 =-1.71$

$\lambda_{2}$ give us (x,y) values negatives, outside the region, so we do not take it in account

For $\lambda_{1}$ : S(x,y)=(0.70, 070)

and

f(S)=0.70+0.70+p1-0.70^2-0.70^2=0.42+p1

We do the same reflexion as for P. S is a maximum relative

Points limits between g1, g2 y g3

we need also to analyse the points limits between g1, g2 y g3, that means U(0,0), V(1,0), W(0,1)

f(U)=p1

f(V)=p1

f(W)=p1

We can see that this 3 points are minimums relatives.

Conclusion:

We compare all the critical points P,Q,R,S,T,U,V,W an their respective values f(x,y). We find that:

absolute max: f(x,y)=1/2+p1 ; at P(1/2,1/2)

absolute min: f(x,y)=p1 ; at U(0,0), V(1,0) and W(0,1)

4 0

3 years ago

Given M is the midpoint ABA. The coordinate of A are (-4,2) and the cordinates of M are (1,1). Find the coordinate of B. Choose

4vir4ik [10]

Answer:

The coordinate of B is (6,0)

Step-by-step explanation:

Given;

coordinate of A = (-4,2) = (x₁, y₁)

coordinate of M = (1,1)

let coordinate of B = (x₂, y₂)

Mid-point is given by;

$M = \frac{X_1 + X_2}{2} , \frac{Y_1 + Y_2}{2}\\\\M= (1,1)\\\\1,1 =\frac{X_1 + X_2}{2} , \frac{Y_1 + Y_2}{2}\\\\ \frac{X_1 + X_2}{2} = 1 ----equation (1)\\\\\frac{Y_1 + Y_2}{2} = 1 -----equation(2)\\\\Solving \ equation(1)\\\\\frac{-4+ X_2}{2} = 1\\\\-4+ X_2 = 2\\\\X_2 = 2+4\\\\X_2 = 6\\\\Solving \ equation(2)\\\\\frac{Y_1 + Y_2}{2} = 1\\\\\frac{2+ Y_2}{2} = 1\\\\2+Y_2 = 2\\\\Y_2 = 2-2\\\\Y_2 = 0\\\\B = (X_2, \ Y_2)\\\\B = (6, \ 0)$

Therefore, the coordinate of B is (6,0)

5 0

3 years ago

Which of the following is NOT a requirement of testing a claim about a population proportion using a formal method of hypothesis

Pie

Answer:

A. The lowercase symbol, p, represents the probability of getting a test statistic at least as extreme as the one representing sample data and is needed to test the claim.

Step-by-step explanation:

The conditions required for testing of a claim about a population proportion using a formal method of hypothesis testing are:

1) The sample observations are a simple random sample.

2) The conditions for a binomial distribution are satisfied

3) The conditions np5 and nq5 are both satisfied. i.e n: p≥ 5and q≥ 5

These conditions are given in th options b,c and d.

Option A is not a condition for testing of a claim about a population proportion using a formal method of hypothesis testing.

8 0

3 years ago

Please answer below :-)

mestny [16]

Answer:

A: 18

Step-by-step explanation:

Hopefully this helps!

3 0

3 years ago

Agustín fabrica artículos de cuero y los vende con un incremento del 8% de su costo. Si en su última campaña utilizó un cuero de

nadezda [96]

Answer:

El aumento del precio de costo único equivalente es un aumento del 20,96% en el precio de costo

Step-by-step explanation:

Los parámetros dados son;

El porcentaje de aumento en el precio de costo que Augustin vende artículos de cuero = 8%

El aumento adicional de Augustin se aplica cuando usó cuero de mayor calidad = 12%

Sea x el precio de costo

Tenemos, el primer precio de venta = 1.08 × x

El precio de venta del cuero de alta calidad = 12% adicional al precio de venta anterior

∴ El precio de venta del cuero de alta calidad = 1,12 × 1,08 × x = 1,2096 × x

Por lo tanto, el precio de venta del cuero de alta calidad = (1.2096 × 100)% de x = 120.96% de x, lo que da;

El aumento del precio de costo único equivalente = (120,96% - 100%) aumento en x = 20,96%

∴ El aumento del precio de costo único equivalente = 20,96% de aumento en el precio de costo.

6 0

3 years ago