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Serhud [2]
3 years ago
11

Find the coordinates of the midpoint of the segment given at its endpoints A(5,8) and B(-1, -4)

Mathematics
1 answer:
Studentka2010 [4]3 years ago
7 0

(x_{m},y_{m} )  = (\frac{X_{A} +X_{B} }{2}  ,\frac{Y_{A} + Y_{B} }{2})

(x_{m},y_{m} )  = (\frac{5-1}{2}  ,\frac{8-4 }{2})

(x_{m},y_{m} )  = (\frac{4}{2}  ,\frac{4 }{2})

(x_{m},y_{m} ) = (2, 2)

Answer: (2, 2)

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Find all points on the portion of the plane x+y+z=5 in the first octant at which f(x, y, z) = xy2z2 has a maximum value.
irina1246 [14]
Lagrange multipliers:

L(x,y,z,\lambda)=xy^2z^2+\lambda(x+y+z-5)

L_x=y^2z^2+\lambda=0
L_y=2xyz^2+\lambda=0
L_z=2xy^2z+\lambda=0
L_\lambda=x+y+z-5=0

\lambda=-y^2z^2=-2xyz^2=-2xy^2z

-y^2z^2=-2xyz^2\implies y=2x (if y,z\neq0)

-y^2z^2=-2xy^2z\implies z=2x (if y,z\neq0)

-2xyz^2=-2xy^2z\implies z=y (if x,y,z\neq0)

In the first octant, we assume x,y,z>0, so we can ignore the caveats above. Now,

x+y+z=5\iff x+2x+2x=5x=5\implies x=1\implies y=z=2

so that the only critical point in the region of interest is (1, 2, 2), for which we get a maximum value of f(1,2,2)=16.

We also need to check the boundary of the region, i.e. the intersection of x+y+z=5 with the three coordinate axes. But in each case, we would end up setting at least one of the variables to 0, which would force f(x,y,z)=0, so the point we found is the only extremum.
4 0
3 years ago
the length of a rectangle is equal to triple the width . if the perimeter is 8 centimeters .what is the length and the width of
madam [21]

Answer:

Length = 3 cm

Width = 1 cm

Step-by-step explanation:

Let the length of rectangle be l and width of rectangle be w.

According to problem,

l = 3w  {Length of rectangle is equal to triple the width}

And Perimeter,P = 8 cm

Since, P = 2 ( l + w )

or 8 = 2( l + w)

Plug l =3w in the above perimeter equation.

We get:

8 = 2( 3w + w)

8 = 2(4w)

8 = 8w

or    w = 1 cm

Then length ,l = 3w =3 * 1 = 3 cm

Hence length of rectangle is 3cm and width of rectangle is 1cm.

3 0
3 years ago
Help (pic attached!!)
mash [69]
( \sqrt[5]{ 3^{2} } )^{ \frac{1}{3} } =  (3^{ \frac{2}{5} } ) ^{ \frac{1}{3} } = 3^{ \frac{2}{15} }

Answer: C.
7 0
3 years ago
Find the value of x so that the function has the given value. t(x) = 3x; t(x)<br> 24*
valentina_108 [34]

8 is the answer because we all ķnow 3 ×8=24

5 0
3 years ago
Suppose that we want to test the hypothesis with a significance level (alpha) of .05 that the climate has changed since industri
saw5 [17]

Answer: we reject the initial claim

(50. 3802, 51.6198)

Step-by-step explanation:

A)

The initial statement is that the mean temperature throughout history is 50°

This is the null hypothesis and it is denoted as H'

H': u = 50.

After taking a sample of 40 years, we realized that the mean temperature is 51°.

This is the alternative hypothesis, in contradiction to the alternative.

The alternative hypothesis is denoted as H1

H1: u > 50 ( upper tailed).

Sample size (n) = 40

Sample mean (x) = 51

Population standard deviation (σ) = 2

We use a z test to get the value of the test statistics.

We are using a z test because sample size is greater than 30 ( n = 40) and population standard deviation is given.

Z score = x - u/ (σ/√n)

Z score = 51 - 50 / (2/√40)

Z score = 1 / 0.3162

Z score = 3.16

Our level of significance is 5%, and the critical value at this level of significance is 1.645.

By comparing the z score relative to the critical value, since our z score is greater than 1.645, it implies that we are in the rejection region hence we reject the initial claim.

B)

We are to construct a 95% confidence interval for population mean temperature.

For upper limit

u = x + Zα/2 ×(σ/√n)

Where Zα/2 = 1.96 which is the critical value for a two tailed test at 5% level of significance.

For upper limit, we have that

u = 51 + 1.96 × (2/√40)

u = 51 + 1.96 (0.3162)

u = 51 + 0.6198

u = 51.6198.

For lower limit, we have that

u = 51 - 1.96 × (2/√40)

u = 51 - 1.96 (0.3162)

u = 51 - 0.6198

u = 50. 3802.

Hence the 95% confidence level for mean temperature is given as (50. 3802°, 51.6198°)

7 0
3 years ago
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