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3 years ago

What is the value of b?

Mathematics

1 answer:

jek_recluse [69]3 years ago

5 0

Answer:

b = 10

Step-by-step explanation:

We know

x^2 / (24^2) - y^2/ ( b^2) = 1

equation: directrix x = a^2 / c = 576/26

a = 24

24^2/ c = 576/26

c = 26

so a^2 + b^2 = c^2

so 24^2 + b^2 = 26^2

b^2 = 26^2 - 24^2 = 100

b = 10

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Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt.

qwelly [4]

Answer:

$\displaystyle \frac{dy}{dt}=-\frac{33}{8}$

$\displaystyle \frac{dx}{dt}=\frac{3}{2}$

Step-by-step explanation:

x and y are differentiable functions of t, and we are given the equation:

$xy=6$

First, let's differentiate both sides of the equation with respect to t. So:

$\displaystyle \frac{d}{dt}\left[xy\right]=\frac{d}{dt}[6]$

By the Product Rule and rewriting:

$\displaystyle \frac{d}{dt}[x(t)]y+x\frac{d}{dt}[y(t)]=0$

Therefore:

$\displaystyle y\frac{dx}{dt}+x\frac{dy}{dt}=0$

We want to find dy/dt when x = 4 and dx/dt = 11.

Using our original equation, find y when x = 4:

$\displaystyle (4)y=6\Rightarrow y=\frac{3}{2}$

Therefore:

$\displaystyle \frac{3}{2}\left(11\right)+(4)\frac{dy}{dt}=0$

Solve for dy/dt:

$\displaystyle \frac{dy}{dt}=-\frac{33}{8}$

We want to find dx/dt when x = 1 and dy/dt = -9.

Again, using our original equation, find y when x = 1:

$(1)y=6\Rightarrow y=6$

Therefore:

$\displaystyle (6)\frac{dx}{dt}+(1)\left(-9)=0$

Solve for dx/dt:

$\displaystyle \frac{dx}{dt}=\frac{3}{2}$

5 0

3 years ago

PLSSSSSSSSSSS HELP OMGGGGGGGG

oksano4ka [1.4K]

Answer:

have a good day for you

8 0

3 years ago

A large van is being used for a picnic. There will be 8 people riding in the van, and there are 8 seats in the van (one of which

Julli [10]

Answer:

The people can seat themselves in 20160 different ways.

Step-by-step explanation:

Arrangements of n elements:

The number of arrangements of n elements is given by the following formula:

$A_{n} = n!$

In this question:

For the first seat, that is, the driver seat, 4 possible people can sit.

For the remaining 7 seats, the number of ways in which the people can sit is an arrangement of 7 elements. So

$T = 4*7! = 4*5040 = 20160$

The people can seat themselves in 20160 different ways.

6 0

3 years ago

The function f (x comma y )equals 3 xy has an absolute maximum value and absolute minimum value subject to the constraint 3 x sq

zmey [24]

Answer:

The maximum value of f is 363, which is reached in (11,11) and (-11,-11) and the minimum value of f is -33, which is reached in (√11,-√11) and (-√11,√11)

Step-by-step explanation:

f(x,y) = 3xy, lets find the gradient of f. First lets compute the derivate of f in terms of x, thinking of y like a constant.

$f_x(x,y) = 3y$

In a similar way

$f_y(x,y) = 3x$

Thus,

$\nabla{f} = (3y,3x)$

The restriction is given by g(x,y) = 121, with g(x,y) = 3x²+3y²-5xy. The partial derivates of g are

[ŧex] g_x(x,y) = 6x-5y [/tex]

$g_y(x,y) = 6y - 5x$

Thus,

$\nabla g(x,y) = (6x-5y,6y-5x)$

For the Langrange multipliers theorem, we have that for an extreme (x0,y0) with the restriction g(x,y) = 121, we have that for certain λ,

$f_x(x_0,y_0) = \lambda \, g_x(x0,y0)$
$f_y(x_0,y_0) = \lambda \, g_y(x_0,y_0)$
$g(x_0,y_0) = 121$

This can be translated into

$3y = \lambda (6x-5y)$
$3x = \lambda (-5x+6y)$
$3 (x_0)^2 + 3(y_0)^2 - 5\,x_0y_0 = 121$

If we sum the first two expressions, we obtain

$3x + 3y = \lambda (x+y)$

Thus, x = -y or λ=3.

If x were -y, then we can replace x for -y in both equations

3y = -11 λ y

-3y = 11 λ y, and therefore

y = 0, or λ = -3/11.

Note that y cant take the value 0 because, since x = -y, we have that x = y = y, and g(x,y) = 0. Therefore, equation 3 wouldnt hold.

Now, lets suppose that λ=3, if that is the case, we can replace in the first 2 equations obtaining

3y = 3(6x-5y) = 18x -15y

thus, 18y = 18x

y = x

and also,

3x = 3(6y-5x) = 18y-15x

18x = 18y

x = y

Therefore, x = y or x = -y.

If x = -y:

Lets evaluate g in (-y,y) and try to find y

g(-y,y) = 3(-y)² + 3y*2 - 5(-y)y = 11y² = 121

Therefore,

y² = 121/11 = 11

y = √11 or y = -√11

The candidates to extremes are, as a result (√11,-√11), (-√11, √11). In both cases, f(x,y) = 3 √11 (-√11) = -33

If x = y:

g(y,y) = 3y²+3y²-5y² = y² = 121, then y = 11 or y = -11

In both cases f(11,11) = f(-11,-11) = 363.

We conclude that the maximum value of f is 363, which is reached in (11,11) and (-11,-11) and the minimum value of f is -33, which is reached in (√11,-√11) and (-√11,√11)

5 0

4 years ago

Could someone give me a hand

Minchanka [31]

The answer is thirty two

4 0

3 years ago