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Pani-rosa [81]
2 years ago
5

3. Part way through his party, Patrick found that there was 5 2/3 pizzas left. If his friends eat 3/4 of the pizzas, how much of

the 5 will they eat? Write and solve an equation that will help you figure out how much of the remaining pizzas will be eaten. Explain the steps you took to solve the problem.​
Mathematics
1 answer:
NARA [144]2 years ago
8 0

Answer:

4 1/4

Step-by-step explanation:

Convert 5 2/3 to an improper fraction. 3 x 5 = 15 + 2=17

17/3 x 3/4 = 51/12= 4 1/4 of the remaining pizza will be eaten

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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)

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Question 9<br> Which inequality corresponds to the graph shown?
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Answer:

B) |x-1|≤3

Step-by-step explanation:

B) |x-1|≤3

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