20 POINTS You are travelling in New York City and you need to take a cab from the airport to your hotel. A taxi company charges
a flat fee of $2.00 and an additional $0.75 per mile. The graph below shows points representing the total cost of the taxi ride.
Which equation best represents the total cost of the cab ride, y, and the distance, x, that you have traveled?
A. y = 0.75x + 2
B. y = 2x + 0.75
C. y = 0.75x - 2
D. y = 4.25x
Which equation best represents the total cost of the cab ride, y, and the distance, x, that you have traveled?
A. y = 0.75x + 2
B. y = 2x + 0.75
C. y = 0.75x - 2
D. y = 4.25x
2 answers:
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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.
In a similar way
Thus,
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]
Thus,
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,
This can be translated into
If we sum the first two expressions, we obtain
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)