An airport has two long term parking lots. the cost to park,y, in each lot for x days is shown in the tables

Mathematics

1 answer:

Licemer1 [7]2 years ago

8 0

Answer:

y = 4x + 12 will be the other equation.

Step-by-step explanation:

Data given in the tables show a linear relation (has a common data).

To get the linear relation, we will choose the two points from table (1) .

Let the points are (1, 16) and (2, 20).

Slope of the line 'm' = $\frac{\triangle{y}}{\triangle{x}}$

m = $\frac{20-16}{2-1}$

m = $\frac{4}{1}$

m = 4

Equation of the line passing through (1, 16) having slope = 4

y - 16 = 4(x - 1)

y = 4(x - 1) + 16

y = 4x - 4 + 16

y = 4x + 12

Now we take second set of data,

We choose two points (1, 6) and (2, 12).

Slope 'm' = $\frac{\triangle{y}}{\triangle{x}}$

m = $\frac{12-6}{2-1}$

m = 6

Equation of the line passing through (1, 6) having slope = 6

y - 6 = 6(x - 1)

y = 6x - 6 + 6

y = 6x

Therefore, other equation of the system of equations will be,

y = 4x + 12

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Find three real numbers x, y, and z whose sum is 6 and the sum of whose squares is as small as possible. g

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We're minimizing $x^2+y^2+z^2$ subject to $x+y+z=6$ . Using Lagrange multipliers, we have the Lagrangian

$L(x,y,z,\lambda)=x^2+y^2+z^2+\lambda(x+y+z-6)$

with partial derivatives

$\begin{cases}L_x=2x+\lambda\\L_y=2y+\lambda\\L_z=2z+\lambda\\L_\lambda=x+y+z-6\end{cases}$

Set each partial derivative equal to 0:

$\begin{cases}2x+\lambda=0\\2y+\lambda=0\\2z+\lambda=0\\x+y+z=6\end{cases}$

Subtracting the second equation from the first, we find

$2x-2y=0\implies x=y$

Similarly, we can determine that $x=z$ and $y=z$ by taking any two of the first three equations. So if $x=y=z$ determines a critical point, then

$x+y+z=3x=6\implies x=y=z=2$

So the smallest value for the sum of squares is $2^2+2^2+2^2=12$ when $(x,y,z)=(2,2,2)$ .