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4 years ago

Please answer soon.

Mathematics

1 answer:

Eva8 [605]4 years ago

3 0

<u>Answer-</u>

Set of constraints to model the problem are,

$12x+9y\geq 510\\y \leq 2x\\y \geq 25$

<u>Solution-</u>

Let us assume,

x = the number of lawns weeded by Gwen,

y = the number of dogs walked by Fabio.

Gwen charges $12 each time she weeds a yard and Fabio charges $9 each time he walks a dog,

$\text{Earnings of Gwen} = 12x\\\text{Earnings of Fabio} = 9y\\\text{Total earnings of Gwen and Fabio} = 12x+9y$

They need at least $510 to purchase the new gaming station, means they need $510 or more than $510.

An equation for this situation will be,

$12x+9y\geq 510$

The number of dog walked by Fabio has scheduled is no more than twice the number of yards Gwen has scheduled to weed, means y must be less than or equal to 2x.

An equation for this situation will be,

$y \leq 2x$

Fabio will walk at least 25 dogs, means y must be greater than of equal to 25.

An equation for this situation will be,

$y \geq 25$

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The area of the parallelogram is;

32 square units

Step-by-step explanation:

The given parameters are;

The coordinates of the parallelogram RSTU = R(-4, 4), S(2, 6), T(6, 2), and U(0, 0)

We note that the area of a parallelogram = Base length × Height

From the drawing of the parallelogram RSTU, we have;

The base length = The length of $\overline {TU}$ = The length of $\overline {SR}$ = √((2 - (-4))² + (6 - 4)²) = 2·√10

The height of a parallelogram is perpendicular to its base length = The line $\overline {VT}$

∴ Where, the slope of the base length = m, the slope of the height = -1/m

The slope, 'm' of $\overline {SR}$ = (6 - 4)/(2 - (-4)) = 1/3

Therefore, the slope of the height = -1/(1/3) = -3

We note that a point on the height is the point 'T', therefore, the equation of the line in point and slope form is therefore;

y - 0 = -3·(x - 0)

∴ y = -3·x

Therefore, the coordinates of the point 'V' is given by the simultaneous solution of the equations of $\overline {SR}$ and $\overline {VT}$

The equation of the line $\overline {SR}$ in point and slope form from the point 'R' and the slope 'm = 1/3' is given as follows;

y - 4 = (1/3) × (x - (-4)) = (1/3) × (x + 4)

y = x/3 + 4/3 + 4 = x/3 + 16/3

y = x/3 + 16/3

We then have the coordinate at the point 'V' (x, y) is given as follows;

-3·x = x/3 + 16/3

-9·x = x + 16

-10·x = 16

x = -16/10 = -1.6

x = -1.6

∴ y = -3·x = -3 × -1.6 = -4.8

y = 4.8

The coordinate at the point, V = (-1.6, 4.8)

The length of the line $\overline {VT}$ = The height of the parallelogram = √((-1.6 - 0)² + (4.8 - 0)²) = 8/5·√10

The height of the parallelogram = 8/5·√10

The area of the parallelogram, A = Base length × Height

∴ A = 2·√(10) × 8/5·√(10) = (16/5) × 10 = 32

The area of the parallelogram, A = 32 square units.

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