He campus of a college has plans to construct a rectangular parking lot on land bordered on one side by a highway. there are 720 ft of fencing available to fence the other three sides. let x represent the length of each of the two parallel sides of fencing. a rectangle has width x. x x (a) express the length of the remaining side to be fenced in terms of x. (b) what are the restrictions on x? (c) determine a function a that represents the area of the parking lot in terms of x. (d) determine the values of x that will give an area between 20 comma 000 and 40 comma 000 ftsquared. (e) what dimensions will give a maximum area, and what will this area be?

  (a) 720 -2x

  (b) 0 ≤ x ≤ 360

  (c) A = x(720 -2x)

  (d) (30.334, 68.645) ∪ (291.355, 329.666) (two disjoint intervals)

  (e) x = 180 ft, the other side = 360 ft; total area 64,800 ft²

Step-by-step explanation:

(a) The two parallel sides of the fenced area are each x feet, so the remaining amount of fence available for the third side is (720 -2x) ft. Then ...

  length = 720 -2x

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(b) The two parallel sides cannot be negative, and they cannot exceed half the length of the fence available, so ...

  0 ≤ x ≤ 360

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(c) Area is the product of the length (720-2x) and the width (x). The desired function is ...

  A = x(720 -2x)

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(d) For an area of 20,000 ft², the values of x will be ...

  20000 = x(720 -2x)

  2x² -720x +20000 = 0

  x = (-(-720) ±√((-720)² -4(2)(20000)))/(2(2)) = (720±√358400)/4

  x = 180 ±40√14 = {30.334, 329.666} . . . feet

For an area of 40,000 ft², the values of x will be ...

  x = 180 ±20√31 ≈ {68.645, 291.355} . . . feet

The values of x producing areas between 20,000 and 40,000 ft² will be values of x in the intervals (30.334, 68.645) or (291.355, 329.666) feet.

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(e) The vertex of the area function is at the axis of symmetry: x = 180. The corresponding dimensions are ...

  180 ft × 360 ft

and the area of that is 64,800 ft².

The scale factor of dilation is One-third

Step-by-step explanation:

If point (x , y) is dilated by a scale factor k, then its image is (kx , ky)

∵ The vertices of the square ABCD are:

A = (-21 , 18)

B = (-12 , 18)

C = (-12 , 9)

D = (-21 , 9)

∵ A' = (-7 , 6)

∴ (kx , ky) = (-7 , 6)

∵ x , y are the coordinates of Point A

∵ x = -21 and kx = -7

∴ k(-21) = - 7

- Divide both sides by -21

∴ k =

∵ y = 18 and ky = 6

∴ k(18) = 6

- Divide both sides by 18

∴ k =

∵ C' = (-4 , 3)

∴ (kx , ky) = (-4 , 3)

∵ x , y are the coordinates of Point C

∵ x = -12 and kx = - 4

∴ k(-12) = - 4

- Divide both sides by -12

∴ k =

∵ y = 9 and ky = 3

∴ k(9) = 3

- Divide both sides by 9

∴ k =

∴ The scale factor of dilation is One-third