A walkway is being installed around a rectangular playground. The playground is 30 feet by 12 feet, and the total area of the playground and the walkway is 1,288 ft^2. What is the width of the walkway?

10.3

Step-by-step explanation:

From these question, we can illustrate the playground has a small rectangle with 30x12ft

the walkway is around it with an area of 1288ft²

the playground area = 30 x 12 = 360ft²

the length dimension of the walkway is 30+y+y( these is because the walkway is round the playground in two sides) = 30 +2y

the breadth dimension of the walkway is 12+y+y ( these is because the walkway is round the playground in two sides) = 12+2y

Area of walkway = (30+2y)(12+2y)= 30(12+2y)+2y(12+2y) = 360+60y+24y+4y²= 360+84y+4y²

The total area = 1288ft²

The total area = area of walkway - area of playground

1288ft² = 360+84y+4y² - 360

choose like terms

1288 = 360-360+84y+4y²

1288 = 84y+4y²

divide by 4

322= 21y+y²

y²+21y-322 =0

using quadratic formula

x= (-b±√b²-4ac)/2a = {-21±√21²-4(1)(-322)}/2x1 =  {-21±√441+1288}/2x1  = -21±√1729/2

-21+√1729/2   -21-√1729/2

√1729 = 41.5812457726

-21+√1729/2   -21-√1729/2

{-21+41.5812457726}/2   {-21-41.5812457726}/2

20.5812457726/2          -62.5812457726/2

10.2906228863              -31.2906228863

we disregard the negative sign

our width is 10.2906228863 ≅10.3

the width of the walkway is 8 feet.

Step-by-step explanation:

Let x represent the width of the walkway.

If the length of the playground is 30 feet, then the total length of the playground and the walkway is 30 + x + x = 30 + 2x

If the width of the playground is 12 feet, then the total width of the playground and the walkway is 12 + x + x = 12 + 2x

If the total area of the playground and the walkway is 1,288 ft^2, it means that

(30 + 2x)(12 + 2x) = 1288

360 + 60x + 24x + 4x² = 1288

4x² + 84x + 360 - 1288 = 0

4x² + 84x - 928 = 0

Dividing through by 4, it becomes

x² + 21x - 232 = 0

x² + 29x - 8x - 232 = 0

x(x + 29) - 8(x + 29) = 0

x - 8 = 0 or x + 29 = 0

x = 8 or x = x = - 29

Since the width of the walkway cannot be negative, then x = 8

the answer is a) y= 2 csc [0.5 (x+pi/2)]-1

step-by-step explanation: