Question

A rectangular picture is 12 by 16 inches. If a frame of uniform width contains an area of 165 square inches, what is the width of the frame?

Answer 1

Answer:

2.5

Step-by-step explanation:

A rectangular picture is 12 by 16 inches.

the area of the frame is 165 sq inches

the frame borders the pictures round the 4 corners

for the length, Let  x  be the frame  distance to the picture which is 12+x+x= 12+2x

and for the breadth, Let  x  be the frame  distance to the picture which is 16+x+x = 16+2x

(12+2x) x (16+2x) = length x breadth of frame

12(16+2x)+2x(16+2x)

192+24x+32x+4x²

192+56x+4x² is the area of frame

165 is area of frame - area of rectangular picture

165 = 192+56x+4x² - (12 x16)

165 = 192+56x+4x² - 192

165 = 56x+4x²

subtract 165 from both sides to form a quadratic equation

4x²+56x-165= 0

using almighty formula

x=-b±√b²-4ac/2a

a= 4

b=56

c=-165

x=-b±√b²-4ac/2a = -56±√56²-4(4)(-165)/2(4)

= -56±√3136+2640/8= -56±√5776/8 =  -56±76/8

splitting the sign±

-56+76/8 ,-56-76/8

20/8, -132/8

5/2,  -16.5

2.5, -16.5

we need to neglect the neagtive answer because the width cant be negative

the width is 2.5