Question

A person has a rectangular board 12 inches by 16 inches around which she wants to put a uniform border of shells. If she has enough shells for a border whose area is 380 square​ inches, determine the width of the border.

Answer 1

The width of the border is 5 inches.

Explanation:

Let x be the width of the border,

Given,

The rectangular board has the dimension 12 inches by 16 inches,

Thus, the dimension of the figure by joining the border with the board is,

(12+2x) inches by (16+2x) inches,

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

Hence, the area of the border = area of the figure by joining the border with the board - area of the board

= (12+2x)(16+2x) - 12 × 16

According to the question,

The area of the border = 380 square​ inches,

`(12+2x)(16+2x) - 12* 16=380`

`192+24x+32x+4x^2-192=380`

`4x^2+56x-380=0`

`4x^2+76x-20x-380=0`

`4x(x+19)-20(x+19)=0`

`(4x-20)(x+19)=0`

By zero product property,

`x=5\text{ or }x=-19`

Since, width can not be negative,

Hence, the width of the border is 5 inches.

Answer 2

so width of border is 10 inches

Explanation:

Given data

board size = 12 * 16 inches

border area = 380 square inches

to find out

width of border

solution

let us assume width of border is x

first we find area of board i.e. 12 × 16 = 192 sq inches

and border area is given = 380 square inches

so total area of border and board is 192 + 380 = 572 sq inches

so we can say ( x+ 12 ) (x +16 ) = 572

solve this equation we get

x² +16x + 12x + 192 = 572

x² + 28x + 192 = 572

x² + 28x - 380 = 0

solve this equation and we get two value of x i.e.

x = 10 and x = -38

we will consider positive value here

so width of border is 10 inches