A rectangular garden measuring 12\text{ m} by 8\text{ m} has a paved path of uniform width… — JAMB UTME

A rectangular garden measuring $12\text{ m}$ by $8\text{ m}$ has a paved path of uniform width $x\text{ m}$ constructed inside it along its perimeter. If the area of the remaining inner garden is $60\text{ m}^2$ , what is the width $x$ of the path?

$1\text{ m}$ Cevap
B
$2\text{ m}$
C
$1.5\text{ m}$
D
$3\text{ m}$

Cevap

The width of the path is

1\text{ m}

The total outer garden area is

12 \times 8 = 96\text{ m}^2

. Subtracting the path width

x

from both ends gives inner dimensions

(12 - 2x)

and

(8 - 2x)

. Equating the inner area

(12 - 2x)(8 - 2x) = 60

yields

x^2 - 10x + 9 = 0

. The realistic physical solution is

x = 1\text{ m}

Adım Adım Çözüm

Express the inner dimensions in terms of path width

x

Length =

(12 - 2x)\text{ m}

, Width =

(8 - 2x)\text{ m}

The path reduces each side dimension by

x

at both ends.

Set up the area equation for the inner rectangular garden

(12 - 2x)(8 - 2x) = 60

The area of a rectangle is length multiplied by width.

Expand and simplify the quadratic equation

96 - 40x + 4x^2 = 60 \implies 4x^2 - 40x + 36 = 0 \implies x^2 - 10x + 9 = 0

Divide the whole equation by 4 to simplify quadratic terms.

Solve for

x

by factoring

(x - 1)(x - 9) = 0 \implies x = 1

x = 9

Since the width

x

cannot exceed half of the smaller side (

x < 4\text{ m}

x = 9

is extraneous.

Anahtar Kavram

Perimeter and Area of Rectangles with Uniform Borders

Tahmini Süre:1m 30s

Soru

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