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Zorluk: KolayQuadratic Equations and Polynomial Factoring

A rectangular garden has an area of 4848 square meters. If the length of the garden is 22 meters greater than its width, what is the perimeter of the garden, in meters?

Cevap: 28 meters

Cevap

The perimeter of the garden is 28 meters.
By setting up the area equation w(w+2)=48w(w + 2) = 48, we obtain the quadratic equation w2+2w−48=0w^2 + 2w - 48 = 0. Factoring gives (w+8)(w−6)=0(w + 8)(w - 6) = 0. Since width must be positive, w=6w = 6 meters. The length is 6+2=86 + 2 = 8 meters. Thus, the perimeter is 2(6+8)=282(6 + 8) = 28 meters.

Adım Adım Çözüm

1
Define variables for width and length.
Width = ww, Length = w+2w + 2.
The length is given as 2 meters greater than the width.
2
Formulate and rearrange the quadratic equation for area.
w2+2w−48=0w^2 + 2w - 48 = 0
Area is length multiplied by width, set equal to 48.
3
Factor the quadratic polynomial.
(w+8)(w−6)=0(w + 8)(w - 6) = 0
Find two numbers that multiply to -48 and add up to +2.
4
Determine the valid physical dimension.
w=6w = 6 meters and length l=8l = 8 meters.
A physical dimension cannot be negative, so w=−8w = -8 is discarded.
5
Compute the perimeter.
Perimeter = 2(6+8)=282(6 + 8) = 28 meters.
Perimeter of a rectangle is twice the sum of its length and width.

Anahtar Kavram

Solving quadratic equations by factoring in word problem contexts.
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Konular

Quadratic Equations and Polynomial Factoring

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