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Zorluk: OrtaQuadratic Equations and Factoring

The length of a rectangular plot of land is 33 meters less than twice its width. If the area of the plot is 9090 square meters, what is the perimeter of the plot, in meters?

Cevap: 39 meters

Cevap

The perimeter of the plot of land is 39 meters.
Setting the length to 2w−32w - 3 gives an area equation of w(2w−3)=90w(2w - 3) = 90, which expands and rearranges to 2w2−3w−90=02w^2 - 3w - 90 = 0. Factoring this quadratic equation yields (2w−15)(w+6)=0(2w - 15)(w + 6) = 0. Since width must be positive, w=7.5w = 7.5 meters, which means the length is 1212 meters. The perimeter is 2(12+7.5)=392(12 + 7.5) = 39 meters.

Adım Adım Çözüm

1
Express the length in terms of width and set up the area equation.
Let ww be the width of the rectangle. Length l=2w−3l = 2w - 3. Area equation: w(2w−3)=90w(2w - 3) = 90.
The area of a rectangle is equal to length multiplied by width.
2
Rearrange into standard quadratic form ax2+bx+c=0ax^2 + bx + c = 0.
2w2−3w−90=02w^2 - 3w - 90 = 0
Distributing ww and subtracting 9090 from both sides puts the equation in standard quadratic form.
3
Factor the quadratic equation.
(2w−15)(w+6)=0(2w - 15)(w + 6) = 0
Finding two numbers with a product of 2×(−90)=−1802 \times (-90) = -180 and a sum of −3-3 gives −15-15 and 1212.
4
Determine the valid physical dimensions.
w=7.5w = 7.5 meters and l=12l = 12 meters.
The root w=−6w = -6 is discarded because physical length cannot be negative. Thus w=152=7.5w = \frac{15}{2} = 7.5 meters.
5
Calculate the perimeter.
Perimeter =2(l+w)=2(12+7.5)=39= 2(l + w) = 2(12 + 7.5) = 39 meters.
The perimeter of a rectangle is given by 2×(length+width)2 \times (\text{length} + \text{width}).

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