Sasha works at the botanical gardens and is planning a new rectangular rose garden 20m x 30m. she plans to build a walkway with a uniform width all around the garden. Her budget is $6,000 and she knows it will cost $10/m² to construct the walkway. How wide can the walkway be?
thanks for your help. i'm doing summer school with great difficulty..no sarcastic remarks please, teacher always had other students
Math, help...?
On either side, you'll have a rectangle measuring 20 × w for the walkway. ("w" is the width of the walkway.) On the other pair of sides, you'll also have rectangles measuring 30 × w. Finally, you'll have four squares at the corners of the garden, each measuring w × w.
The total walkway area is 2(20w) + 2(30w) + 4(w²).
With $6,000 at $10/m², you have 600m² for your maximum walkway area.
4(w²) + 2(20w) + 2(30w) = 600
4w² + 40w + 60w = 600
4w² + 100w = 600. [Divide each term by 4.]
w² + 25w = 150. [Subtract 150 to set the expression equal to 0.]
w² + 25w - 150 = 0. [Factor]
(w + 30)(w - 5) = 0
w + 30 = 0 or w - 5 = 0
w = -30 or w = 5.
Since w represents a width, and cannot be negative, disregard the -30.
w = 5 meters.
(You could also solve w² + 25w - 150 = 0 using the quadratic formula, too, and you'd get the same solutions.)
Reply:you have to reduce it to this formula:
4w^2+40w+60w=6000/10
then you can get
4w^2+100w-600=0
and solve like a normal quadratic
Reply:First you need to calculate the perimeter of the garden: in this case it is 100 m. Since you know the budget amount (i.e. the total) and the price per square meter, you can find the number of square meters you can construct : $6000 / $10 sq. meters = 600 sq meters.
Area = length x width.
Therefore Width = Area / length
Now simply divide the area by the single dimension (600 sq meters / 100 meters in length) and voila!
Sasha can afford to construct a walkway of 6 foot uniform width all the away around her garden.
Reply:The rectangle has a perimeter of 100 m.
w*100 is the area of the walkway's 4 rectangles.
If each m^2 is10$ then her walkway is 1000 *w $ just the 4 rectangles on the sides.
Also 4 squares, that is 4*w^2.
So 40*w^2+1000w=6000
w^2+25w-150 =0
w= [-25+sqrt(625+600)]/2
w= (-25+35)/2 =5 this is the walkway width.
Reply:The area of the entire garden is 600 sq m (20 m * 30 m).
$6000 divided by $10 per sq m means 600 sq m can be walkway.
So, unless there is something about this that I don't understand it seems that the entire garden could be walkway.
The width would be 10 m -- which would cover the entire garden!
Reply:Well, let's figure out what we know:
- The area of the garden is 20m x 30m, or 600 m².
- Sasha's budget is $6,000, which will buy her 600 m² of walkway.
This means that the total area of the garden AND the walkway is going to be 1200 m². Since the walkway is going to be a uniform width, one outer edge of the walkway is going to be 10 m longer than another edge, the following must be true.
x * (x + 10) = 1200
x is going to represent the short side of the outer perimeter of the walkway. Shuffling around the equation, you get:
x² + 10x = 1200
Now it's just a matter of working through the quadratic equation.
x² + 10x + (10 / 2)² = 1200 + (10 / 2)²
x² + 10x + 25 = 1200 + 25
(x + 5)² = 1225
x + 5 = 35
x = 35 - 5
x = 30
Since x represents the short side of the whole area, the long side is x + 10, so the total area will be 30m x 40m. Simple subtraction from the area of the garden, 20m x 30m, tells you that the width of the walkway must be 10m.
Reply:First find an expression for the area of the walk way.
Call it's width "x".
The area of the entire garden is: 600 m^2
The area of the garden plus walkway is: (20 + 2x)(30 + 2x) m^2
The walk way area = [(20 + 2x)(30 + 2x) - 600] m^2
=600 + 60x + 40x + 4x^2 - 600
= 4x^2 + 100x
Budget limit is $6000 and the unit cost is $10/m^2
Cost = [Cost/area] * [area]
$6000 = [$10 / m^2] * [4x^2 + 100x] m^2
6000 = 40x^2 + 1000x
150 = x^2 + 25x
x^2 + 25x - 150 = 0
(x + 30)(x - 5) = 0
x = -30 or 5 meters
Width has to be positive, so the maximum width of the walkway is 5 meters.
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