A Gardener has 40 m of fencing to make 3 equal rectangular plots. The Total area of the plots are 30m2. Determine the Constraints on the Dimensions of the recatngles and find the dimensions of each triangle
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Answer:
Since we know we only have 80 feet of fence available, we know that ... We know the area of a rectangle is length multiplied by width
Answer:
we label the dimensions of each rectangle as x and y, then each rectangle has an area of xy and a perimeter of 2x + 2y.
The total area of the three rectangles is 30, so 3xy = 30
The total fencing for the three rectangles gives our other equation 3(2x + 2y) = 40
Now we have a system of equations that we can solve.
(1) 3xy = 30
(2) 3(2x + 2y) = 40
Simplifying each equation gives
(1) xy = 10
(2) 6x + 6y = 40
Solving equation (2) for y gives
y = 20/3 - x
Substitute this expression for y into equation (1) to get
x(20/3 - x) = 10
Put this quadratic in standard form to get
x2 - (20/3)x + 10 = 0
Use the quadratic formula with a = 1, b = -20/3 and c = 10 to get your answers
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Andy C. answered • 03/14/18
TUTOR 4.9 (27)
Math/Physics Tutor
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Here are the two main equations:
3 * perimeter = 40
3 * area = 30
---------------------------------
3 * perimeter = 40
3 * (2L + 2w) = 40
3*2*(L+w) = 40
3*(L+w) = 20
L+w = 20/3
w = 20/3 - L <--- this is the first equation
---------------------------
3*Area = 30
Area = 30/3 = 10
L * w = 10 <---- this is the second equation
-----------------------------------
L * (20/3 - L) = 10 <--- substitutes first equation into the second equation
L * (3L - 20) = -30 <---- multiplies everything by -3 to clear away the fraction
and changes the signs
3L^2 - 20L + 30 = 0 <---- adds 30 to both sides
L = [20 +or- sqrt( (-20)^2 - 4(3)(30 ) ] / (2*3) <--- quadratic formula with A=3, B= -20, C = 30
L = [ 20 +or- sqrt( 400 - 360) ]/ 6
L = [ 20 +or- sqrt( 40) ]/6
L = [ 20 +or- sqrt(4*10)]/ 6
L = [ 20 +or- sqrt(4)*sqrt(10)]/6
L = [ 20 +or- 2*sqrt(10)]/6
L = [ 20 +or- 2 * sqrt(10)]/6
L = [ 10 +or- sqrt(10)]/3
For the positive branch of the solution:
which is approximately 4.38742588672.... or the latter number you have given as the answer.
The width is the area/length = 10 / [( 10 + sqrt(10)/3]
= 30 / [ 10 + sqrt(10)]
= 30 ( 10 - sqrt(10)) / [ 10 + sqrt(10][10 - sqrt(10]) <--- rationalizes the denominator
= 30 ( 10 - sqrt(10)) / [ 100 - 10 ]
= 30 ( 10 - sqrt(10))/ 90
= ( 10 - sqrt(10) ) / 3
which is approximately 2.279241 or the former number you have listed as the answer
For the negative branch:
L = [ 10 - sqrt(10)]/3 which is the former number you have given as the answer, as just shown in hte above line.
The width in this case is 10/ [( 10 - sqrt(10))/3] =
30 / [ 10 - sqrt(10)] =
30 [ 10 + sqrt(10)] / [ 10 - sqrt(10)][10 + sqrt(10)]
30 [ 10 + sqrt(10)] / [ 100 - 10]
30 [ 10 + sqrt(10)] / 90
[ 10 + sqrt(10) ]/3 which was the original value for length as shown above in bold, the
latter value you have listed as the answer.
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