Respuesta :
Answer:
a) the larger rectangle will be actually a square with sides of 96 ft
b) the pens will have 96 ft of length and 31 ft of width
Step-by-step explanation:
denoting the side lengths of the big rectangle as x and y , we have the following expression for its area A:
A= x*y
that is tied to the constraint that the perimeter should not be larger than the available fencing , thus
2*x + 2*y = 384 (we use all the fencing to maximise the area)
y = (384 -2*x)/2 = 192 - x
replacing in A
A= x*y = x*(192 - x) = 192*x - x²
we can complete the square to rearrange the equation of A
A=192*x - x² + 96² - 96² = 96² - ( x² - 192*x + 96²) = 9216 - (x - 96)²
then A is maximum for x - 96 = 0 → x=96 ft
thus y= 192 - 96 = 96 ft
thus the rectangle with maximum area is actually a square with sides 96 ft ( if a rectangle is required, then diminish any of the lengths by the smallest amount possible. For example x= 95.99999 and y=96.00001)
Then since we maximised the area of the bigger rectangle , we have maximised the area of the smaller pens. The dimensions will be
x small = 96 ft /3 = 31 ft , y small = 96 ft
The dimensions of the rectangle are 48ft and 92 feet.
How to calculate the dimensions of the rectangle?
The area of the larger rectangle will be calculated as:
= Length × Width
In this case, 2y + 4x = 384. Therefore, the equation divided by 2 will be:
y + 2x = 192
y = 192 - 2x
Area will now be:
= (192 - 2x) × x
= 192x² - 2x²
The area is computed as 4608 feet². Therefore, the dimensions will be:
= 4608/48
= 92
In conclusion, the dimensions are 48ft and 92 feet.
Learn more about rectangles on:
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