Questions: A rectangular enclosure is built using a long existing fence and three new sides that use a total of 40 meters of fencing. Denote the length of the two new sides perpendicular to the existing fence by x m, and the length of the new side parallel to the existing fence by y m, so that 2x + y = 40. Find the largest possible area of the enclosure. 200 m^2 150 m^2 220 m^2 160 m^2 192 m^2

A rectangular enclosure is built using a long existing fence and three new sides that use a total of 40 meters of fencing. Denote the length of the two new sides perpendicular to the existing fence by x m, and the length of the new side parallel to the existing fence by y m, so that 2x + y = 40. Find the largest possible area of the enclosure. 200 m^2 150 m^2 220 m^2 160 m^2 192 m^2
Transcript text: 20. A rectangular enclosure is built using a long existing fence and three new sides that use a total of 40 metres of fencing. existing long fence Denote the length of the two new sides perpendicular to the existing fence by $x \mathrm{~m}$, and the length of the new side parallel to the existing fence by $y \mathrm{~m}$, so that $2 x+y=40$. Find the largest possible area of the enclosure. $200 \mathrm{~m}^{2}$ $150 \mathrm{~m}^{2}$ $220 \mathrm{~m}^{2}$ $160 \mathrm{~m}^{2}$ $192 \mathrm{~m}^{2}$
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Solution

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Solution Steps

Step 1: Express y in terms of x

Given the equation 2x + y = 40, we can express y in terms of x: y = 40 - 2x.

Step 2: Express the area in terms of x

The area of the rectangular enclosure is given by A = xy. Substituting the expression for y from Step 1, we get A(x) = x(40 - 2x) = 40x - 2x².

Step 3: Find the maximum area

To find the maximum area, we can complete the square for the quadratic equation or find the vertex. The x-coordinate of the vertex is given by x = -b/2a = -40/(2*-2) = 10. Substituting x = 10 back into the area equation, we get A(10) = 40(10) - 2(10)² = 400 - 200 = 200.

Final Answer

200 m²

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