15. mr. coley has 2200 yards of fencing to enclose a rectangular area for his pet snail. express the area a…
15. mr. coley has 2200 yards of fencing to enclose a rectangular area for his pet snail. express the area a of the rectangle as a function of the width w of the rectangle. for what value of w is the area largest? what is the maximum area?
Answer
Explanation:
Step1: Find the length in terms of width
The perimeter of a rectangle is (P = 2L+2W). Given (P = 2200), we have (2L+2W=2200), so (L=\frac{2200 - 2W}{2}=1100 - W).
Step2: Express the area as a function of width
The area of a rectangle is (A = L\times W). Substitute (L = 1100 - W) into the area formula: (A(W)=(1100 - W)W=1100W - W^{2}).
Step3: Find the value of (W) for maximum area
For a quadratic function (y = ax^{2}+bx + c) ((a=- 1), (b = 1100), (c = 0)), the vertex of the parabola (x=-\frac{b}{2a}). Using the formula (W=-\frac{1100}{2\times(-1)} = 550).
Step4: Calculate the maximum area
Substitute (W = 550) into the area function (A(W)). (A(550)=1100\times550-550^{2}=550\times(1100 - 550)=550\times550 = 302500)
Answer:
The value of (W) for which the area is largest is (550) yards. The maximum area is (302500) square yards.