A rectangle has perimeter 14 m.Express the area of the rectangle as a function of the length of one of its sides.

Question

A rectangle has perimeter 14 m.Express the area of the rectangle as a function  of the length  of one of its sides.

Quizplus · Accepted Answer

Let the length and width of the rectangle be $l$ and $w$, respectively. We are given that $2(l+w)=11eaae29_f559_b34e_87fe_e1145403ae0a_TB2067_11$, so $l+w=\frac{11eaae29_f559_b34e_87fe_e1145403ae0a_TB2067_11}{2}$. Since the perimeter of a rectangle is twice the sum of its length and width, we can also write $2l+2w=11eaae29_f559_b34e_87fe_e1145403ae0a_TB2067_11$, or $l+w=\frac{11eaae29_f559_b34e_87fe_e1145403ae0a_TB2067_11}{2}$ (which is the same equation as before).

Solving for $w$, we get $w=\frac{11eaae29_f559_b34e_87fe_e1145403ae0a_TB2067_11}{2}-l$. The area of the rectangle is $A=lw$, which we can write as $A=l\left(\frac{11eaae29_f559_b34e_87fe_e1145403ae0a_TB2067_11}{2}-light)$. Simplifying this expression, we have $A=-\frac{1}{2}l^2+\frac{11eaae29_f559_b34e_87fe_e1145403ae0a_TB2067_11}{2}l$. This is a quadratic function with negative leading coefficient, which means it has a maximum value when $l$ is halfway between its roots. Since the coefficient of $l^2$ is $-\frac{1}{2}$, the maximum value occurs when $l=\frac{-\frac{11eaae29_f559_b34e_87fe_e1145403ae0a_TB2067_11}{2}}{-2}=11eaae29_f55a_0172_87fe_89d3ba43a3a0_TB2067_11$.

Therefore, the area is maximized when the length is $11eaae29_f55a_0172_87fe_89d3ba43a3a0_TB2067_11$, and this maximum area is $A=\frac{1}{2}(11eaae29_f55a_0172_87fe_89d3ba43a3a0_TB2067_11)^2=\boxed{	extbf{(B) }11eaae29_f55a_0172_87fe_89d3ba43a3a0_TB2067_11}$.

A Rectangle Has Perimeter 14 M