Question

A rectangular lot is 110 yard long and 60 yards wide. Give the length and width of another rectangular lot that has the same perimeter but larger area.

Answer 1

Answer:

Length and width of 85 yards will give the same perimeter and larger area.

Step-by-step explanation:

Let x represent length and y represent width of rectangle.

The perimeter of the rectangle would be $2x+2y\Rightarrow 2(x+y)$.

We have been given that a rectangular lot is 110 yard long and 60 yards wide. The perimeter of the given rectangle would be 2 times the width and length.

$\text{Perimeter}=2(110+60)$  

$\text{Perimeter}=2(170)$

$\text{Perimeter}=340$

Upon equating both perimeters, we will get:

$2(x+y)=340$

Divide both sides by 2:

$x+y=170$

$y=170-x$

We know that area of rectangle is length times width.

$\text{Area}=x\cdot y$

$A(x)=x\cdot (170-x)$

$A(x)=170x-x^2$

Now, we will take the derivative of area function as:

$A'(x)=170-2x$

Now we will equate derivative with 0 and solve for x.

$0=170-2x$

$2x=170$

$\frac{2x}{2}=\frac{170}{2}$

$x=85$

Therefore, the length of rectangle would be 85 yards.

Upon substituting $x=85$ in equation $y=170-x$, we will get:

$y=170-85=85$

Therefore, the width of rectangle would be 85 yards.

This means that we will get a square. Since each square is a rectangle, therefore, length and width of 85 yards will give the same perimeter and larger area.

We can verify our answers.

$\text{New area}=85\times 85=7225$

$\text{Original area}=110\cdot 60=6600$

$\text{New perimeter}=2(85+85)$

$\text{New perimeter}=2(170)=340$

Answer 2

Answer:

85 yards

Step-by-step explanation:

Length = 110 yard

width = 60 yards

Perimeter of rectangle = 2 ( length + width)

P = 2 (110 + 60) = 340 yards

Now let the length is L and width is W.

P = 340 = 2 ( L + W)

L + W = 170 W = 170 - L ..... (1)

Area, A = L x W

A = L (170 - L)

A = 170 L - L²

Differentiate with respect to L

dA/dL = 170 - 2 L

Put it equal to zero for maxima and minima

2 L = 170

L = 85 yards

So, W = 170 - L = 170 - 85 = 85 yards

So, A = 85 x 85 = 7225 yard²

So, when the length and width is same and equal to 85 yards the perimeter is same and the area is largest.