Question

A rectangular deck is to be constructed using a rock wall as one side and fencing for the other three sides. There are 24 yards of fencing available. Determine the dimensions that would create the deck of maximum area. What is the maximum area

Answer 1

Answer:

A = 72 yard²

Step-by-step explanation:

Let x be length of the rectangular deck and y be the width of the deck.

Given,

x + 2 y = 24

x = 24 -2y

we know,

Area = x × y

A = y (24-2y)

A = 24 y -2y²

Using double derivative method to find maximum area:

$\dfrac{dA}{dy}=\dfrac{d}{dy}(24y-2y^2)$

$\dfrac{dA}{dy}=24-4y$

equating differential equation to zero

$24-4y = 0$

y = 6 is the critical point

now, double differentiating the equation, if double differentiation comes out to be negative then area will be maximum at critical point.

$\dfrac{d^2A}{dy^2}=-4$

so, maxima at y = 6

Maximum area =

A = 24 y -2y²

A = 24 x 6 - 2 x 6²

A = 72 yard²

Maximum area is equal to 72 yard².

Answer 2

Answer:

Maxim area: $72\text{ yards}^2$

Step-by-step explanation:

Let x represent side opposite to rock wall and y represent each of other two sides of the fencing.

We have been given that that a rectangular deck is to be constructed using a rock wall as one side and fencing for the other three sides. There are 24 yards of fencing available.

The perimeter of the fencing is needed for 3 sides that is $x+y+y=x+2y$.

Since available fencing is 24 yards, so we will equate perimeter of 3 sides with 24 as:

$x+2y=24...(1)$

$x=24-2y...(1)$

We know that area of rectangle is length times width, so our area function would be:

$A=x\cdot y$

Upon substituting equation (1) in equation (2), we will get:

$A(y)=(24-2y)\cdot y$

$A(y)=24y-2y^2$

Since we need to find the maximum area, so we will find the derivative.  

$A'(y)=24-4y$

Now, we will equate derivative with 0 and solve for y as:

$24-4y=0$

$24=4y$

$\frac{24}{4}=\frac{4y}{4}$

$6=y$

Upon substituting $y=6$ in equation (1), we will get:

$x=24-2(6)$

$x=24-12$

$x=12$

Therefore, the dimension of side opposite to rock wall will be 12 yards and other sides will be 6 yards each.

To find the maximum area, we will substitute  $y=6$  in area function as:

$A(6)=24(6)-2(6)^2$

$A(6)=144-2(36)$

$A(6)=144-72$

$A(6)=72$

Therefore, the maximum area would be 72 square yards.