Question

The function A(x) = -x(10 – x) describes the area A of a rectangular flower garden, where x is the width in yards. What is the maximum area of the garden?

Answer 1

Answer:

No, maxima point found!

Step-by-step explanation:

Given function:

$A(x)=-x(10-x)$

where:

A = area of rectangular flower garden

x= side of rectangle

Now for the function to yield extrema:

$\frac{d}{dx}(A) =0$

$\frac{d}{dx} (-10x+x^2)=0$

$-10+2x=0$

$x=5$ will be the point of extrema.

Now we check for second derivative A" at x=5:

$A"=\frac{d}{dx} (A')$

$A"=x$

$A"=5>0$ so it yields minima at this point and hence we do not have a maxima for this function.

Answer 2

Answer:

Maximum area of the rectangular park is 75 square yards.

Step-by-step explanation:

The function A(x) = -x(10 - x) describes the area of a rectangular flower garden.

A(x) = -x(10 - x)

Where x = width of the garden

For the maximum area we will find the derivative of the given function and equate it to zero.

$\frac{dA}{dx}=0$

$\frac{dA}{dx}=\frac{d}{dx}[-x(10-x)]$ = 0

$A'(x)=-10+2x$

For A'(x) = 0,

2x - 10 = 0

x = 5 yards

For A(5) = 5(10 + 5)

             = 75 square yards

Therefore, the maximum area of the rectangular garden is 75 yards².