Question

I have a calculus math problem I need help with

Answer 1

See below

Explanation:

Long side = x

Other long side = x - 20

short side = y

Area enclosed = xy

x + x -20 + y = 1000 or y = 1020 -2x <=====sub into first equation

area = x * ( 1020-2x) = -2x^2 + 1020x

this is a dome shaped parabola

max will occur at x = -b/2a = - 1020/ (2 * -2) = 255 ft

then y = 1020 -2x = 510 ft

(area = xy = 130 050 ft^2 )

Answer 2

The area of the field is:

A = (x + 20)y

The length of fence needed is:

x + y + x + 20

(remember that 20 ft are not needed because the building, and y ft are not needed because the river)

We have 1000 ft of fencing, then:

1000 = x + y + x + 20

1000 - 20 = 2x + y

980 = 2x + y

Isolating y from the preceding equation:

y = 980 - 2x

Substituting this into the area equation:

A = (x + 20)(980 - 2x)

Distributing:

`\begin{gathered} A=980x-2x^2+20\cdot980-40x \\ A=-2x^2+940x+19600 \end{gathered}`

At the maximum, the derivative of A with respect to x is zero, then:

`\begin{gathered} (dA)/(dx)=(d)/(dx)(-2x^2+940x+19600) \\ (dA)/(dx)=-2(d)/(dx)(x^2)+940\cdot(dx)/(dx)+(d)/(dx)(19600) \\ (dA)/(dx)=-4x+940 \\ 0=-4x+940 \\ 4x=940 \\ x=(940)/(4) \\ x=235 \end{gathered}`

Recalling the equation of y and substituting this result:

y = 980 - 2x

y = 980 - 2*235

y = 510

The dimensions are:

length: 255 ft (on the side without the building)

width: 510 ft