Question

A rectangular playground is enclosed on three sides by a fence and one side of a house. The length of the fence is 20 yards. Given that the width of the playground is x yards, find:

an expression for the lenth og the playground in temrs of x
the area of the playground in terms of x
the actual length and width of the playground that will give the maximum area

this is a caculus question for optimization so...yeah

Answer 1

\[l=20-2x\]

Answer 2

how do you know that?

Answer 3

\[A=x(20-2x)=20x-2x^2\]\]

Answer 4

how do you know that...?

Answer 5

i know it because the total is 20.

Answer 6

yeah...

Answer 7

you have 20 yards and one side is against the house. if we call this L then
\[l+2x=20\] and so
\[l=20-2x\]

Answer 8

ok

Answer 9

you only have to use up 3 sides for your 20 because one is free (against the house)

Answer 10

and width is 2x and area is l*w so A=2(20-x)...?

Answer 11

normally perimeter of a rectangle would be
\[2l+2w\] but in this case one side is free so it is

\[l+2x=20\]

Answer 12

area is width times length so next answer is
\[A=x(20-2x)\]

Answer 13

okay

Answer 14

or
\[A(x)=20x-2x^2\]

Answer 15

now you sure as heck don't need calculus to find the maximum area because this is a parabola that faces down and its maximum is at the vertex.

Answer 16

if you recall that the vertex is given by
\[-\frac{b}{2a}\] you are done.

Answer 17

dont we find the derivative and then solve for x?

Answer 18

but since you are supposed to use calculus take the derivative, set = 0 and solve. you will still get
\[-\frac{b}{2a}\]

Answer 19

you mean "take the derivative, set = 0 and solve for x" yes

Answer 20

yes...

Answer 21

\[A'(x)=20-4x\]

Answer 22

set = 0 get
\[20-4x=0\]
\[x=5\]

Answer 23

but i repeat:
you do not need calculus for this. the vertex is
\[-\frac{b}{2a}\] whether you take the derivative, set = 0 and solve, or just remember that the vertex of a parabola is given by that.

Answer 24

in any case x = 5 and the other side is 5 and the side against the house it 10

Answer 25

okay lol thanks for the tip...-b/2a

Answer 26

now i you want to impress your math teacher... use P instead of 20

Answer 27

you will see that in this case the side against the house should be
\[\frac{p}{2}\] and the other two sides should be
\[\frac{P}{4}\]

Answer 28

its a practice question for finals...not for a grade but okay let me hear it...lol

Answer 29

and then do it without the side against the house being free and you will see that each side should be
\[\frac{P}{4}\] i.e. a square.

Answer 30

btw
\[-\frac{b}{2a}\] is for
\[f(x)=ax^2+bx+c\] for higher degrees you will need calculus

Answer 31

okay..

Answer 32

im trying to understand what you said about the P/4 and P/2 i'll just work it out

Answer 33

thanks so much satellite! :)