Question

Find a formula for

1) the area of the figure
**is it xy+(πr^2/2) ? ****
2) the perimeter of the figure
****is it x + x + y + (π * y / 2 ) ? ****

And then find the dimensions of x and y that maximize the area given that the perimeter is 500.

write EXACT answers
x=____ ; y= _______

thanks!! @ganeshie8 ?

Answer 1

FIGURE! :)

Answer 2

so you have a rectangle and a semicircle

Answer 3

whats the radius of semicircle ?

Answer 4

1/2 diameter?

Answer 5

yes, so r = y/2

right ?

Answer 6

that gives area of semicircle = \(\huge \frac{\pi (\frac{y}{2})^2}{2}\)

Answer 7

yes:)

Answer 8

so area of this figure is xy + (pi(y/2)^2 / 2)) ?

Answer 9

yup ! simplifying it gives :

Area of entire figure = \(\huge xy + \frac{\pi }{8}y^2\)

Answer 10

ahh ok:)

Answer 11

Area of the rectangle:\[A=lw=xy\] Area of 1/2 circle \[A=\frac{\pi r^{2}}{2}=\frac{\pi (y/2)^{2}}{2}\]

Total area \[A=A_{rect}+A_{1/2circ}=xy+\frac{\pi (y/2)^{2}}{2}\]


From there you can simplify if need be to \[xy+\frac{\pi}{8}y^{2}\]

Answer 12

ahh so what about the perimeter?

Answer 13

perimeter looks good !

Answer 14

yay!! so what about the last part then?

Answer 15

And then find the dimensions of x and y that maximize the area given that the perimeter is 500.

Answer 16

how do we do that?

Answer 17

set the perimeter equal to 500 and solve x or y

Answer 18

x + x + y + (π * y / 2 ) = 500

Answer 19

looks, solving x is easy, so lets sovle x

Answer 20

so 2x + y + (pi * y / 2) = 500

Answer 21

yes, solve x all the way,
then plug that value in Area expression

Answer 22

so 2x + πy/2 = 500 -y ?

Answer 23

so 2x + y + (pi * y / 2) = 500

2x = 500 - y - (pi * y / 2)

x = 250 - y/2 - (pi * y / 4)

Answer 24

^^

Answer 25

plug this x value in earlier Area formula

Answer 26

okay following :)

Answer 27

so \[A=250-\frac{ y }{ 2 } - (\frac{ πy }{ 4 })y + (\frac{ πr^2 }{ 2 })\] like this?

Answer 28

\(\huge A = xy + \frac{\pi }{8}y^2 \)

Answer 29

plugin \(x = 250 - y/2 - (\pi * y / 4) \)

Answer 30

so 250 y - (y^2/2) - (pi*y/4) ?

Answer 31

\(\large A = [250 - \frac{y}{2}- \frac{\pi y }{4} ]y + \frac{\pi }{8}y^2 \)

Answer 32

simplify it if u want, and maximize

Answer 33

okay.. so \[A=250y - \frac{ y^2 }{ 2 } - \frac{ πy^2 }{ 4 } + \frac{ πy }{ 4 }\] ?

Answer 34

hmm check ur last term again

Answer 35

ermm not sure how to simplify that part :/

Answer 36

\(\large A = [250 - \frac{y}{2}- \frac{\pi y }{4} ]y + \frac{\pi }{8}y^2 \)

distribute y :
\(\large A = 250y - \frac{y^2}{2}- \frac{\pi y^2 }{4} + \frac{\pi }{8}y^2 \)

pull out y^2 :

\(\large A = 250y - y^2[\frac{1}{2} + \frac{\pi }{4} - \frac{\pi }{8}] \)

simplify :

\(\large A = 250y - y^2[\frac{4+\pi }{8}] \)

Answer 37

Maximize it !

Answer 38

set it equal to 0?

Answer 39

find the derivative first

Answer 40

then u can set it equal to 0 and solve y

Answer 41

250 + ....

not sure how to derive that second part :/

Answer 42

let me ask u a q :

whats the derivative of y^2*10 ?

Answer 43

wait would it be A' = 250 + 1/4(4+π)y ? :/

Answer 44

y^(2*10) ?

Answer 45

Perfect !
set it equal to 0 and sovle y

Answer 46

oh just 2y+0 ?

Answer 47

forget about my q,
u got the derivative correct ! set it equal to 0 and sovle y :)

Answer 48

250 + 1/4(4+π)y = 0

so 250 +y+πy = 0 ?

Answer 49

wait a sec,
u have flipped hte sign

Answer 50

\(\large A = 250y - y^2[\frac{4+\pi }{8}]\)


\(\large A' = 250 - 2y[\frac{4+\pi }{8}]\)

Answer 51

set it equal to 0 and sovle y

Answer 52

\(\large 250 - 2y[\frac{4+\pi }{8}] = 0 \)

\(\large y[\frac{4+\pi }{4}] = 250 \)

\(\large y[4+\pi ] = 1000 \)

\(\large y = \frac{1000}{4 + \pi} \)

Answer 53

So, the max value occurs at \(\large y = \frac{1000}{4 + \pi} \)

Answer 54

ahh okay so that's it?

x= 250 y - (y^2/2) - (pi*y/4) and y = what you just typed?

Answer 55

@ganeshie8 did i get that x value right? :/

Answer 56

u need to sovle x

Answer 57

u have :

x + x + y + (π * y / 2 ) = 500

Answer 58

plugin y value in this, and solve x

Answer 59

you should get :


\(\large x = \frac{500}{4 + \pi}\)

Answer 60

2x + (1000/4+π) + (π*(1000/4+π / 2) = 500 ?

Answer 61

simplify it in any way u wish !
test for ur algebra skills :P

Answer 62

ahh okay:P

thank you!!!

Answer 63

i got 500/4+pi on my paper :)

it got messy!! hahahaa

Answer 64

good :)
wolfram agrees wid our answer :
http://www.wolframalpha.com/input/?i=maximize+xy+%2B++++%5Cfrac%7B%5Cpi+%7D%7B8%7Dy%5E2%2C++x+%2B+x+%2B+y+%2B+%28%5Cpi+*+y+%2F+2+%29+%3D+500%2C+x+%3E+0%2C+y+%3E+0

Answer 65

hehe thanks!! :D

Answer 66

np :)