Question

A window frame is in the shape of a semicircle joined to a rectangle.
Find the maximum area of a window using 300 cm of framework.

Answer 1

amogh come to chat

Answer 2

if the radius is r
the area function
f(r,b) = (pi)r^2 + 2rb

and
300 = (pi)r + 2(r+b)

now replace b to make f a function of r and tell me

Answer 3

df/dr = 2(pi)r + 2rdb/dr + 2b

now put in db/dr frm eqn 2

Answer 4

sorry the second eqn is a little off track

Answer 5

i have Area= 1/2(pi)^2 + 2rh

Answer 6

if the radius is r
the area function
f(r,b) = (pi)r^2 + 2rb

Answer 7

perimeter = pi(r) + 2h + 2r

Answer 8

yes

Answer 9

i tried to use http://tutorial.math.lamar.edu/Classes/CalcI/MoreOptimization.aspx but the conclusion is unclear

Answer 10

weve got the 2 functions havent we?

Answer 11

u have the two functions right?

Answer 12

Y DONT U RESPOND??

Answer 13

Let the breadth of the rectangle be x,
radius of the semi-circle= x/2 [If I'm imagining it correctly]
Area of semi-circle=(pi/8)x^2
Area of rectangle= x.y
Total area= (pi/8)x^2 + xy
And,
300=pi.x/2 + x + 2y

We are supposed to maximize the area, so we'll substitute the value of y in eqn 1 and differentiate it

dA/dx = (pi/4)x + xdy/dx + y

Differentiating the other eqn we get,
0=pi/2 + 2dy/dx
dy/dx = -pi/4

and 300 - (pi/2+1)x = 2y

Therefore putting the values of dy/dx and y in eqn 3

dA/dx = (pi/4)x + x. -pi/4 + 150 - (pi/2+1)x/2

Now equate this to 0 to get the max area

150 = x.(pi/2+1)/2
thus x = 300/(pi/2+1)

Answer 14

sorry what does breadth mean?

Answer 15

Yes I have two equations but I cant seem to get the maximun area correct!

Answer 16

the length of the side which is attached to the semi-circle

Answer 17

it meant to be 6298.9cm^2

Answer 18

How did u try

Answer 19

i made an equation for the perimeter and the area
Area= 1/2(pi)^2 + 2rh, perimeter = pi(r) + 2h + 2r
got to the point where Radius = 300/4-(pi)

Answer 20

to get area i should be able to substitue the radius and h into area right?

Answer 21

plug in h into the first eqn

Answer 22

i dont know what h is

Answer 23

calculate it from the second eqn

Answer 24

wait h=150-(pi)/2r-r

Answer 25

yeah

Answer 26

now plug into the first one

Answer 27

I get a negative number! height cannot be negative

Answer 28

no just plug it in terms of r

Answer 29

what do you mean?