Question

A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 31 feet?

Answer 1

looks familiar

Answer 2

what do you mean

Answer 3

hey mubzz can you help me pls

Answer 4

Area of rectangle = (2r)*h = 2rh
Area of semicircle = 1/2 pi r^2
So we want to maximize A = 2rh + 1/2 pi r^2
but there are two variables, so we'll first try to eliminate one of them.

Note that perimeter = 2h + 2r + pi r = 31
so h = 31 - r - pi/2 r = 31 - (1 + pi/2)r

Substitute that into our equation for area to get:
A = 2r(31-(1+pi/2)r) + 1/2 pi r^2
= 62r - 2r^2 - 1/2 pi r^2

Take the derivative:
dA/dr = 62 - 4r - pi r
and set it equal to zero:
0 = 62 - 4r - pi r
(4 + pi) r = 62
r = 62 / (4 + pi)

Now that you have the radius, you can find the area A by using the equation
A = 62r - 2r^2 - 1/2 pi r^2

Answer 5

my program wants a numerical value

Answer 6

sooo

Answer 7

i just substitute r?

Answer 8

just set r = 31 feet ans solve it

Answer 9

A = 62r - 2r^2 - 1/2 pi r^2

= 62*31 - 2(31)^2 - (1/2)*pi*(31)^2

= 1922 - 1922 - 1510.14

|A| = 1510.14

Answer 10

right?

Answer 11

my program says no

Answer 12

:(

Answer 13

@satellite73

Answer 14

this person probably know how to solve this problem

Answer 15

I'm sorry but I really can't get it

Answer 16

ok what about this

A box is to be made out of a 8 by 16 piece of cardboard. Squares of equal size will be cut out of each corner, and then the ends and sides will be folded up to form a box with an open top. Find the length L, width W, and height H of the resulting box that maximizes the volume. (Assume that W≤L).

Answer 17

@smr_kml

Note that perimeter = 2h + 2r + pi r = 31
so h = 31 - r - pi/2 r = 31 - (1 + pi/2)r


You forgot to divide 31 by 2. otherwise your work is fine.

Answer 18

A box is to be made out of a 8 by 16 piece of cardboard. Squares of equal size will be cut out of each corner, and then the ends and sides will be folded up to form a box with an open top. Find the length L, width W, and height H of the resulting box that maximizes the volume. (Assume that W≤L).

replace W and L in the equation for the volume and take the derivative with respect to H